284.88/91.24 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))

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↳2 CdtProblem

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↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳4 CdtProblem

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↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳8 CdtProblem

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↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳10 CdtProblem

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↳11 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳12 CdtProblem

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↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳14 CdtProblem

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↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳16 CdtProblem

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↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳18 CdtProblem

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↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳20 CdtProblem

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↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳22 CdtProblem

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↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳24 CdtProblem

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↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳26 CdtProblem

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↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳28 CdtProblem

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↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳30 CdtProblem

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↳31 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳32 CdtProblem

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↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳34 CdtProblem

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↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳36 CdtProblem

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↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳38 CdtProblem

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↳39 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳40 CdtProblem

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↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳42 CdtProblem

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↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳44 CdtProblem

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↳45 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳46 CdtProblem

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↳47 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳48 CdtProblem

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↳49 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳50 CdtProblem

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↳51 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳52 CdtProblem

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↳53 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳54 CdtProblem

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↳55 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳56 CdtProblem

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↳57 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳58 CdtProblem

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↳59 CdtUnreachableProof (⇔)

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↳60 CdtProblem

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↳61 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳62 CdtProblem

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↳63 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳64 CdtProblem

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↳65 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳66 CdtProblem

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↳67 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳68 CdtProblem

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↳69 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳70 CdtProblem

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↳71 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳72 CdtProblem

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↳73 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳74 CdtProblem

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↳75 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳76 CdtProblem

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↳77 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳78 CdtProblem

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↳79 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳80 CdtProblem

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↳81 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳82 CdtProblem

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↳83 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳84 CdtProblem

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↳85 SIsEmptyProof (BOTH BOUNDS(ID, ID))

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↳86 BOUNDS(O(1), O(1))

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(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X))))) 285.26/91.33
active(add(0, X)) → mark(X) 285.26/91.33
active(add(s(X), Y)) → mark(s(add(X, Y))) 285.26/91.33
active(prod(0, X)) → mark(0) 285.26/91.33
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y))) 285.26/91.33
active(if(true, X, Y)) → mark(X) 285.26/91.33
active(if(false, X, Y)) → mark(Y) 285.26/91.33
active(zero(0)) → mark(true) 285.26/91.33
active(zero(s(X))) → mark(false) 285.26/91.33
active(p(s(X))) → mark(X) 285.26/91.33
active(fact(X)) → fact(active(X)) 285.26/91.33
active(if(X1, X2, X3)) → if(active(X1), X2, X3) 285.26/91.33
active(zero(X)) → zero(active(X)) 285.26/91.33
active(s(X)) → s(active(X)) 285.26/91.33
active(prod(X1, X2)) → prod(active(X1), X2) 285.26/91.33
active(prod(X1, X2)) → prod(X1, active(X2)) 285.26/91.33
active(p(X)) → p(active(X)) 285.26/91.33
active(add(X1, X2)) → add(active(X1), X2) 285.26/91.33
active(add(X1, X2)) → add(X1, active(X2)) 285.26/91.33
fact(mark(X)) → mark(fact(X)) 285.26/91.33
if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) 285.26/91.33
zero(mark(X)) → mark(zero(X)) 285.26/91.33
s(mark(X)) → mark(s(X)) 285.26/91.33
prod(mark(X1), X2) → mark(prod(X1, X2)) 285.26/91.33
prod(X1, mark(X2)) → mark(prod(X1, X2)) 285.26/91.33
p(mark(X)) → mark(p(X)) 285.26/91.33
add(mark(X1), X2) → mark(add(X1, X2)) 285.26/91.33
add(X1, mark(X2)) → mark(add(X1, X2)) 285.26/91.33
proper(fact(X)) → fact(proper(X)) 285.26/91.33
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3)) 285.26/91.33
proper(zero(X)) → zero(proper(X)) 285.26/91.33
proper(s(X)) → s(proper(X)) 285.26/91.33
proper(0) → ok(0) 285.26/91.33
proper(prod(X1, X2)) → prod(proper(X1), proper(X2)) 285.26/91.33
proper(p(X)) → p(proper(X)) 285.26/91.33
proper(add(X1, X2)) → add(proper(X1), proper(X2)) 285.26/91.33
proper(true) → ok(true) 285.26/91.33
proper(false) → ok(false) 285.26/91.33
fact(ok(X)) → ok(fact(X)) 285.26/91.33
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) 285.26/91.33
zero(ok(X)) → ok(zero(X)) 285.26/91.33
s(ok(X)) → ok(s(X)) 285.26/91.33
prod(ok(X1), ok(X2)) → ok(prod(X1, X2)) 285.26/91.33
p(ok(X)) → ok(p(X)) 285.26/91.33
add(ok(X1), ok(X2)) → ok(add(X1, X2)) 285.26/91.33
top(mark(X)) → top(proper(X)) 285.26/91.33
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.26/91.33
active(add(0, z0)) → mark(z0) 285.26/91.33
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.26/91.33
active(prod(0, z0)) → mark(0) 285.26/91.33
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.26/91.33
active(if(true, z0, z1)) → mark(z0) 285.26/91.33
active(if(false, z0, z1)) → mark(z1) 285.26/91.33
active(zero(0)) → mark(true) 285.26/91.33
active(zero(s(z0))) → mark(false) 285.26/91.33
active(p(s(z0))) → mark(z0) 285.26/91.33
active(fact(z0)) → fact(active(z0)) 285.26/91.33
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.26/91.33
active(zero(z0)) → zero(active(z0)) 285.26/91.33
active(s(z0)) → s(active(z0)) 285.26/91.33
active(prod(z0, z1)) → prod(active(z0), z1) 285.26/91.33
active(prod(z0, z1)) → prod(z0, active(z1)) 285.26/91.33
active(p(z0)) → p(active(z0)) 285.26/91.33
active(add(z0, z1)) → add(active(z0), z1) 285.26/91.33
active(add(z0, z1)) → add(z0, active(z1)) 285.26/91.33
fact(mark(z0)) → mark(fact(z0)) 285.26/91.33
fact(ok(z0)) → ok(fact(z0)) 285.26/91.33
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.26/91.33
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.26/91.33
zero(mark(z0)) → mark(zero(z0)) 285.26/91.33
zero(ok(z0)) → ok(zero(z0)) 285.26/91.33
s(mark(z0)) → mark(s(z0)) 285.26/91.33
s(ok(z0)) → ok(s(z0)) 285.26/91.33
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.26/91.33
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.26/91.33
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.26/91.33
p(mark(z0)) → mark(p(z0)) 285.26/91.33
p(ok(z0)) → ok(p(z0)) 285.26/91.33
add(mark(z0), z1) → mark(add(z0, z1)) 285.26/91.33
add(z0, mark(z1)) → mark(add(z0, z1)) 285.26/91.33
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.26/91.33
proper(fact(z0)) → fact(proper(z0)) 285.26/91.33
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.26/91.33
proper(zero(z0)) → zero(proper(z0)) 285.26/91.33
proper(s(z0)) → s(proper(z0)) 285.26/91.33
proper(0) → ok(0) 285.26/91.33
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.26/91.33
proper(p(z0)) → p(proper(z0)) 285.26/91.33
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.26/91.33
proper(true) → ok(true) 285.26/91.33
proper(false) → ok(false) 285.26/91.33
top(mark(z0)) → top(proper(z0)) 285.26/91.33
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(fact(z0)) → c(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), ZERO(z0), S(0), PROD(z0, fact(p(z0))), FACT(p(z0)), P(z0)) 285.26/91.33
ACTIVE(add(s(z0), z1)) → c2(S(add(z0, z1)), ADD(z0, z1)) 285.26/91.33
ACTIVE(prod(s(z0), z1)) → c4(ADD(z1, prod(z0, z1)), PROD(z0, z1)) 285.26/91.33
ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.26/91.33
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
FACT(mark(z0)) → c19(FACT(z0)) 285.26/91.33
FACT(ok(z0)) → c20(FACT(z0)) 285.26/91.33
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.26/91.33
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.26/91.33
ZERO(mark(z0)) → c23(ZERO(z0)) 285.26/91.33
ZERO(ok(z0)) → c24(ZERO(z0)) 285.26/91.33
S(mark(z0)) → c25(S(z0)) 285.26/91.33
S(ok(z0)) → c26(S(z0)) 285.26/91.33
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.26/91.33
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.26/91.33
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.26/91.33
P(mark(z0)) → c30(P(z0)) 285.26/91.33
P(ok(z0)) → c31(P(z0)) 285.26/91.33
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.26/91.33
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.26/91.33
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.26/91.33
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.26/91.33
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.33
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.33
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.26/91.33
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(fact(z0)) → c(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), ZERO(z0), S(0), PROD(z0, fact(p(z0))), FACT(p(z0)), P(z0)) 285.26/91.33
ACTIVE(add(s(z0), z1)) → c2(S(add(z0, z1)), ADD(z0, z1)) 285.26/91.33
ACTIVE(prod(s(z0), z1)) → c4(ADD(z1, prod(z0, z1)), PROD(z0, z1)) 285.26/91.33
ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.26/91.33
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
FACT(mark(z0)) → c19(FACT(z0)) 285.26/91.33
FACT(ok(z0)) → c20(FACT(z0)) 285.26/91.33
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.26/91.33
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.26/91.33
ZERO(mark(z0)) → c23(ZERO(z0)) 285.26/91.33
ZERO(ok(z0)) → c24(ZERO(z0)) 285.26/91.33
S(mark(z0)) → c25(S(z0)) 285.26/91.33
S(ok(z0)) → c26(S(z0)) 285.26/91.33
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.26/91.33
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.26/91.33
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.26/91.33
P(mark(z0)) → c30(P(z0)) 285.26/91.33
P(ok(z0)) → c31(P(z0)) 285.26/91.33
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.26/91.33
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.26/91.33
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.26/91.33
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.26/91.33
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.33
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.33
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.26/91.33
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c, c2, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46

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(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.26/91.33
active(add(0, z0)) → mark(z0) 285.26/91.33
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.26/91.33
active(prod(0, z0)) → mark(0) 285.26/91.33
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.26/91.33
active(if(true, z0, z1)) → mark(z0) 285.26/91.33
active(if(false, z0, z1)) → mark(z1) 285.26/91.33
active(zero(0)) → mark(true) 285.26/91.33
active(zero(s(z0))) → mark(false) 285.26/91.33
active(p(s(z0))) → mark(z0) 285.26/91.33
active(fact(z0)) → fact(active(z0)) 285.26/91.33
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.26/91.33
active(zero(z0)) → zero(active(z0)) 285.26/91.33
active(s(z0)) → s(active(z0)) 285.26/91.33
active(prod(z0, z1)) → prod(active(z0), z1) 285.26/91.33
active(prod(z0, z1)) → prod(z0, active(z1)) 285.26/91.33
active(p(z0)) → p(active(z0)) 285.26/91.33
active(add(z0, z1)) → add(active(z0), z1) 285.26/91.33
active(add(z0, z1)) → add(z0, active(z1)) 285.26/91.33
fact(mark(z0)) → mark(fact(z0)) 285.26/91.33
fact(ok(z0)) → ok(fact(z0)) 285.26/91.33
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.26/91.33
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.26/91.33
zero(mark(z0)) → mark(zero(z0)) 285.26/91.33
zero(ok(z0)) → ok(zero(z0)) 285.26/91.33
s(mark(z0)) → mark(s(z0)) 285.26/91.33
s(ok(z0)) → ok(s(z0)) 285.26/91.33
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.26/91.33
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.26/91.33
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.26/91.33
p(mark(z0)) → mark(p(z0)) 285.26/91.33
p(ok(z0)) → ok(p(z0)) 285.26/91.33
add(mark(z0), z1) → mark(add(z0, z1)) 285.26/91.33
add(z0, mark(z1)) → mark(add(z0, z1)) 285.26/91.33
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.26/91.33
proper(fact(z0)) → fact(proper(z0)) 285.26/91.33
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.26/91.33
proper(zero(z0)) → zero(proper(z0)) 285.26/91.33
proper(s(z0)) → s(proper(z0)) 285.26/91.33
proper(0) → ok(0) 285.26/91.33
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.26/91.33
proper(p(z0)) → p(proper(z0)) 285.26/91.33
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.26/91.33
proper(true) → ok(true) 285.26/91.33
proper(false) → ok(false) 285.26/91.33
top(mark(z0)) → top(proper(z0)) 285.26/91.33
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.26/91.33
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
FACT(mark(z0)) → c19(FACT(z0)) 285.26/91.33
FACT(ok(z0)) → c20(FACT(z0)) 285.26/91.33
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.26/91.33
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.26/91.33
ZERO(mark(z0)) → c23(ZERO(z0)) 285.26/91.33
ZERO(ok(z0)) → c24(ZERO(z0)) 285.26/91.33
S(mark(z0)) → c25(S(z0)) 285.26/91.33
S(ok(z0)) → c26(S(z0)) 285.26/91.33
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.26/91.33
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.26/91.33
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.26/91.33
P(mark(z0)) → c30(P(z0)) 285.26/91.33
P(ok(z0)) → c31(P(z0)) 285.26/91.33
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.26/91.33
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.26/91.33
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.26/91.33
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.26/91.33
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.33
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.26/91.33
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.33
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.26/91.33
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(fact(z0)) → c(ZERO(z0), P(z0)) 285.26/91.33
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.26/91.33
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1))

S tuples:

ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.26/91.33
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.26/91.33
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.26/91.33
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.26/91.35
FACT(mark(z0)) → c19(FACT(z0)) 285.26/91.35
FACT(ok(z0)) → c20(FACT(z0)) 285.26/91.35
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.26/91.35
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.26/91.35
ZERO(mark(z0)) → c23(ZERO(z0)) 285.26/91.35
ZERO(ok(z0)) → c24(ZERO(z0)) 285.26/91.35
S(mark(z0)) → c25(S(z0)) 285.26/91.35
S(ok(z0)) → c26(S(z0)) 285.26/91.35
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.26/91.35
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.26/91.35
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.26/91.35
P(mark(z0)) → c30(P(z0)) 285.26/91.35
P(ok(z0)) → c31(P(z0)) 285.26/91.35
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.26/91.35
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.26/91.35
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.26/91.35
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.26/91.35
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.35
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.35
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.26/91.35
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(fact(z0)) → c(ZERO(z0), P(z0)) 285.26/91.35
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.26/91.35
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c, c2, c4

285.26/91.35

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

285.26/91.35

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.26/91.35
active(add(0, z0)) → mark(z0) 285.26/91.35
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.26/91.35
active(prod(0, z0)) → mark(0) 285.26/91.35
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.26/91.35
active(if(true, z0, z1)) → mark(z0) 285.26/91.35
active(if(false, z0, z1)) → mark(z1) 285.26/91.35
active(zero(0)) → mark(true) 285.26/91.35
active(zero(s(z0))) → mark(false) 285.26/91.35
active(p(s(z0))) → mark(z0) 285.26/91.35
active(fact(z0)) → fact(active(z0)) 285.26/91.35
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.26/91.35
active(zero(z0)) → zero(active(z0)) 285.26/91.35
active(s(z0)) → s(active(z0)) 285.26/91.35
active(prod(z0, z1)) → prod(active(z0), z1) 285.26/91.35
active(prod(z0, z1)) → prod(z0, active(z1)) 285.26/91.35
active(p(z0)) → p(active(z0)) 285.26/91.35
active(add(z0, z1)) → add(active(z0), z1) 285.26/91.35
active(add(z0, z1)) → add(z0, active(z1)) 285.26/91.35
fact(mark(z0)) → mark(fact(z0)) 285.26/91.35
fact(ok(z0)) → ok(fact(z0)) 285.26/91.35
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.26/91.35
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.26/91.35
zero(mark(z0)) → mark(zero(z0)) 285.26/91.35
zero(ok(z0)) → ok(zero(z0)) 285.26/91.35
s(mark(z0)) → mark(s(z0)) 285.26/91.35
s(ok(z0)) → ok(s(z0)) 285.26/91.35
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.26/91.35
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.26/91.35
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.26/91.35
p(mark(z0)) → mark(p(z0)) 285.26/91.35
p(ok(z0)) → ok(p(z0)) 285.26/91.35
add(mark(z0), z1) → mark(add(z0, z1)) 285.26/91.35
add(z0, mark(z1)) → mark(add(z0, z1)) 285.26/91.35
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.26/91.35
proper(fact(z0)) → fact(proper(z0)) 285.26/91.35
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.26/91.35
proper(zero(z0)) → zero(proper(z0)) 285.26/91.35
proper(s(z0)) → s(proper(z0)) 285.26/91.35
proper(0) → ok(0) 285.26/91.35
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.26/91.35
proper(p(z0)) → p(proper(z0)) 285.26/91.35
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.26/91.35
proper(true) → ok(true) 285.26/91.35
proper(false) → ok(false) 285.26/91.35
top(mark(z0)) → top(proper(z0)) 285.26/91.35
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.26/91.35
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.26/91.35
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.26/91.35
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.26/91.35
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.26/91.35
FACT(mark(z0)) → c19(FACT(z0)) 285.26/91.35
FACT(ok(z0)) → c20(FACT(z0)) 285.26/91.35
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.26/91.35
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.26/91.35
ZERO(mark(z0)) → c23(ZERO(z0)) 285.26/91.35
ZERO(ok(z0)) → c24(ZERO(z0)) 285.26/91.35
S(mark(z0)) → c25(S(z0)) 285.26/91.35
S(ok(z0)) → c26(S(z0)) 285.26/91.35
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.26/91.35
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.26/91.35
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.26/91.35
P(mark(z0)) → c30(P(z0)) 285.26/91.35
P(ok(z0)) → c31(P(z0)) 285.26/91.35
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.26/91.35
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.26/91.35
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.26/91.35
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.26/91.35
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.35
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.35
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.26/91.35
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.26/91.35
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.26/91.35
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.26/91.35
ACTIVE(fact(z0)) → c1(P(z0))

S tuples:

ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.26/91.35
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.26/91.35
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.26/91.35
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.26/91.35
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.26/91.35
FACT(mark(z0)) → c19(FACT(z0)) 285.26/91.35
FACT(ok(z0)) → c20(FACT(z0)) 285.26/91.35
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.26/91.35
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.26/91.35
ZERO(mark(z0)) → c23(ZERO(z0)) 285.26/91.35
ZERO(ok(z0)) → c24(ZERO(z0)) 285.26/91.35
S(mark(z0)) → c25(S(z0)) 285.26/91.35
S(ok(z0)) → c26(S(z0)) 285.26/91.35
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.26/91.35
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.26/91.35
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.26/91.35
P(mark(z0)) → c30(P(z0)) 285.26/91.35
P(ok(z0)) → c31(P(z0)) 285.26/91.35
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.26/91.35
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.26/91.35
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.26/91.35
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.26/91.35
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.35
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.26/91.35
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.26/91.35
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.26/91.35
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.26/91.35
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.26/91.35
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.26/91.35
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.26/91.35
ACTIVE(fact(z0)) → c1(P(z0))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1

285.62/91.49

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(fact(z0)) → c10(FACT(active(z0)), ACTIVE(z0)) by

ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1)))

285.62/91.49

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.62/91.49
active(add(0, z0)) → mark(z0) 285.62/91.49
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.62/91.49
active(prod(0, z0)) → mark(0) 285.62/91.49
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.62/91.49
active(if(true, z0, z1)) → mark(z0) 285.62/91.49
active(if(false, z0, z1)) → mark(z1) 285.62/91.49
active(zero(0)) → mark(true) 285.62/91.49
active(zero(s(z0))) → mark(false) 285.62/91.49
active(p(s(z0))) → mark(z0) 285.62/91.49
active(fact(z0)) → fact(active(z0)) 285.62/91.49
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.62/91.49
active(zero(z0)) → zero(active(z0)) 285.62/91.49
active(s(z0)) → s(active(z0)) 285.62/91.49
active(prod(z0, z1)) → prod(active(z0), z1) 285.62/91.49
active(prod(z0, z1)) → prod(z0, active(z1)) 285.62/91.49
active(p(z0)) → p(active(z0)) 285.62/91.49
active(add(z0, z1)) → add(active(z0), z1) 285.62/91.49
active(add(z0, z1)) → add(z0, active(z1)) 285.62/91.49
fact(mark(z0)) → mark(fact(z0)) 285.62/91.49
fact(ok(z0)) → ok(fact(z0)) 285.62/91.49
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.62/91.49
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.62/91.49
zero(mark(z0)) → mark(zero(z0)) 285.62/91.49
zero(ok(z0)) → ok(zero(z0)) 285.62/91.49
s(mark(z0)) → mark(s(z0)) 285.62/91.49
s(ok(z0)) → ok(s(z0)) 285.62/91.49
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.62/91.49
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.62/91.49
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.62/91.49
p(mark(z0)) → mark(p(z0)) 285.62/91.49
p(ok(z0)) → ok(p(z0)) 285.62/91.49
add(mark(z0), z1) → mark(add(z0, z1)) 285.62/91.49
add(z0, mark(z1)) → mark(add(z0, z1)) 285.62/91.49
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.62/91.49
proper(fact(z0)) → fact(proper(z0)) 285.62/91.49
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.62/91.49
proper(zero(z0)) → zero(proper(z0)) 285.62/91.49
proper(s(z0)) → s(proper(z0)) 285.62/91.49
proper(0) → ok(0) 285.62/91.49
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.62/91.49
proper(p(z0)) → p(proper(z0)) 285.62/91.49
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.62/91.49
proper(true) → ok(true) 285.62/91.49
proper(false) → ok(false) 285.62/91.49
top(mark(z0)) → top(proper(z0)) 285.62/91.49
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.62/91.49
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1)))

S tuples:

ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 285.62/91.49
ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10

285.62/91.49

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1)))

285.62/91.49

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.62/91.49
active(add(0, z0)) → mark(z0) 285.62/91.49
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.62/91.49
active(prod(0, z0)) → mark(0) 285.62/91.49
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.62/91.49
active(if(true, z0, z1)) → mark(z0) 285.62/91.49
active(if(false, z0, z1)) → mark(z1) 285.62/91.49
active(zero(0)) → mark(true) 285.62/91.49
active(zero(s(z0))) → mark(false) 285.62/91.49
active(p(s(z0))) → mark(z0) 285.62/91.49
active(fact(z0)) → fact(active(z0)) 285.62/91.49
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.62/91.49
active(zero(z0)) → zero(active(z0)) 285.62/91.49
active(s(z0)) → s(active(z0)) 285.62/91.49
active(prod(z0, z1)) → prod(active(z0), z1) 285.62/91.49
active(prod(z0, z1)) → prod(z0, active(z1)) 285.62/91.49
active(p(z0)) → p(active(z0)) 285.62/91.49
active(add(z0, z1)) → add(active(z0), z1) 285.62/91.49
active(add(z0, z1)) → add(z0, active(z1)) 285.62/91.49
fact(mark(z0)) → mark(fact(z0)) 285.62/91.49
fact(ok(z0)) → ok(fact(z0)) 285.62/91.49
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.62/91.49
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.62/91.49
zero(mark(z0)) → mark(zero(z0)) 285.62/91.49
zero(ok(z0)) → ok(zero(z0)) 285.62/91.49
s(mark(z0)) → mark(s(z0)) 285.62/91.49
s(ok(z0)) → ok(s(z0)) 285.62/91.49
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.62/91.49
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.62/91.49
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.62/91.49
p(mark(z0)) → mark(p(z0)) 285.62/91.49
p(ok(z0)) → ok(p(z0)) 285.62/91.49
add(mark(z0), z1) → mark(add(z0, z1)) 285.62/91.49
add(z0, mark(z1)) → mark(add(z0, z1)) 285.62/91.49
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.62/91.49
proper(fact(z0)) → fact(proper(z0)) 285.62/91.49
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.62/91.49
proper(zero(z0)) → zero(proper(z0)) 285.62/91.49
proper(s(z0)) → s(proper(z0)) 285.62/91.49
proper(0) → ok(0) 285.62/91.49
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.62/91.49
proper(p(z0)) → p(proper(z0)) 285.62/91.49
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.62/91.49
proper(true) → ok(true) 285.62/91.49
proper(false) → ok(false) 285.62/91.49
top(mark(z0)) → top(proper(z0)) 285.62/91.49
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1)))

S tuples:

ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11

285.62/91.49

(11) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

285.62/91.49

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.62/91.49
active(add(0, z0)) → mark(z0) 285.62/91.49
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.62/91.49
active(prod(0, z0)) → mark(0) 285.62/91.49
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.62/91.49
active(if(true, z0, z1)) → mark(z0) 285.62/91.49
active(if(false, z0, z1)) → mark(z1) 285.62/91.49
active(zero(0)) → mark(true) 285.62/91.49
active(zero(s(z0))) → mark(false) 285.62/91.49
active(p(s(z0))) → mark(z0) 285.62/91.49
active(fact(z0)) → fact(active(z0)) 285.62/91.49
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.62/91.49
active(zero(z0)) → zero(active(z0)) 285.62/91.49
active(s(z0)) → s(active(z0)) 285.62/91.49
active(prod(z0, z1)) → prod(active(z0), z1) 285.62/91.49
active(prod(z0, z1)) → prod(z0, active(z1)) 285.62/91.49
active(p(z0)) → p(active(z0)) 285.62/91.49
active(add(z0, z1)) → add(active(z0), z1) 285.62/91.49
active(add(z0, z1)) → add(z0, active(z1)) 285.62/91.49
fact(mark(z0)) → mark(fact(z0)) 285.62/91.49
fact(ok(z0)) → ok(fact(z0)) 285.62/91.49
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.62/91.49
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.62/91.49
zero(mark(z0)) → mark(zero(z0)) 285.62/91.49
zero(ok(z0)) → ok(zero(z0)) 285.62/91.49
s(mark(z0)) → mark(s(z0)) 285.62/91.49
s(ok(z0)) → ok(s(z0)) 285.62/91.49
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.62/91.49
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.62/91.49
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.62/91.49
p(mark(z0)) → mark(p(z0)) 285.62/91.49
p(ok(z0)) → ok(p(z0)) 285.62/91.49
add(mark(z0), z1) → mark(add(z0, z1)) 285.62/91.49
add(z0, mark(z1)) → mark(add(z0, z1)) 285.62/91.49
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.62/91.49
proper(fact(z0)) → fact(proper(z0)) 285.62/91.49
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.62/91.49
proper(zero(z0)) → zero(proper(z0)) 285.62/91.49
proper(s(z0)) → s(proper(z0)) 285.62/91.49
proper(0) → ok(0) 285.62/91.49
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.62/91.49
proper(p(z0)) → p(proper(z0)) 285.62/91.49
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.62/91.49
proper(true) → ok(true) 285.62/91.49
proper(false) → ok(false) 285.62/91.49
top(mark(z0)) → top(proper(z0)) 285.62/91.49
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.62/91.49
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.62/91.49
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))

S tuples:

ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.62/91.49
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.62/91.49
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11

285.62/91.49

(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(zero(z0)) → c12(ZERO(active(z0)), ACTIVE(z0)) by

ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1)))

285.62/91.49

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.62/91.49
active(add(0, z0)) → mark(z0) 285.62/91.49
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.62/91.49
active(prod(0, z0)) → mark(0) 285.62/91.49
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.62/91.49
active(if(true, z0, z1)) → mark(z0) 285.62/91.49
active(if(false, z0, z1)) → mark(z1) 285.62/91.49
active(zero(0)) → mark(true) 285.62/91.49
active(zero(s(z0))) → mark(false) 285.62/91.49
active(p(s(z0))) → mark(z0) 285.62/91.49
active(fact(z0)) → fact(active(z0)) 285.62/91.49
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.62/91.49
active(zero(z0)) → zero(active(z0)) 285.62/91.49
active(s(z0)) → s(active(z0)) 285.62/91.49
active(prod(z0, z1)) → prod(active(z0), z1) 285.62/91.49
active(prod(z0, z1)) → prod(z0, active(z1)) 285.62/91.49
active(p(z0)) → p(active(z0)) 285.62/91.49
active(add(z0, z1)) → add(active(z0), z1) 285.62/91.49
active(add(z0, z1)) → add(z0, active(z1)) 285.62/91.49
fact(mark(z0)) → mark(fact(z0)) 285.62/91.49
fact(ok(z0)) → ok(fact(z0)) 285.62/91.49
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.62/91.49
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.62/91.49
zero(mark(z0)) → mark(zero(z0)) 285.62/91.49
zero(ok(z0)) → ok(zero(z0)) 285.62/91.49
s(mark(z0)) → mark(s(z0)) 285.62/91.49
s(ok(z0)) → ok(s(z0)) 285.62/91.49
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.62/91.49
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.62/91.49
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.62/91.49
p(mark(z0)) → mark(p(z0)) 285.62/91.49
p(ok(z0)) → ok(p(z0)) 285.62/91.49
add(mark(z0), z1) → mark(add(z0, z1)) 285.62/91.49
add(z0, mark(z1)) → mark(add(z0, z1)) 285.62/91.49
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.62/91.49
proper(fact(z0)) → fact(proper(z0)) 285.62/91.49
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.62/91.49
proper(zero(z0)) → zero(proper(z0)) 285.62/91.49
proper(s(z0)) → s(proper(z0)) 285.62/91.49
proper(0) → ok(0) 285.62/91.49
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.62/91.49
proper(p(z0)) → p(proper(z0)) 285.62/91.49
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.62/91.49
proper(true) → ok(true) 285.62/91.49
proper(false) → ok(false) 285.62/91.49
top(mark(z0)) → top(proper(z0)) 285.62/91.49
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.62/91.49
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.62/91.49
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.62/91.49
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.62/91.49
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.49
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.49
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1)))

S tuples:

ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.49
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.49
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.49
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.49
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.49
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.49
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.49
S(mark(z0)) → c25(S(z0)) 285.62/91.49
S(ok(z0)) → c26(S(z0)) 285.62/91.49
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.49
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.49
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.49
P(mark(z0)) → c30(P(z0)) 285.62/91.49
P(ok(z0)) → c31(P(z0)) 285.62/91.49
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.49
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.49
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.49
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.49
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.49
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.49
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.49
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.49
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.49
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.49
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.49
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.49
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.49
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.49
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.49
ACTIVE(fact(zero(0))) → c10(FACT(mark(true)), ACTIVE(zero(0))) 285.62/91.49
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.49
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.49
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.49
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.49
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.49
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.49
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.49
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.50
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(zero(0))) 285.62/91.50
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.50
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.50
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.50
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.62/91.50
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.62/91.50
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.62/91.50
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.62/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.50
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.50
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.50
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.50
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0)), ACTIVE(if(true, z0, z1))) 285.62/91.50
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1)), ACTIVE(if(false, z0, z1))) 285.62/91.50
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true)), ACTIVE(zero(0))) 285.62/91.50
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.50
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.50
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.50
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.62/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.62/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12

285.62/91.50

(15) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

285.62/91.50

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.62/91.50
active(add(0, z0)) → mark(z0) 285.62/91.50
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.62/91.50
active(prod(0, z0)) → mark(0) 285.62/91.50
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.62/91.50
active(if(true, z0, z1)) → mark(z0) 285.62/91.50
active(if(false, z0, z1)) → mark(z1) 285.62/91.50
active(zero(0)) → mark(true) 285.62/91.50
active(zero(s(z0))) → mark(false) 285.62/91.50
active(p(s(z0))) → mark(z0) 285.62/91.50
active(fact(z0)) → fact(active(z0)) 285.62/91.50
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.62/91.50
active(zero(z0)) → zero(active(z0)) 285.62/91.50
active(s(z0)) → s(active(z0)) 285.62/91.50
active(prod(z0, z1)) → prod(active(z0), z1) 285.62/91.50
active(prod(z0, z1)) → prod(z0, active(z1)) 285.62/91.50
active(p(z0)) → p(active(z0)) 285.62/91.50
active(add(z0, z1)) → add(active(z0), z1) 285.62/91.50
active(add(z0, z1)) → add(z0, active(z1)) 285.62/91.50
fact(mark(z0)) → mark(fact(z0)) 285.62/91.50
fact(ok(z0)) → ok(fact(z0)) 285.62/91.50
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.62/91.50
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.62/91.50
zero(mark(z0)) → mark(zero(z0)) 285.62/91.50
zero(ok(z0)) → ok(zero(z0)) 285.62/91.50
s(mark(z0)) → mark(s(z0)) 285.62/91.50
s(ok(z0)) → ok(s(z0)) 285.62/91.50
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.62/91.50
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.62/91.50
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.62/91.50
p(mark(z0)) → mark(p(z0)) 285.62/91.50
p(ok(z0)) → ok(p(z0)) 285.62/91.50
add(mark(z0), z1) → mark(add(z0, z1)) 285.62/91.50
add(z0, mark(z1)) → mark(add(z0, z1)) 285.62/91.50
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.62/91.50
proper(fact(z0)) → fact(proper(z0)) 285.62/91.50
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.62/91.50
proper(zero(z0)) → zero(proper(z0)) 285.62/91.50
proper(s(z0)) → s(proper(z0)) 285.62/91.50
proper(0) → ok(0) 285.62/91.50
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.62/91.50
proper(p(z0)) → p(proper(z0)) 285.62/91.50
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.62/91.50
proper(true) → ok(true) 285.62/91.50
proper(false) → ok(false) 285.62/91.50
top(mark(z0)) → top(proper(z0)) 285.62/91.50
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.62/91.50
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.62/91.50
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.62/91.50
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.62/91.50
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.62/91.50
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.62/91.50
FACT(mark(z0)) → c19(FACT(z0)) 285.62/91.50
FACT(ok(z0)) → c20(FACT(z0)) 285.62/91.50
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.62/91.50
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.62/91.50
ZERO(mark(z0)) → c23(ZERO(z0)) 285.62/91.50
ZERO(ok(z0)) → c24(ZERO(z0)) 285.62/91.50
S(mark(z0)) → c25(S(z0)) 285.62/91.50
S(ok(z0)) → c26(S(z0)) 285.62/91.50
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.62/91.50
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.62/91.50
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.62/91.50
P(mark(z0)) → c30(P(z0)) 285.62/91.50
P(ok(z0)) → c31(P(z0)) 285.62/91.50
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.62/91.50
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.62/91.50
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.62/91.50
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.62/91.50
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.62/91.50
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.62/91.50
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.62/91.50
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.50
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.62/91.50
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.62/91.50
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.62/91.50
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.62/91.50
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.62/91.50
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.62/91.50
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.62/91.50
ACTIVE(fact(z0)) → c1(P(z0)) 285.62/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.62/91.50
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.62/91.50
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.62/91.50
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.62/91.50
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.62/91.50
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.62/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.62/91.50
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.62/91.50
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.62/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.62/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.62/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.62/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.62/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.62/91.50
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.62/91.50
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.62/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.62/91.50
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.62/91.50
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.62/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.62/91.50
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.62/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.62/91.50
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.62/91.50
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.62/91.50
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.62/91.50
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.62/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.62/91.50
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.50
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.50
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.50
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.50
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true)))

S tuples:

ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.50
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.50
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.50
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.50
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.50
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.50
S(mark(z0)) → c25(S(z0)) 285.98/91.50
S(ok(z0)) → c26(S(z0)) 285.98/91.50
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.50
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.50
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.50
P(mark(z0)) → c30(P(z0)) 285.98/91.50
P(ok(z0)) → c31(P(z0)) 285.98/91.50
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.50
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.50
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.50
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.50
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.50
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.50
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.50
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.50
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.50
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.50
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.50
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.50
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.50
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.50
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.50
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12

285.98/91.50

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c13(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0)), ACTIVE(if(true, z0, z1))) 285.98/91.50
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1)), ACTIVE(if(false, z0, z1))) 285.98/91.50
ACTIVE(s(zero(0))) → c13(S(mark(true)), ACTIVE(zero(0))) 285.98/91.50
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))

285.98/91.50

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.98/91.50
active(add(0, z0)) → mark(z0) 285.98/91.50
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.98/91.50
active(prod(0, z0)) → mark(0) 285.98/91.50
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.98/91.50
active(if(true, z0, z1)) → mark(z0) 285.98/91.50
active(if(false, z0, z1)) → mark(z1) 285.98/91.50
active(zero(0)) → mark(true) 285.98/91.50
active(zero(s(z0))) → mark(false) 285.98/91.50
active(p(s(z0))) → mark(z0) 285.98/91.50
active(fact(z0)) → fact(active(z0)) 285.98/91.50
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.98/91.50
active(zero(z0)) → zero(active(z0)) 285.98/91.50
active(s(z0)) → s(active(z0)) 285.98/91.50
active(prod(z0, z1)) → prod(active(z0), z1) 285.98/91.50
active(prod(z0, z1)) → prod(z0, active(z1)) 285.98/91.50
active(p(z0)) → p(active(z0)) 285.98/91.50
active(add(z0, z1)) → add(active(z0), z1) 285.98/91.50
active(add(z0, z1)) → add(z0, active(z1)) 285.98/91.50
fact(mark(z0)) → mark(fact(z0)) 285.98/91.50
fact(ok(z0)) → ok(fact(z0)) 285.98/91.50
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.98/91.50
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.98/91.50
zero(mark(z0)) → mark(zero(z0)) 285.98/91.50
zero(ok(z0)) → ok(zero(z0)) 285.98/91.50
s(mark(z0)) → mark(s(z0)) 285.98/91.50
s(ok(z0)) → ok(s(z0)) 285.98/91.50
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.98/91.50
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.98/91.50
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.98/91.50
p(mark(z0)) → mark(p(z0)) 285.98/91.50
p(ok(z0)) → ok(p(z0)) 285.98/91.50
add(mark(z0), z1) → mark(add(z0, z1)) 285.98/91.50
add(z0, mark(z1)) → mark(add(z0, z1)) 285.98/91.50
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.98/91.50
proper(fact(z0)) → fact(proper(z0)) 285.98/91.50
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.98/91.50
proper(zero(z0)) → zero(proper(z0)) 285.98/91.50
proper(s(z0)) → s(proper(z0)) 285.98/91.50
proper(0) → ok(0) 285.98/91.50
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.98/91.50
proper(p(z0)) → p(proper(z0)) 285.98/91.50
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.98/91.50
proper(true) → ok(true) 285.98/91.50
proper(false) → ok(false) 285.98/91.50
top(mark(z0)) → top(proper(z0)) 285.98/91.50
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.50
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.50
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.50
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.50
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.50
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.50
S(mark(z0)) → c25(S(z0)) 285.98/91.50
S(ok(z0)) → c26(S(z0)) 285.98/91.50
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.50
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.50
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.50
P(mark(z0)) → c30(P(z0)) 285.98/91.50
P(ok(z0)) → c31(P(z0)) 285.98/91.50
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.50
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.50
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.50
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.50
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.50
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.50
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.50
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.50
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.50
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.50
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.50
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.50
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.50
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.50
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.50
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0)), ACTIVE(if(true, z0, z1))) 285.98/91.50
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1)), ACTIVE(if(false, z0, z1))) 285.98/91.50
ACTIVE(s(zero(0))) → c13(S(mark(true)), ACTIVE(zero(0))) 285.98/91.50
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.50
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.50
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.50
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.50
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.50
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.50
S(mark(z0)) → c25(S(z0)) 285.98/91.50
S(ok(z0)) → c26(S(z0)) 285.98/91.50
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.50
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.50
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.50
P(mark(z0)) → c30(P(z0)) 285.98/91.50
P(ok(z0)) → c31(P(z0)) 285.98/91.50
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.50
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.50
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.50
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.50
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.50
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.50
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.50
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.50
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.50
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.50
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.50
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.50
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.50
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.50
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.50
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0)), ACTIVE(if(true, z0, z1))) 285.98/91.50
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1)), ACTIVE(if(false, z0, z1))) 285.98/91.50
ACTIVE(s(zero(0))) → c13(S(mark(true)), ACTIVE(zero(0))) 285.98/91.50
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13

285.98/91.50

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

285.98/91.50

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.98/91.50
active(add(0, z0)) → mark(z0) 285.98/91.50
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.98/91.50
active(prod(0, z0)) → mark(0) 285.98/91.50
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.98/91.50
active(if(true, z0, z1)) → mark(z0) 285.98/91.50
active(if(false, z0, z1)) → mark(z1) 285.98/91.50
active(zero(0)) → mark(true) 285.98/91.50
active(zero(s(z0))) → mark(false) 285.98/91.50
active(p(s(z0))) → mark(z0) 285.98/91.50
active(fact(z0)) → fact(active(z0)) 285.98/91.50
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.98/91.50
active(zero(z0)) → zero(active(z0)) 285.98/91.50
active(s(z0)) → s(active(z0)) 285.98/91.50
active(prod(z0, z1)) → prod(active(z0), z1) 285.98/91.50
active(prod(z0, z1)) → prod(z0, active(z1)) 285.98/91.50
active(p(z0)) → p(active(z0)) 285.98/91.50
active(add(z0, z1)) → add(active(z0), z1) 285.98/91.50
active(add(z0, z1)) → add(z0, active(z1)) 285.98/91.50
fact(mark(z0)) → mark(fact(z0)) 285.98/91.50
fact(ok(z0)) → ok(fact(z0)) 285.98/91.50
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.98/91.50
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.98/91.50
zero(mark(z0)) → mark(zero(z0)) 285.98/91.50
zero(ok(z0)) → ok(zero(z0)) 285.98/91.50
s(mark(z0)) → mark(s(z0)) 285.98/91.50
s(ok(z0)) → ok(s(z0)) 285.98/91.50
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.98/91.50
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.98/91.50
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.98/91.50
p(mark(z0)) → mark(p(z0)) 285.98/91.50
p(ok(z0)) → ok(p(z0)) 285.98/91.50
add(mark(z0), z1) → mark(add(z0, z1)) 285.98/91.50
add(z0, mark(z1)) → mark(add(z0, z1)) 285.98/91.50
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.98/91.50
proper(fact(z0)) → fact(proper(z0)) 285.98/91.50
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.98/91.50
proper(zero(z0)) → zero(proper(z0)) 285.98/91.50
proper(s(z0)) → s(proper(z0)) 285.98/91.50
proper(0) → ok(0) 285.98/91.50
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.98/91.50
proper(p(z0)) → p(proper(z0)) 285.98/91.50
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.98/91.50
proper(true) → ok(true) 285.98/91.50
proper(false) → ok(false) 285.98/91.50
top(mark(z0)) → top(proper(z0)) 285.98/91.50
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.50
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.50
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.50
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.50
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.50
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.50
S(mark(z0)) → c25(S(z0)) 285.98/91.50
S(ok(z0)) → c26(S(z0)) 285.98/91.50
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.50
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.50
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.50
P(mark(z0)) → c30(P(z0)) 285.98/91.50
P(ok(z0)) → c31(P(z0)) 285.98/91.50
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.50
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.50
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.50
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.50
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.50
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.50
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.50
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.50
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.50
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.50
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.50
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.50
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.50
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.50
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.50
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 285.98/91.50
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 285.98/91.50
ACTIVE(s(zero(0))) → c13(S(mark(true)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.50
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.50
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.50
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.50
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.50
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.50
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.50
S(mark(z0)) → c25(S(z0)) 285.98/91.50
S(ok(z0)) → c26(S(z0)) 285.98/91.50
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.50
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.50
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.50
P(mark(z0)) → c30(P(z0)) 285.98/91.50
P(ok(z0)) → c31(P(z0)) 285.98/91.50
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.50
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.50
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.50
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.50
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.50
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.50
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.50
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.50
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.50
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.50
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.50
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.50
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.50
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.50
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.50
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.50
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.50
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.50
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.50
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.50
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.50
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.50
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.52
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.52
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.52
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.52
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.52
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.52
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.52
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.52
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.52
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.52
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.52
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.52
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.52
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.52
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.52
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.52
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.52
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.52
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.52
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.52
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.52
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.52
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.52
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.52
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.52
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 285.98/91.52
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.52
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.52
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.52
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.52
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.52
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.52
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.52
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.52
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.52
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.52
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 285.98/91.52
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 285.98/91.52
ACTIVE(s(zero(0))) → c13(S(mark(true)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13

285.98/91.52

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(p(z0)) → c16(P(active(z0)), ACTIVE(z0)) by

ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.52
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.52
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.52
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.52
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0)), ACTIVE(if(true, z0, z1))) 285.98/91.52
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1)), ACTIVE(if(false, z0, z1))) 285.98/91.52
ACTIVE(p(zero(0))) → c16(P(mark(true)), ACTIVE(zero(0))) 285.98/91.52
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.52
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.52
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.52
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.52
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 285.98/91.52
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.52
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 285.98/91.52
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.52
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1)))

285.98/91.52

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 285.98/91.52
active(add(0, z0)) → mark(z0) 285.98/91.52
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 285.98/91.52
active(prod(0, z0)) → mark(0) 285.98/91.52
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 285.98/91.52
active(if(true, z0, z1)) → mark(z0) 285.98/91.52
active(if(false, z0, z1)) → mark(z1) 285.98/91.52
active(zero(0)) → mark(true) 285.98/91.52
active(zero(s(z0))) → mark(false) 285.98/91.52
active(p(s(z0))) → mark(z0) 285.98/91.52
active(fact(z0)) → fact(active(z0)) 285.98/91.52
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 285.98/91.52
active(zero(z0)) → zero(active(z0)) 285.98/91.52
active(s(z0)) → s(active(z0)) 285.98/91.52
active(prod(z0, z1)) → prod(active(z0), z1) 285.98/91.52
active(prod(z0, z1)) → prod(z0, active(z1)) 285.98/91.52
active(p(z0)) → p(active(z0)) 285.98/91.52
active(add(z0, z1)) → add(active(z0), z1) 285.98/91.52
active(add(z0, z1)) → add(z0, active(z1)) 285.98/91.52
fact(mark(z0)) → mark(fact(z0)) 285.98/91.52
fact(ok(z0)) → ok(fact(z0)) 285.98/91.52
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 285.98/91.52
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 285.98/91.52
zero(mark(z0)) → mark(zero(z0)) 285.98/91.52
zero(ok(z0)) → ok(zero(z0)) 285.98/91.52
s(mark(z0)) → mark(s(z0)) 285.98/91.52
s(ok(z0)) → ok(s(z0)) 285.98/91.52
prod(mark(z0), z1) → mark(prod(z0, z1)) 285.98/91.52
prod(z0, mark(z1)) → mark(prod(z0, z1)) 285.98/91.52
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 285.98/91.52
p(mark(z0)) → mark(p(z0)) 285.98/91.52
p(ok(z0)) → ok(p(z0)) 285.98/91.52
add(mark(z0), z1) → mark(add(z0, z1)) 285.98/91.52
add(z0, mark(z1)) → mark(add(z0, z1)) 285.98/91.52
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 285.98/91.52
proper(fact(z0)) → fact(proper(z0)) 285.98/91.52
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 285.98/91.52
proper(zero(z0)) → zero(proper(z0)) 285.98/91.52
proper(s(z0)) → s(proper(z0)) 285.98/91.52
proper(0) → ok(0) 285.98/91.52
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 285.98/91.52
proper(p(z0)) → p(proper(z0)) 285.98/91.52
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 285.98/91.52
proper(true) → ok(true) 285.98/91.52
proper(false) → ok(false) 285.98/91.52
top(mark(z0)) → top(proper(z0)) 285.98/91.52
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.52
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.52
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.52
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.52
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.52
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.52
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.52
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.52
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.52
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.52
S(mark(z0)) → c25(S(z0)) 285.98/91.52
S(ok(z0)) → c26(S(z0)) 285.98/91.52
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.52
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.52
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.52
P(mark(z0)) → c30(P(z0)) 285.98/91.52
P(ok(z0)) → c31(P(z0)) 285.98/91.52
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.52
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.52
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.52
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.52
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.52
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.52
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.52
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.52
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.52
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.52
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.52
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.52
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.52
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.52
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.52
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.52
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.52
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.52
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.52
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.52
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.52
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.53
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.53
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.53
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.53
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.53
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.53
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.53
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.53
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.53
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.53
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.53
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.53
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.53
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.53
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.53
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.53
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 285.98/91.53
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.53
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.53
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.53
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.53
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 285.98/91.53
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 285.98/91.53
ACTIVE(s(zero(0))) → c13(S(mark(true))) 285.98/91.53
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.53
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.53
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.53
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.53
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0)), ACTIVE(if(true, z0, z1))) 285.98/91.53
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1)), ACTIVE(if(false, z0, z1))) 285.98/91.53
ACTIVE(p(zero(0))) → c16(P(mark(true)), ACTIVE(zero(0))) 285.98/91.53
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 285.98/91.53
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 285.98/91.53
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 285.98/91.53
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 285.98/91.53
FACT(mark(z0)) → c19(FACT(z0)) 285.98/91.53
FACT(ok(z0)) → c20(FACT(z0)) 285.98/91.53
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 285.98/91.53
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 285.98/91.53
ZERO(mark(z0)) → c23(ZERO(z0)) 285.98/91.53
ZERO(ok(z0)) → c24(ZERO(z0)) 285.98/91.53
S(mark(z0)) → c25(S(z0)) 285.98/91.53
S(ok(z0)) → c26(S(z0)) 285.98/91.53
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 285.98/91.53
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 285.98/91.53
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 285.98/91.53
P(mark(z0)) → c30(P(z0)) 285.98/91.53
P(ok(z0)) → c31(P(z0)) 285.98/91.53
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 285.98/91.53
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 285.98/91.53
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 285.98/91.53
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 285.98/91.53
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 285.98/91.53
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 285.98/91.53
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 285.98/91.53
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.53
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 285.98/91.53
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 285.98/91.53
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 285.98/91.53
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 285.98/91.53
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 285.98/91.53
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 285.98/91.53
ACTIVE(fact(z0)) → c1(ZERO(z0)) 285.98/91.53
ACTIVE(fact(z0)) → c1(P(z0)) 285.98/91.53
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.53
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.53
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.53
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.53
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.53
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 285.98/91.53
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 285.98/91.53
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 285.98/91.53
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 285.98/91.53
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 285.98/91.53
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 285.98/91.53
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 285.98/91.53
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 285.98/91.53
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 285.98/91.53
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 285.98/91.53
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 285.98/91.53
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 285.98/91.53
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.55
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 285.98/91.55
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 285.98/91.55
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 285.98/91.55
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 285.98/91.55
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 285.98/91.55
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.55
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.55
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.55
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.55
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.55
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.55
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.55
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.55
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.55
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.55
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 285.98/91.55
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.55
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.55
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 285.98/91.55
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.55
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.55
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 285.98/91.55
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 285.98/91.55
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 285.98/91.55
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 285.98/91.55
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 285.98/91.55
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.55
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.55
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.55
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.55
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.55
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 285.98/91.55
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 285.98/91.55
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 285.98/91.55
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 285.98/91.55
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 285.98/91.55
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 285.98/91.55
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 285.98/91.55
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 285.98/91.55
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 285.98/91.55
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 285.98/91.55
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 285.98/91.55
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 285.98/91.55
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 285.98/91.55
ACTIVE(s(zero(0))) → c13(S(mark(true))) 285.98/91.55
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 285.98/91.55
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 285.98/91.55
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 285.98/91.55
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 285.98/91.55
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 285.98/91.55
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0)), ACTIVE(if(true, z0, z1))) 286.36/91.60
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1)), ACTIVE(if(false, z0, z1))) 286.36/91.60
ACTIVE(p(zero(0))) → c16(P(mark(true)), ACTIVE(zero(0))) 286.36/91.60
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.60
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.60
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.60
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.60
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.60
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.60
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.60
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.60
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.60
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.60
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16

286.36/91.60

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.60

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.60
active(add(0, z0)) → mark(z0) 286.36/91.60
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.60
active(prod(0, z0)) → mark(0) 286.36/91.60
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.60
active(if(true, z0, z1)) → mark(z0) 286.36/91.60
active(if(false, z0, z1)) → mark(z1) 286.36/91.60
active(zero(0)) → mark(true) 286.36/91.60
active(zero(s(z0))) → mark(false) 286.36/91.61
active(p(s(z0))) → mark(z0) 286.36/91.61
active(fact(z0)) → fact(active(z0)) 286.36/91.61
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.61
active(zero(z0)) → zero(active(z0)) 286.36/91.61
active(s(z0)) → s(active(z0)) 286.36/91.61
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.61
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.61
active(p(z0)) → p(active(z0)) 286.36/91.61
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.61
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.61
fact(mark(z0)) → mark(fact(z0)) 286.36/91.61
fact(ok(z0)) → ok(fact(z0)) 286.36/91.61
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.61
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.61
zero(mark(z0)) → mark(zero(z0)) 286.36/91.61
zero(ok(z0)) → ok(zero(z0)) 286.36/91.61
s(mark(z0)) → mark(s(z0)) 286.36/91.61
s(ok(z0)) → ok(s(z0)) 286.36/91.61
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.61
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.61
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.61
p(mark(z0)) → mark(p(z0)) 286.36/91.61
p(ok(z0)) → ok(p(z0)) 286.36/91.61
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.61
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.61
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.61
proper(fact(z0)) → fact(proper(z0)) 286.36/91.61
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.61
proper(zero(z0)) → zero(proper(z0)) 286.36/91.61
proper(s(z0)) → s(proper(z0)) 286.36/91.61
proper(0) → ok(0) 286.36/91.61
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.61
proper(p(z0)) → p(proper(z0)) 286.36/91.61
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.61
proper(true) → ok(true) 286.36/91.61
proper(false) → ok(false) 286.36/91.61
top(mark(z0)) → top(proper(z0)) 286.36/91.61
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16

286.36/91.61

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(fact(z0)) → c35(FACT(proper(z0)), PROPER(z0)) by

PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.61
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.61
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.61
PROPER(fact(0)) → c35(FACT(ok(0)), PROPER(0)) 286.36/91.61
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.61
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.61
PROPER(fact(true)) → c35(FACT(ok(true)), PROPER(true)) 286.36/91.61
PROPER(fact(false)) → c35(FACT(ok(false)), PROPER(false))

286.36/91.61

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.61
active(add(0, z0)) → mark(z0) 286.36/91.61
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.61
active(prod(0, z0)) → mark(0) 286.36/91.61
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.61
active(if(true, z0, z1)) → mark(z0) 286.36/91.61
active(if(false, z0, z1)) → mark(z1) 286.36/91.61
active(zero(0)) → mark(true) 286.36/91.61
active(zero(s(z0))) → mark(false) 286.36/91.61
active(p(s(z0))) → mark(z0) 286.36/91.61
active(fact(z0)) → fact(active(z0)) 286.36/91.61
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.61
active(zero(z0)) → zero(active(z0)) 286.36/91.61
active(s(z0)) → s(active(z0)) 286.36/91.61
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.61
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.61
active(p(z0)) → p(active(z0)) 286.36/91.61
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.61
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.61
fact(mark(z0)) → mark(fact(z0)) 286.36/91.61
fact(ok(z0)) → ok(fact(z0)) 286.36/91.61
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.61
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.61
zero(mark(z0)) → mark(zero(z0)) 286.36/91.61
zero(ok(z0)) → ok(zero(z0)) 286.36/91.61
s(mark(z0)) → mark(s(z0)) 286.36/91.61
s(ok(z0)) → ok(s(z0)) 286.36/91.61
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.61
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.61
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.61
p(mark(z0)) → mark(p(z0)) 286.36/91.61
p(ok(z0)) → ok(p(z0)) 286.36/91.61
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.61
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.61
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.61
proper(fact(z0)) → fact(proper(z0)) 286.36/91.61
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.61
proper(zero(z0)) → zero(proper(z0)) 286.36/91.61
proper(s(z0)) → s(proper(z0)) 286.36/91.61
proper(0) → ok(0) 286.36/91.61
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.61
proper(p(z0)) → p(proper(z0)) 286.36/91.61
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.61
proper(true) → ok(true) 286.36/91.61
proper(false) → ok(false) 286.36/91.61
top(mark(z0)) → top(proper(z0)) 286.36/91.61
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.61
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.61
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.61
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.61
PROPER(fact(0)) → c35(FACT(ok(0)), PROPER(0)) 286.36/91.61
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.61
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.61
PROPER(fact(true)) → c35(FACT(ok(true)), PROPER(true)) 286.36/91.61
PROPER(fact(false)) → c35(FACT(ok(false)), PROPER(false))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.61
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.61
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.61
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.61
PROPER(fact(0)) → c35(FACT(ok(0)), PROPER(0)) 286.36/91.61
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.61
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.61
PROPER(fact(true)) → c35(FACT(ok(true)), PROPER(true)) 286.36/91.61
PROPER(fact(false)) → c35(FACT(ok(false)), PROPER(false))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35

286.36/91.61

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.61

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.61
active(add(0, z0)) → mark(z0) 286.36/91.61
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.61
active(prod(0, z0)) → mark(0) 286.36/91.61
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.61
active(if(true, z0, z1)) → mark(z0) 286.36/91.61
active(if(false, z0, z1)) → mark(z1) 286.36/91.61
active(zero(0)) → mark(true) 286.36/91.61
active(zero(s(z0))) → mark(false) 286.36/91.61
active(p(s(z0))) → mark(z0) 286.36/91.61
active(fact(z0)) → fact(active(z0)) 286.36/91.61
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.61
active(zero(z0)) → zero(active(z0)) 286.36/91.61
active(s(z0)) → s(active(z0)) 286.36/91.61
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.61
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.61
active(p(z0)) → p(active(z0)) 286.36/91.61
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.61
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.61
fact(mark(z0)) → mark(fact(z0)) 286.36/91.61
fact(ok(z0)) → ok(fact(z0)) 286.36/91.61
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.61
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.61
zero(mark(z0)) → mark(zero(z0)) 286.36/91.61
zero(ok(z0)) → ok(zero(z0)) 286.36/91.61
s(mark(z0)) → mark(s(z0)) 286.36/91.61
s(ok(z0)) → ok(s(z0)) 286.36/91.61
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.61
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.61
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.61
p(mark(z0)) → mark(p(z0)) 286.36/91.61
p(ok(z0)) → ok(p(z0)) 286.36/91.61
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.61
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.61
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.61
proper(fact(z0)) → fact(proper(z0)) 286.36/91.61
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.61
proper(zero(z0)) → zero(proper(z0)) 286.36/91.61
proper(s(z0)) → s(proper(z0)) 286.36/91.61
proper(0) → ok(0) 286.36/91.61
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.61
proper(p(z0)) → p(proper(z0)) 286.36/91.61
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.61
proper(true) → ok(true) 286.36/91.61
proper(false) → ok(false) 286.36/91.61
top(mark(z0)) → top(proper(z0)) 286.36/91.61
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.61
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.61
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.61
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.61
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.61
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.61
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.61
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.61
PROPER(fact(false)) → c35(FACT(ok(false)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.61
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.61
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.61
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.61
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.61
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.61
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.61
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.61
PROPER(fact(false)) → c35(FACT(ok(false)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c36, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35

286.36/91.61

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(if(z0, z1, z2)) → c36(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.61
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.61
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.61
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 286.36/91.61
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.61
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.61
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 286.36/91.61
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 286.36/91.61
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 286.36/91.61
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 286.36/91.61
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 286.36/91.61
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))

286.36/91.61

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.61
active(add(0, z0)) → mark(z0) 286.36/91.61
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.61
active(prod(0, z0)) → mark(0) 286.36/91.61
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.61
active(if(true, z0, z1)) → mark(z0) 286.36/91.61
active(if(false, z0, z1)) → mark(z1) 286.36/91.61
active(zero(0)) → mark(true) 286.36/91.61
active(zero(s(z0))) → mark(false) 286.36/91.61
active(p(s(z0))) → mark(z0) 286.36/91.61
active(fact(z0)) → fact(active(z0)) 286.36/91.61
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.61
active(zero(z0)) → zero(active(z0)) 286.36/91.61
active(s(z0)) → s(active(z0)) 286.36/91.61
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.61
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.61
active(p(z0)) → p(active(z0)) 286.36/91.61
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.61
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.61
fact(mark(z0)) → mark(fact(z0)) 286.36/91.61
fact(ok(z0)) → ok(fact(z0)) 286.36/91.61
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.61
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.61
zero(mark(z0)) → mark(zero(z0)) 286.36/91.61
zero(ok(z0)) → ok(zero(z0)) 286.36/91.61
s(mark(z0)) → mark(s(z0)) 286.36/91.61
s(ok(z0)) → ok(s(z0)) 286.36/91.61
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.61
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.61
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.61
p(mark(z0)) → mark(p(z0)) 286.36/91.61
p(ok(z0)) → ok(p(z0)) 286.36/91.61
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.61
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.61
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.61
proper(fact(z0)) → fact(proper(z0)) 286.36/91.61
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.61
proper(zero(z0)) → zero(proper(z0)) 286.36/91.61
proper(s(z0)) → s(proper(z0)) 286.36/91.61
proper(0) → ok(0) 286.36/91.61
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.61
proper(p(z0)) → p(proper(z0)) 286.36/91.61
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.61
proper(true) → ok(true) 286.36/91.61
proper(false) → ok(false) 286.36/91.61
top(mark(z0)) → top(proper(z0)) 286.36/91.61
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.61
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.61
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.61
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.61
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.61
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.61
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.61
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.61
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.61
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.61
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.61
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.61
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.61
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.61
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.61
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.61
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.61
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.61
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.61
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.61
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.61
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.61
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.61
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.61
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 286.36/91.61
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.61
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.61
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.61
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 286.36/91.61
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 286.36/91.61
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 286.36/91.61
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.61
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 286.36/91.61
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 286.36/91.61
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 286.36/91.61
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.61
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.61
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.61
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.61
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.61
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.61
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.61
S(mark(z0)) → c25(S(z0)) 286.36/91.61
S(ok(z0)) → c26(S(z0)) 286.36/91.61
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.61
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.61
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.61
P(mark(z0)) → c30(P(z0)) 286.36/91.61
P(ok(z0)) → c31(P(z0)) 286.36/91.61
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.61
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.61
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.61
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.61
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.61
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.61
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.61
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.61
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.61
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.61
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.61
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.61
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.61
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.61
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.61
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.61
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.61
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.61
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.61
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.61
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.62
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.62
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.62
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.62
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.62
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.62
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.62
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.62
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.62
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.62
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.62
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.62
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.62
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.62
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.62
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.62
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.62
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.62
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 286.36/91.62
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.62
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 286.36/91.62
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 286.36/91.62
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 286.36/91.62
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 286.36/91.62
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36

286.36/91.62

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts

286.36/91.62

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.62
active(add(0, z0)) → mark(z0) 286.36/91.62
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.62
active(prod(0, z0)) → mark(0) 286.36/91.62
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.62
active(if(true, z0, z1)) → mark(z0) 286.36/91.62
active(if(false, z0, z1)) → mark(z1) 286.36/91.62
active(zero(0)) → mark(true) 286.36/91.62
active(zero(s(z0))) → mark(false) 286.36/91.62
active(p(s(z0))) → mark(z0) 286.36/91.62
active(fact(z0)) → fact(active(z0)) 286.36/91.62
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.62
active(zero(z0)) → zero(active(z0)) 286.36/91.62
active(s(z0)) → s(active(z0)) 286.36/91.62
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.62
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.62
active(p(z0)) → p(active(z0)) 286.36/91.62
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.62
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.62
fact(mark(z0)) → mark(fact(z0)) 286.36/91.62
fact(ok(z0)) → ok(fact(z0)) 286.36/91.62
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.62
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.62
zero(mark(z0)) → mark(zero(z0)) 286.36/91.62
zero(ok(z0)) → ok(zero(z0)) 286.36/91.62
s(mark(z0)) → mark(s(z0)) 286.36/91.62
s(ok(z0)) → ok(s(z0)) 286.36/91.62
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.62
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.62
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.62
p(mark(z0)) → mark(p(z0)) 286.36/91.62
p(ok(z0)) → ok(p(z0)) 286.36/91.62
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.62
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.62
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.62
proper(fact(z0)) → fact(proper(z0)) 286.36/91.62
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.62
proper(zero(z0)) → zero(proper(z0)) 286.36/91.62
proper(s(z0)) → s(proper(z0)) 286.36/91.62
proper(0) → ok(0) 286.36/91.62
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.62
proper(p(z0)) → p(proper(z0)) 286.36/91.62
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.62
proper(true) → ok(true) 286.36/91.62
proper(false) → ok(false) 286.36/91.62
top(mark(z0)) → top(proper(z0)) 286.36/91.62
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.62
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.62
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.62
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.62
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.62
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.62
S(mark(z0)) → c25(S(z0)) 286.36/91.62
S(ok(z0)) → c26(S(z0)) 286.36/91.62
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.62
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.62
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.62
P(mark(z0)) → c30(P(z0)) 286.36/91.62
P(ok(z0)) → c31(P(z0)) 286.36/91.62
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.62
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.62
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.62
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.62
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.62
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.62
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.62
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.62
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.62
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.62
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.62
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.62
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.62
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.62
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.62
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.62
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.62
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.62
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.62
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.62
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.62
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.62
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.62
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.62
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.62
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.62
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.62
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.62
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.62
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.62
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.62
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.62
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.62
S(mark(z0)) → c25(S(z0)) 286.36/91.62
S(ok(z0)) → c26(S(z0)) 286.36/91.62
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.62
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.62
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.62
P(mark(z0)) → c30(P(z0)) 286.36/91.62
P(ok(z0)) → c31(P(z0)) 286.36/91.62
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.62
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.62
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.62
PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.62
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.62
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.62
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.62
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.62
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.62
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.62
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.62
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.62
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.62
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.62
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.62
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.62
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.62
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.62
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.62
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.62
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.62
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.62
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.62
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.62
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.62
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.62
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.62
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c37, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36

286.36/91.62

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(zero(z0)) → c37(ZERO(proper(z0)), PROPER(z0)) by

PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(zero(0)) → c37(ZERO(ok(0)), PROPER(0)) 286.36/91.62
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(zero(true)) → c37(ZERO(ok(true)), PROPER(true)) 286.36/91.62
PROPER(zero(false)) → c37(ZERO(ok(false)), PROPER(false))

286.36/91.62

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.62
active(add(0, z0)) → mark(z0) 286.36/91.62
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.62
active(prod(0, z0)) → mark(0) 286.36/91.62
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.62
active(if(true, z0, z1)) → mark(z0) 286.36/91.62
active(if(false, z0, z1)) → mark(z1) 286.36/91.62
active(zero(0)) → mark(true) 286.36/91.62
active(zero(s(z0))) → mark(false) 286.36/91.62
active(p(s(z0))) → mark(z0) 286.36/91.62
active(fact(z0)) → fact(active(z0)) 286.36/91.62
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.62
active(zero(z0)) → zero(active(z0)) 286.36/91.62
active(s(z0)) → s(active(z0)) 286.36/91.62
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.62
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.62
active(p(z0)) → p(active(z0)) 286.36/91.62
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.62
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.62
fact(mark(z0)) → mark(fact(z0)) 286.36/91.62
fact(ok(z0)) → ok(fact(z0)) 286.36/91.62
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.62
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.62
zero(mark(z0)) → mark(zero(z0)) 286.36/91.62
zero(ok(z0)) → ok(zero(z0)) 286.36/91.62
s(mark(z0)) → mark(s(z0)) 286.36/91.62
s(ok(z0)) → ok(s(z0)) 286.36/91.62
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.62
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.62
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.62
p(mark(z0)) → mark(p(z0)) 286.36/91.62
p(ok(z0)) → ok(p(z0)) 286.36/91.62
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.62
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.62
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.62
proper(fact(z0)) → fact(proper(z0)) 286.36/91.62
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.62
proper(zero(z0)) → zero(proper(z0)) 286.36/91.62
proper(s(z0)) → s(proper(z0)) 286.36/91.62
proper(0) → ok(0) 286.36/91.62
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.62
proper(p(z0)) → p(proper(z0)) 286.36/91.62
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.62
proper(true) → ok(true) 286.36/91.62
proper(false) → ok(false) 286.36/91.62
top(mark(z0)) → top(proper(z0)) 286.36/91.62
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.62
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.62
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.62
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.62
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.62
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.62
S(mark(z0)) → c25(S(z0)) 286.36/91.62
S(ok(z0)) → c26(S(z0)) 286.36/91.62
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.62
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.62
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.62
P(mark(z0)) → c30(P(z0)) 286.36/91.62
P(ok(z0)) → c31(P(z0)) 286.36/91.62
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.62
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.62
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.62
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.62
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.62
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.62
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.62
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.62
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.62
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.62
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.62
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.62
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.62
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.62
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.62
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.62
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.62
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.62
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.62
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.62
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.62
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.62
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.62
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.62
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.62
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.62
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.62
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(zero(0)) → c37(ZERO(ok(0)), PROPER(0)) 286.36/91.62
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(zero(true)) → c37(ZERO(ok(true)), PROPER(true)) 286.36/91.62
PROPER(zero(false)) → c37(ZERO(ok(false)), PROPER(false))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.62
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.62
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.62
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.62
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.62
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.62
S(mark(z0)) → c25(S(z0)) 286.36/91.62
S(ok(z0)) → c26(S(z0)) 286.36/91.62
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.62
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.62
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.62
P(mark(z0)) → c30(P(z0)) 286.36/91.62
P(ok(z0)) → c31(P(z0)) 286.36/91.62
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.62
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.62
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.62
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.62
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.62
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.62
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.62
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.62
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.62
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.62
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.62
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.62
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.62
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.62
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.62
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.62
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.62
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.62
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.62
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.62
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.62
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.62
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.62
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.62
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.62
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.62
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.62
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(zero(0)) → c37(ZERO(ok(0)), PROPER(0)) 286.36/91.62
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(zero(true)) → c37(ZERO(ok(true)), PROPER(true)) 286.36/91.62
PROPER(zero(false)) → c37(ZERO(ok(false)), PROPER(false))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37

286.36/91.62

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.62

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.62
active(add(0, z0)) → mark(z0) 286.36/91.62
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.62
active(prod(0, z0)) → mark(0) 286.36/91.62
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.62
active(if(true, z0, z1)) → mark(z0) 286.36/91.62
active(if(false, z0, z1)) → mark(z1) 286.36/91.62
active(zero(0)) → mark(true) 286.36/91.62
active(zero(s(z0))) → mark(false) 286.36/91.62
active(p(s(z0))) → mark(z0) 286.36/91.62
active(fact(z0)) → fact(active(z0)) 286.36/91.62
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.62
active(zero(z0)) → zero(active(z0)) 286.36/91.62
active(s(z0)) → s(active(z0)) 286.36/91.62
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.62
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.62
active(p(z0)) → p(active(z0)) 286.36/91.62
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.62
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.62
fact(mark(z0)) → mark(fact(z0)) 286.36/91.62
fact(ok(z0)) → ok(fact(z0)) 286.36/91.62
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.62
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.62
zero(mark(z0)) → mark(zero(z0)) 286.36/91.62
zero(ok(z0)) → ok(zero(z0)) 286.36/91.62
s(mark(z0)) → mark(s(z0)) 286.36/91.62
s(ok(z0)) → ok(s(z0)) 286.36/91.62
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.62
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.62
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.62
p(mark(z0)) → mark(p(z0)) 286.36/91.62
p(ok(z0)) → ok(p(z0)) 286.36/91.62
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.62
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.62
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.62
proper(fact(z0)) → fact(proper(z0)) 286.36/91.62
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.62
proper(zero(z0)) → zero(proper(z0)) 286.36/91.62
proper(s(z0)) → s(proper(z0)) 286.36/91.62
proper(0) → ok(0) 286.36/91.62
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.62
proper(p(z0)) → p(proper(z0)) 286.36/91.62
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.62
proper(true) → ok(true) 286.36/91.62
proper(false) → ok(false) 286.36/91.62
top(mark(z0)) → top(proper(z0)) 286.36/91.62
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.62
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.62
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.62
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.62
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.62
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.62
S(mark(z0)) → c25(S(z0)) 286.36/91.62
S(ok(z0)) → c26(S(z0)) 286.36/91.62
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.62
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.62
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.62
P(mark(z0)) → c30(P(z0)) 286.36/91.62
P(ok(z0)) → c31(P(z0)) 286.36/91.62
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.62
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.62
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.62
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.62
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.62
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.62
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.62
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.62
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.62
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.62
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.62
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.62
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.62
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.62
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.62
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.62
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.62
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.62
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.62
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.62
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.62
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.62
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.62
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.62
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.62
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.62
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.62
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.62
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.62
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.62
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.62
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.62
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.62
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.62
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.62
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.62
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.62
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.62
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.62
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.62
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.62
PROPER(zero(false)) → c37(ZERO(ok(false)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.62
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.62
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.62
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.62
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.62
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.62
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.62
S(mark(z0)) → c25(S(z0)) 286.36/91.62
S(ok(z0)) → c26(S(z0)) 286.36/91.62
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.62
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.62
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.62
P(mark(z0)) → c30(P(z0)) 286.36/91.62
P(ok(z0)) → c31(P(z0)) 286.36/91.62
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.62
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.62
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.62
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.62
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.62
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.62
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.62
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.62
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.62
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.62
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.62
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.62
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.62
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.62
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.62
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.62
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.62
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.62
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.62
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.62
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.62
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.63
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.63
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.63
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.63
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.63
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.63
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.63
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.63
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.63
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.63
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.63
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.63
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.63
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.63
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.63
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.63
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.63
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.63
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.63
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.63
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.63
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.63
PROPER(zero(false)) → c37(ZERO(ok(false)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c38, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37

286.36/91.63

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0)) by

PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(s(0)) → c38(S(ok(0)), PROPER(0)) 286.36/91.63
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(s(true)) → c38(S(ok(true)), PROPER(true)) 286.36/91.63
PROPER(s(false)) → c38(S(ok(false)), PROPER(false))

286.36/91.63

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.63
active(add(0, z0)) → mark(z0) 286.36/91.63
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.63
active(prod(0, z0)) → mark(0) 286.36/91.63
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.63
active(if(true, z0, z1)) → mark(z0) 286.36/91.63
active(if(false, z0, z1)) → mark(z1) 286.36/91.63
active(zero(0)) → mark(true) 286.36/91.63
active(zero(s(z0))) → mark(false) 286.36/91.63
active(p(s(z0))) → mark(z0) 286.36/91.63
active(fact(z0)) → fact(active(z0)) 286.36/91.63
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.63
active(zero(z0)) → zero(active(z0)) 286.36/91.63
active(s(z0)) → s(active(z0)) 286.36/91.63
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.63
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.63
active(p(z0)) → p(active(z0)) 286.36/91.63
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.63
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.63
fact(mark(z0)) → mark(fact(z0)) 286.36/91.63
fact(ok(z0)) → ok(fact(z0)) 286.36/91.63
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.63
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.63
zero(mark(z0)) → mark(zero(z0)) 286.36/91.63
zero(ok(z0)) → ok(zero(z0)) 286.36/91.63
s(mark(z0)) → mark(s(z0)) 286.36/91.63
s(ok(z0)) → ok(s(z0)) 286.36/91.63
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.63
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.63
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.63
p(mark(z0)) → mark(p(z0)) 286.36/91.63
p(ok(z0)) → ok(p(z0)) 286.36/91.63
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.63
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.63
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.63
proper(fact(z0)) → fact(proper(z0)) 286.36/91.63
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.63
proper(zero(z0)) → zero(proper(z0)) 286.36/91.63
proper(s(z0)) → s(proper(z0)) 286.36/91.63
proper(0) → ok(0) 286.36/91.63
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.63
proper(p(z0)) → p(proper(z0)) 286.36/91.63
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.63
proper(true) → ok(true) 286.36/91.63
proper(false) → ok(false) 286.36/91.63
top(mark(z0)) → top(proper(z0)) 286.36/91.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.63
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.63
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.63
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.63
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.63
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.63
S(mark(z0)) → c25(S(z0)) 286.36/91.63
S(ok(z0)) → c26(S(z0)) 286.36/91.63
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.63
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.63
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.63
P(mark(z0)) → c30(P(z0)) 286.36/91.63
P(ok(z0)) → c31(P(z0)) 286.36/91.63
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.63
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.63
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.63
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.63
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.63
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.63
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.63
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.63
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.63
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.63
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.63
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.63
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.63
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.63
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.63
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.63
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.63
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.63
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.63
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.63
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.63
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.63
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.63
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.63
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.63
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.63
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.63
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.63
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.63
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.63
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.63
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(s(0)) → c38(S(ok(0)), PROPER(0)) 286.36/91.63
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(s(true)) → c38(S(ok(true)), PROPER(true)) 286.36/91.63
PROPER(s(false)) → c38(S(ok(false)), PROPER(false))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.63
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.63
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.63
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.63
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.63
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.63
S(mark(z0)) → c25(S(z0)) 286.36/91.63
S(ok(z0)) → c26(S(z0)) 286.36/91.63
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.63
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.63
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.63
P(mark(z0)) → c30(P(z0)) 286.36/91.63
P(ok(z0)) → c31(P(z0)) 286.36/91.63
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.63
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.63
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.63
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.63
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.63
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.63
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.63
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.63
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.63
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.63
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.63
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.63
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.63
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.63
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.63
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.63
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.63
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.63
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.63
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.63
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.63
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.63
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.63
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.63
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.63
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.63
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.63
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.63
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.63
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.63
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.63
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(s(0)) → c38(S(ok(0)), PROPER(0)) 286.36/91.63
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(s(true)) → c38(S(ok(true)), PROPER(true)) 286.36/91.63
PROPER(s(false)) → c38(S(ok(false)), PROPER(false))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38

286.36/91.63

(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.63

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.63
active(add(0, z0)) → mark(z0) 286.36/91.63
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.63
active(prod(0, z0)) → mark(0) 286.36/91.63
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.63
active(if(true, z0, z1)) → mark(z0) 286.36/91.63
active(if(false, z0, z1)) → mark(z1) 286.36/91.63
active(zero(0)) → mark(true) 286.36/91.63
active(zero(s(z0))) → mark(false) 286.36/91.63
active(p(s(z0))) → mark(z0) 286.36/91.63
active(fact(z0)) → fact(active(z0)) 286.36/91.63
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.63
active(zero(z0)) → zero(active(z0)) 286.36/91.63
active(s(z0)) → s(active(z0)) 286.36/91.63
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.63
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.63
active(p(z0)) → p(active(z0)) 286.36/91.63
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.63
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.63
fact(mark(z0)) → mark(fact(z0)) 286.36/91.63
fact(ok(z0)) → ok(fact(z0)) 286.36/91.63
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.63
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.63
zero(mark(z0)) → mark(zero(z0)) 286.36/91.63
zero(ok(z0)) → ok(zero(z0)) 286.36/91.63
s(mark(z0)) → mark(s(z0)) 286.36/91.63
s(ok(z0)) → ok(s(z0)) 286.36/91.63
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.63
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.63
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.63
p(mark(z0)) → mark(p(z0)) 286.36/91.63
p(ok(z0)) → ok(p(z0)) 286.36/91.63
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.63
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.63
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.63
proper(fact(z0)) → fact(proper(z0)) 286.36/91.63
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.63
proper(zero(z0)) → zero(proper(z0)) 286.36/91.63
proper(s(z0)) → s(proper(z0)) 286.36/91.63
proper(0) → ok(0) 286.36/91.63
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.63
proper(p(z0)) → p(proper(z0)) 286.36/91.63
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.63
proper(true) → ok(true) 286.36/91.63
proper(false) → ok(false) 286.36/91.63
top(mark(z0)) → top(proper(z0)) 286.36/91.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.63
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.63
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.63
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.63
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.63
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.63
S(mark(z0)) → c25(S(z0)) 286.36/91.63
S(ok(z0)) → c26(S(z0)) 286.36/91.63
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.63
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.63
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.63
P(mark(z0)) → c30(P(z0)) 286.36/91.63
P(ok(z0)) → c31(P(z0)) 286.36/91.63
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.63
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.63
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.63
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.63
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.63
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.63
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.63
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.63
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.63
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.63
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.63
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.63
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.63
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.63
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.63
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.63
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.63
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.63
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.63
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.63
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.63
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.63
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.63
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.63
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.63
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.63
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.63
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.63
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.63
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.63
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.63
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.63
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.63
PROPER(s(false)) → c38(S(ok(false)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.63
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.63
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.63
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.63
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.63
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.63
S(mark(z0)) → c25(S(z0)) 286.36/91.63
S(ok(z0)) → c26(S(z0)) 286.36/91.63
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.63
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.63
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.63
P(mark(z0)) → c30(P(z0)) 286.36/91.63
P(ok(z0)) → c31(P(z0)) 286.36/91.63
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.63
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.63
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.63
PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.63
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.63
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.63
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.63
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.63
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.63
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.63
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.63
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.63
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.63
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.63
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.63
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.63
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.63
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.63
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.63
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.63
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.63
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.63
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.63
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.63
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.63
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.63
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.63
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.63
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.63
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.63
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.63
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.63
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.63
PROPER(s(false)) → c38(S(ok(false)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c40, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38

286.36/91.63

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(prod(z0, z1)) → c40(PROD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.63
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.63
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.63
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 286.36/91.63
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.63
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.63
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 286.36/91.63
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 286.36/91.63
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.63
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.63
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.63
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.63
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 286.36/91.63
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.63
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.63
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.63
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 286.36/91.63
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(false), PROPER(x1))

286.36/91.63

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.63
active(add(0, z0)) → mark(z0) 286.36/91.63
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.63
active(prod(0, z0)) → mark(0) 286.36/91.63
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.63
active(if(true, z0, z1)) → mark(z0) 286.36/91.63
active(if(false, z0, z1)) → mark(z1) 286.36/91.63
active(zero(0)) → mark(true) 286.36/91.63
active(zero(s(z0))) → mark(false) 286.36/91.63
active(p(s(z0))) → mark(z0) 286.36/91.63
active(fact(z0)) → fact(active(z0)) 286.36/91.63
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.63
active(zero(z0)) → zero(active(z0)) 286.36/91.63
active(s(z0)) → s(active(z0)) 286.36/91.63
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.63
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.63
active(p(z0)) → p(active(z0)) 286.36/91.63
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.63
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.63
fact(mark(z0)) → mark(fact(z0)) 286.36/91.63
fact(ok(z0)) → ok(fact(z0)) 286.36/91.63
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.63
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.63
zero(mark(z0)) → mark(zero(z0)) 286.36/91.63
zero(ok(z0)) → ok(zero(z0)) 286.36/91.63
s(mark(z0)) → mark(s(z0)) 286.36/91.63
s(ok(z0)) → ok(s(z0)) 286.36/91.63
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.63
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.63
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.63
p(mark(z0)) → mark(p(z0)) 286.36/91.63
p(ok(z0)) → ok(p(z0)) 286.36/91.63
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.63
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.63
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.63
proper(fact(z0)) → fact(proper(z0)) 286.36/91.63
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.63
proper(zero(z0)) → zero(proper(z0)) 286.36/91.63
proper(s(z0)) → s(proper(z0)) 286.36/91.63
proper(0) → ok(0) 286.36/91.63
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.63
proper(p(z0)) → p(proper(z0)) 286.36/91.63
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.63
proper(true) → ok(true) 286.36/91.63
proper(false) → ok(false) 286.36/91.63
top(mark(z0)) → top(proper(z0)) 286.36/91.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.63
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.63
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.63
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.63
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.63
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.63
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.63
S(mark(z0)) → c25(S(z0)) 286.36/91.63
S(ok(z0)) → c26(S(z0)) 286.36/91.63
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.63
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.63
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.63
P(mark(z0)) → c30(P(z0)) 286.36/91.63
P(ok(z0)) → c31(P(z0)) 286.36/91.63
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.63
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.63
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.63
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.63
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.63
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.63
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.63
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.63
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.63
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.63
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.63
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.63
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.63
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.63
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.63
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.63
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.63
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.63
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.63
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.63
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.63
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.63
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.63
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.63
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.63
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.63
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.63
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.63
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.63
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.63
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.63
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.63
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.63
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.63
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.63
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.63
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.63
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.63
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.63
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.63
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.63
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.63
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.63
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.63
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.63
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.63
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.63
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.63
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.63
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.63
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.63
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.63
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.63
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.64
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.64
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.64
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.64
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.64
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.64
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.64
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.64
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.64
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 286.36/91.64
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.64
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.64
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 286.36/91.64
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 286.36/91.64
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.64
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 286.36/91.64
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 286.36/91.64
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(false), PROPER(x1))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.64
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.64
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.64
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.64
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.64
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.64
S(mark(z0)) → c25(S(z0)) 286.36/91.64
S(ok(z0)) → c26(S(z0)) 286.36/91.64
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.64
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.64
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.64
P(mark(z0)) → c30(P(z0)) 286.36/91.64
P(ok(z0)) → c31(P(z0)) 286.36/91.64
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.64
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.64
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.64
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.64
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.64
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.64
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.64
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.64
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.64
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.64
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.64
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.64
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.64
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.64
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.64
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.64
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.64
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.64
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.64
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.64
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.64
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.64
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.64
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.64
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.64
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.64
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.64
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.64
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.64
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.64
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.64
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.64
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.64
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.64
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.64
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.64
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.64
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.64
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 286.36/91.64
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.64
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.64
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 286.36/91.64
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 286.36/91.64
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.64
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 286.36/91.64
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 286.36/91.64
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(false), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40

286.36/91.64

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

286.36/91.64

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.64
active(add(0, z0)) → mark(z0) 286.36/91.64
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.64
active(prod(0, z0)) → mark(0) 286.36/91.64
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.64
active(if(true, z0, z1)) → mark(z0) 286.36/91.64
active(if(false, z0, z1)) → mark(z1) 286.36/91.64
active(zero(0)) → mark(true) 286.36/91.64
active(zero(s(z0))) → mark(false) 286.36/91.64
active(p(s(z0))) → mark(z0) 286.36/91.64
active(fact(z0)) → fact(active(z0)) 286.36/91.64
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.64
active(zero(z0)) → zero(active(z0)) 286.36/91.64
active(s(z0)) → s(active(z0)) 286.36/91.64
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.64
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.64
active(p(z0)) → p(active(z0)) 286.36/91.64
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.64
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.64
fact(mark(z0)) → mark(fact(z0)) 286.36/91.64
fact(ok(z0)) → ok(fact(z0)) 286.36/91.64
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.64
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.64
zero(mark(z0)) → mark(zero(z0)) 286.36/91.64
zero(ok(z0)) → ok(zero(z0)) 286.36/91.64
s(mark(z0)) → mark(s(z0)) 286.36/91.64
s(ok(z0)) → ok(s(z0)) 286.36/91.64
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.64
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.64
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.64
p(mark(z0)) → mark(p(z0)) 286.36/91.64
p(ok(z0)) → ok(p(z0)) 286.36/91.64
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.64
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.64
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.64
proper(fact(z0)) → fact(proper(z0)) 286.36/91.64
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.64
proper(zero(z0)) → zero(proper(z0)) 286.36/91.64
proper(s(z0)) → s(proper(z0)) 286.36/91.64
proper(0) → ok(0) 286.36/91.64
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.64
proper(p(z0)) → p(proper(z0)) 286.36/91.64
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.64
proper(true) → ok(true) 286.36/91.64
proper(false) → ok(false) 286.36/91.64
top(mark(z0)) → top(proper(z0)) 286.36/91.64
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.64
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.64
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.64
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.64
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.64
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.64
S(mark(z0)) → c25(S(z0)) 286.36/91.64
S(ok(z0)) → c26(S(z0)) 286.36/91.64
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.64
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.64
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.64
P(mark(z0)) → c30(P(z0)) 286.36/91.64
P(ok(z0)) → c31(P(z0)) 286.36/91.64
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.64
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.64
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.64
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.64
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.64
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.64
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.64
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.64
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.64
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.64
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.64
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.64
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.64
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.64
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.64
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.64
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.64
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.64
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.64
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.64
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.64
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.64
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.64
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.64
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.64
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.64
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.64
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.64
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.64
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.64
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.64
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.64
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.64
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.64
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.64
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.64
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.64
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.64
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.64
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.64
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.64
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.64
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.64
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.64
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.64
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.64
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.64
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.64
S(mark(z0)) → c25(S(z0)) 286.36/91.64
S(ok(z0)) → c26(S(z0)) 286.36/91.64
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.64
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.64
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.64
P(mark(z0)) → c30(P(z0)) 286.36/91.64
P(ok(z0)) → c31(P(z0)) 286.36/91.64
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.64
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.64
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.64
PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) 286.36/91.64
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.64
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.64
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.64
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.64
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.64
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.64
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.64
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.64
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.64
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.64
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.64
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.64
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.64
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.64
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.64
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.64
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.64
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.64
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.64
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.64
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.64
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.64
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.64
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.64
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.64
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.64
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.64
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.64
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.64
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.64
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.64
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.64
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.64
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.64
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.64
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.64
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.64
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.64
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c41, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40

286.36/91.64

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(p(z0)) → c41(P(proper(z0)), PROPER(z0)) by

PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(p(0)) → c41(P(ok(0)), PROPER(0)) 286.36/91.64
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(p(true)) → c41(P(ok(true)), PROPER(true)) 286.36/91.64
PROPER(p(false)) → c41(P(ok(false)), PROPER(false))

286.36/91.64

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.64
active(add(0, z0)) → mark(z0) 286.36/91.64
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.64
active(prod(0, z0)) → mark(0) 286.36/91.64
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.64
active(if(true, z0, z1)) → mark(z0) 286.36/91.64
active(if(false, z0, z1)) → mark(z1) 286.36/91.64
active(zero(0)) → mark(true) 286.36/91.64
active(zero(s(z0))) → mark(false) 286.36/91.64
active(p(s(z0))) → mark(z0) 286.36/91.64
active(fact(z0)) → fact(active(z0)) 286.36/91.64
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.64
active(zero(z0)) → zero(active(z0)) 286.36/91.64
active(s(z0)) → s(active(z0)) 286.36/91.64
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.64
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.64
active(p(z0)) → p(active(z0)) 286.36/91.64
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.64
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.64
fact(mark(z0)) → mark(fact(z0)) 286.36/91.64
fact(ok(z0)) → ok(fact(z0)) 286.36/91.64
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.64
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.64
zero(mark(z0)) → mark(zero(z0)) 286.36/91.64
zero(ok(z0)) → ok(zero(z0)) 286.36/91.64
s(mark(z0)) → mark(s(z0)) 286.36/91.64
s(ok(z0)) → ok(s(z0)) 286.36/91.64
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.64
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.64
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.64
p(mark(z0)) → mark(p(z0)) 286.36/91.64
p(ok(z0)) → ok(p(z0)) 286.36/91.64
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.64
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.64
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.64
proper(fact(z0)) → fact(proper(z0)) 286.36/91.64
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.64
proper(zero(z0)) → zero(proper(z0)) 286.36/91.64
proper(s(z0)) → s(proper(z0)) 286.36/91.64
proper(0) → ok(0) 286.36/91.64
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.64
proper(p(z0)) → p(proper(z0)) 286.36/91.64
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.64
proper(true) → ok(true) 286.36/91.64
proper(false) → ok(false) 286.36/91.64
top(mark(z0)) → top(proper(z0)) 286.36/91.64
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.64
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.64
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.64
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.64
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.64
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.64
S(mark(z0)) → c25(S(z0)) 286.36/91.64
S(ok(z0)) → c26(S(z0)) 286.36/91.64
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.64
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.64
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.64
P(mark(z0)) → c30(P(z0)) 286.36/91.64
P(ok(z0)) → c31(P(z0)) 286.36/91.64
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.64
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.64
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.64
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.64
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.64
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.64
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.64
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.64
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.64
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.64
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.64
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.64
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.64
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.64
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.64
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.64
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.64
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.64
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.64
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.64
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.64
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.64
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.64
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.64
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.64
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.64
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.64
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.64
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.64
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.64
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.64
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.64
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.64
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.64
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.64
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.64
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.64
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.64
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.64
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.64
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.64
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(p(0)) → c41(P(ok(0)), PROPER(0)) 286.36/91.64
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(p(true)) → c41(P(ok(true)), PROPER(true)) 286.36/91.64
PROPER(p(false)) → c41(P(ok(false)), PROPER(false))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.64
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.64
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.64
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.64
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.64
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.64
S(mark(z0)) → c25(S(z0)) 286.36/91.64
S(ok(z0)) → c26(S(z0)) 286.36/91.64
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.64
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.64
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.64
P(mark(z0)) → c30(P(z0)) 286.36/91.64
P(ok(z0)) → c31(P(z0)) 286.36/91.64
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.64
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.64
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.64
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.64
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.64
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.64
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.64
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.64
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.64
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.64
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.64
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.64
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.64
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.64
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.64
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.64
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.64
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.64
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.64
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.64
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.64
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.64
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.64
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.64
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.64
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.64
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.64
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.64
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.64
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.64
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.64
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.64
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.64
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.64
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.64
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.64
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.64
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.64
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.64
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.64
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.64
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.64
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.64
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.64
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.64
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.64
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.64
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.64
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.64
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.64
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.64
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.64
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.64
PROPER(p(0)) → c41(P(ok(0)), PROPER(0)) 286.36/91.64
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.64
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.64
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.64
PROPER(p(true)) → c41(P(ok(true)), PROPER(true)) 286.36/91.64
PROPER(p(false)) → c41(P(ok(false)), PROPER(false))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41

286.36/91.64

(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.64

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.64
active(add(0, z0)) → mark(z0) 286.36/91.64
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.64
active(prod(0, z0)) → mark(0) 286.36/91.64
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.64
active(if(true, z0, z1)) → mark(z0) 286.36/91.64
active(if(false, z0, z1)) → mark(z1) 286.36/91.64
active(zero(0)) → mark(true) 286.36/91.64
active(zero(s(z0))) → mark(false) 286.36/91.64
active(p(s(z0))) → mark(z0) 286.36/91.64
active(fact(z0)) → fact(active(z0)) 286.36/91.64
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.64
active(zero(z0)) → zero(active(z0)) 286.36/91.64
active(s(z0)) → s(active(z0)) 286.36/91.64
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.64
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.64
active(p(z0)) → p(active(z0)) 286.36/91.64
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.64
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.64
fact(mark(z0)) → mark(fact(z0)) 286.36/91.64
fact(ok(z0)) → ok(fact(z0)) 286.36/91.64
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.64
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.64
zero(mark(z0)) → mark(zero(z0)) 286.36/91.64
zero(ok(z0)) → ok(zero(z0)) 286.36/91.64
s(mark(z0)) → mark(s(z0)) 286.36/91.64
s(ok(z0)) → ok(s(z0)) 286.36/91.64
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.64
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.64
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.64
p(mark(z0)) → mark(p(z0)) 286.36/91.64
p(ok(z0)) → ok(p(z0)) 286.36/91.64
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.64
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.64
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.64
proper(fact(z0)) → fact(proper(z0)) 286.36/91.64
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.64
proper(zero(z0)) → zero(proper(z0)) 286.36/91.64
proper(s(z0)) → s(proper(z0)) 286.36/91.64
proper(0) → ok(0) 286.36/91.64
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.64
proper(p(z0)) → p(proper(z0)) 286.36/91.64
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.64
proper(true) → ok(true) 286.36/91.64
proper(false) → ok(false) 286.36/91.64
top(mark(z0)) → top(proper(z0)) 286.36/91.64
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.64
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.64
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.64
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.64
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.64
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.64
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.64
S(mark(z0)) → c25(S(z0)) 286.36/91.64
S(ok(z0)) → c26(S(z0)) 286.36/91.64
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.64
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.64
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.64
P(mark(z0)) → c30(P(z0)) 286.36/91.64
P(ok(z0)) → c31(P(z0)) 286.36/91.64
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.64
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.64
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.64
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.64
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.64
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.64
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.64
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.64
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.64
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.64
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.64
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.64
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.64
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.64
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.64
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.64
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.64
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.64
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.64
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.64
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.64
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.64
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.64
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.64
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.64
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.64
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.64
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.64
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.64
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.64
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.64
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.64
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.64
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.65
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.65
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.65
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.65
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.65
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.65
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.65
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.65
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.65
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.65
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.65
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.65
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.65
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.65
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.65
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.65
PROPER(p(false)) → c41(P(ok(false)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.65
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.65
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.65
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.65
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.65
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.65
S(mark(z0)) → c25(S(z0)) 286.36/91.65
S(ok(z0)) → c26(S(z0)) 286.36/91.65
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.65
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.65
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.65
P(mark(z0)) → c30(P(z0)) 286.36/91.65
P(ok(z0)) → c31(P(z0)) 286.36/91.65
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.65
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.65
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.65
PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 286.36/91.65
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.65
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.65
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.65
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.65
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.65
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.65
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.65
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.65
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.65
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.65
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.65
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.65
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.65
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.65
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.65
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.65
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.65
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.65
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.65
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.65
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.65
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.65
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.65
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.65
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.65
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.65
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.65
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.65
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.65
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.65
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.65
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.65
PROPER(p(false)) → c41(P(ok(false)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c42, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41, c41

286.36/91.65

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(add(z0, z1)) → c42(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 286.36/91.65
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 286.36/91.65
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 286.36/91.65
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 286.36/91.65
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 286.36/91.65
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(false), PROPER(x1))

286.36/91.65

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.65
active(add(0, z0)) → mark(z0) 286.36/91.65
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.65
active(prod(0, z0)) → mark(0) 286.36/91.65
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.65
active(if(true, z0, z1)) → mark(z0) 286.36/91.65
active(if(false, z0, z1)) → mark(z1) 286.36/91.65
active(zero(0)) → mark(true) 286.36/91.65
active(zero(s(z0))) → mark(false) 286.36/91.65
active(p(s(z0))) → mark(z0) 286.36/91.65
active(fact(z0)) → fact(active(z0)) 286.36/91.65
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.65
active(zero(z0)) → zero(active(z0)) 286.36/91.65
active(s(z0)) → s(active(z0)) 286.36/91.65
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.65
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.65
active(p(z0)) → p(active(z0)) 286.36/91.65
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.65
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.65
fact(mark(z0)) → mark(fact(z0)) 286.36/91.65
fact(ok(z0)) → ok(fact(z0)) 286.36/91.65
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.65
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.65
zero(mark(z0)) → mark(zero(z0)) 286.36/91.65
zero(ok(z0)) → ok(zero(z0)) 286.36/91.65
s(mark(z0)) → mark(s(z0)) 286.36/91.65
s(ok(z0)) → ok(s(z0)) 286.36/91.65
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.65
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.65
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.65
p(mark(z0)) → mark(p(z0)) 286.36/91.65
p(ok(z0)) → ok(p(z0)) 286.36/91.65
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.65
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.65
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.65
proper(fact(z0)) → fact(proper(z0)) 286.36/91.65
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.65
proper(zero(z0)) → zero(proper(z0)) 286.36/91.65
proper(s(z0)) → s(proper(z0)) 286.36/91.65
proper(0) → ok(0) 286.36/91.65
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.65
proper(p(z0)) → p(proper(z0)) 286.36/91.65
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.65
proper(true) → ok(true) 286.36/91.65
proper(false) → ok(false) 286.36/91.65
top(mark(z0)) → top(proper(z0)) 286.36/91.65
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.65
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.65
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.65
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.65
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.65
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.65
S(mark(z0)) → c25(S(z0)) 286.36/91.65
S(ok(z0)) → c26(S(z0)) 286.36/91.65
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.65
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.65
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.65
P(mark(z0)) → c30(P(z0)) 286.36/91.65
P(ok(z0)) → c31(P(z0)) 286.36/91.65
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.65
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.65
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.65
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.65
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.65
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.65
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.65
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.65
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.65
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.65
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.65
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.65
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.65
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.65
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.65
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.65
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.65
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.65
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.65
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.65
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.65
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.65
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.65
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.65
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.65
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.65
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.65
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.65
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.65
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.65
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.65
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.65
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.65
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.65
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.65
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.65
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 286.36/91.65
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 286.36/91.65
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 286.36/91.65
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 286.36/91.65
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 286.36/91.65
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(false), PROPER(x1))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.65
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.65
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.65
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.65
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.65
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.65
S(mark(z0)) → c25(S(z0)) 286.36/91.65
S(ok(z0)) → c26(S(z0)) 286.36/91.65
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.65
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.65
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.65
P(mark(z0)) → c30(P(z0)) 286.36/91.65
P(ok(z0)) → c31(P(z0)) 286.36/91.65
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.65
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.65
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.65
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.65
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.65
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.65
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.65
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.65
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.65
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.65
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.65
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.65
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.65
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.65
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.65
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.65
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.65
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.65
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.65
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.65
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.65
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.65
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.65
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.65
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.65
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.65
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.65
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.65
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.65
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.65
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.65
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.65
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.65
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.65
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.65
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.65
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 286.36/91.65
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 286.36/91.65
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 286.36/91.65
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 286.36/91.65
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 286.36/91.65
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(false), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, TOP, PROPER

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41, c41, c42

286.36/91.65

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

286.36/91.65

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.65
active(add(0, z0)) → mark(z0) 286.36/91.65
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.65
active(prod(0, z0)) → mark(0) 286.36/91.65
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.65
active(if(true, z0, z1)) → mark(z0) 286.36/91.65
active(if(false, z0, z1)) → mark(z1) 286.36/91.65
active(zero(0)) → mark(true) 286.36/91.65
active(zero(s(z0))) → mark(false) 286.36/91.65
active(p(s(z0))) → mark(z0) 286.36/91.65
active(fact(z0)) → fact(active(z0)) 286.36/91.65
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.65
active(zero(z0)) → zero(active(z0)) 286.36/91.65
active(s(z0)) → s(active(z0)) 286.36/91.65
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.65
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.65
active(p(z0)) → p(active(z0)) 286.36/91.65
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.65
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.65
fact(mark(z0)) → mark(fact(z0)) 286.36/91.65
fact(ok(z0)) → ok(fact(z0)) 286.36/91.65
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.65
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.65
zero(mark(z0)) → mark(zero(z0)) 286.36/91.65
zero(ok(z0)) → ok(zero(z0)) 286.36/91.65
s(mark(z0)) → mark(s(z0)) 286.36/91.65
s(ok(z0)) → ok(s(z0)) 286.36/91.65
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.65
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.65
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.65
p(mark(z0)) → mark(p(z0)) 286.36/91.65
p(ok(z0)) → ok(p(z0)) 286.36/91.65
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.65
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.65
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.65
proper(fact(z0)) → fact(proper(z0)) 286.36/91.65
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.65
proper(zero(z0)) → zero(proper(z0)) 286.36/91.65
proper(s(z0)) → s(proper(z0)) 286.36/91.65
proper(0) → ok(0) 286.36/91.65
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.65
proper(p(z0)) → p(proper(z0)) 286.36/91.65
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.65
proper(true) → ok(true) 286.36/91.65
proper(false) → ok(false) 286.36/91.65
top(mark(z0)) → top(proper(z0)) 286.36/91.65
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.65
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.65
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.65
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.65
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.65
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.65
S(mark(z0)) → c25(S(z0)) 286.36/91.65
S(ok(z0)) → c26(S(z0)) 286.36/91.65
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.65
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.65
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.65
P(mark(z0)) → c30(P(z0)) 286.36/91.65
P(ok(z0)) → c31(P(z0)) 286.36/91.65
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.65
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.65
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.65
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.65
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.65
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.65
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.65
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.65
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.65
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.65
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.65
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.65
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.65
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.65
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.65
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.65
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.65
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.65
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.65
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.65
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.65
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.65
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.65
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.65
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.65
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.65
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.65
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.65
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.65
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.65
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.65
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.65
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.65
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.65
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.65
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.65
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.65
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.65
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.65
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.65
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.65
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.65
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.65
S(mark(z0)) → c25(S(z0)) 286.36/91.65
S(ok(z0)) → c26(S(z0)) 286.36/91.65
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.65
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.65
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.65
P(mark(z0)) → c30(P(z0)) 286.36/91.65
P(ok(z0)) → c31(P(z0)) 286.36/91.65
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.65
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.65
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.65
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) 286.36/91.65
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.65
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.65
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.65
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.65
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.65
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.65
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.65
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.65
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.65
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.65
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.65
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.65
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.65
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.65
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.65
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.65
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.65
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.65
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.65
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.65
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.65
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.65
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.65
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.65
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.65
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.65
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.65
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.65
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.65
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.65
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.65
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.65
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.65
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.65
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.65
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.65
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.65
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.65
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.65
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.65
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.65
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.65
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.65
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.65
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.65
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.65
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.65
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.65
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.65
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.65
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.65
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.65
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.65
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.65
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.65
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.65
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.65
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.65
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.65
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.65
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.65
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.65
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.65
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, TOP, PROPER

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c45, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41, c41, c42, c42

286.36/91.65

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.65
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.65
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.65
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.65
TOP(mark(0)) → c45(TOP(ok(0)), PROPER(0)) 286.36/91.65
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.65
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.65
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.65
TOP(mark(true)) → c45(TOP(ok(true)), PROPER(true)) 286.36/91.65
TOP(mark(false)) → c45(TOP(ok(false)), PROPER(false))

286.36/91.65

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.65
active(add(0, z0)) → mark(z0) 286.36/91.65
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.65
active(prod(0, z0)) → mark(0) 286.36/91.65
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.65
active(if(true, z0, z1)) → mark(z0) 286.36/91.65
active(if(false, z0, z1)) → mark(z1) 286.36/91.65
active(zero(0)) → mark(true) 286.36/91.65
active(zero(s(z0))) → mark(false) 286.36/91.65
active(p(s(z0))) → mark(z0) 286.36/91.65
active(fact(z0)) → fact(active(z0)) 286.36/91.65
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.65
active(zero(z0)) → zero(active(z0)) 286.36/91.65
active(s(z0)) → s(active(z0)) 286.36/91.65
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.65
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.65
active(p(z0)) → p(active(z0)) 286.36/91.65
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.65
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.65
fact(mark(z0)) → mark(fact(z0)) 286.36/91.65
fact(ok(z0)) → ok(fact(z0)) 286.36/91.65
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.65
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.65
zero(mark(z0)) → mark(zero(z0)) 286.36/91.65
zero(ok(z0)) → ok(zero(z0)) 286.36/91.65
s(mark(z0)) → mark(s(z0)) 286.36/91.65
s(ok(z0)) → ok(s(z0)) 286.36/91.65
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.65
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.65
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.65
p(mark(z0)) → mark(p(z0)) 286.36/91.65
p(ok(z0)) → ok(p(z0)) 286.36/91.65
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.65
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.65
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.65
proper(fact(z0)) → fact(proper(z0)) 286.36/91.65
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.65
proper(zero(z0)) → zero(proper(z0)) 286.36/91.65
proper(s(z0)) → s(proper(z0)) 286.36/91.65
proper(0) → ok(0) 286.36/91.65
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.65
proper(p(z0)) → p(proper(z0)) 286.36/91.65
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.65
proper(true) → ok(true) 286.36/91.65
proper(false) → ok(false) 286.36/91.65
top(mark(z0)) → top(proper(z0)) 286.36/91.65
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.66
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.66
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.66
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.66
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.66
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.66
S(mark(z0)) → c25(S(z0)) 286.36/91.66
S(ok(z0)) → c26(S(z0)) 286.36/91.66
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.66
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.66
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.66
P(mark(z0)) → c30(P(z0)) 286.36/91.66
P(ok(z0)) → c31(P(z0)) 286.36/91.66
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.66
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.66
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.66
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.66
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.66
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.66
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.66
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.66
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.66
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.66
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.66
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.66
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.66
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.66
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.66
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.66
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.66
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.66
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.66
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.66
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.66
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.66
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.66
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.66
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.66
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.66
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.66
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.66
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.66
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.66
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.66
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.66
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.66
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.66
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.66
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
TOP(mark(0)) → c45(TOP(ok(0)), PROPER(0)) 286.36/91.66
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
TOP(mark(true)) → c45(TOP(ok(true)), PROPER(true)) 286.36/91.66
TOP(mark(false)) → c45(TOP(ok(false)), PROPER(false))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.66
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.66
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.66
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.66
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.66
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.66
S(mark(z0)) → c25(S(z0)) 286.36/91.66
S(ok(z0)) → c26(S(z0)) 286.36/91.66
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.66
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.66
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.66
P(mark(z0)) → c30(P(z0)) 286.36/91.66
P(ok(z0)) → c31(P(z0)) 286.36/91.66
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.66
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.66
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.66
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.66
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.66
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.66
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.66
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.66
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.66
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.66
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.66
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.66
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.66
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.66
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.66
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.66
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.66
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.66
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.66
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.66
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.66
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.66
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.66
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.66
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.66
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.66
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.66
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.66
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.66
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.66
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.66
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.66
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.66
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.66
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.66
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
TOP(mark(0)) → c45(TOP(ok(0)), PROPER(0)) 286.36/91.66
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
TOP(mark(true)) → c45(TOP(ok(true)), PROPER(true)) 286.36/91.66
TOP(mark(false)) → c45(TOP(ok(false)), PROPER(false))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, TOP, PROPER

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41, c41, c42, c42, c45

286.36/91.66

(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.66

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.66
active(add(0, z0)) → mark(z0) 286.36/91.66
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.66
active(prod(0, z0)) → mark(0) 286.36/91.66
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.66
active(if(true, z0, z1)) → mark(z0) 286.36/91.66
active(if(false, z0, z1)) → mark(z1) 286.36/91.66
active(zero(0)) → mark(true) 286.36/91.66
active(zero(s(z0))) → mark(false) 286.36/91.66
active(p(s(z0))) → mark(z0) 286.36/91.66
active(fact(z0)) → fact(active(z0)) 286.36/91.66
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.66
active(zero(z0)) → zero(active(z0)) 286.36/91.66
active(s(z0)) → s(active(z0)) 286.36/91.66
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.66
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.66
active(p(z0)) → p(active(z0)) 286.36/91.66
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.66
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.66
fact(mark(z0)) → mark(fact(z0)) 286.36/91.66
fact(ok(z0)) → ok(fact(z0)) 286.36/91.66
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.66
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.66
zero(mark(z0)) → mark(zero(z0)) 286.36/91.66
zero(ok(z0)) → ok(zero(z0)) 286.36/91.66
s(mark(z0)) → mark(s(z0)) 286.36/91.66
s(ok(z0)) → ok(s(z0)) 286.36/91.66
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.66
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.66
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.66
p(mark(z0)) → mark(p(z0)) 286.36/91.66
p(ok(z0)) → ok(p(z0)) 286.36/91.66
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.66
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.66
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.66
proper(fact(z0)) → fact(proper(z0)) 286.36/91.66
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.66
proper(zero(z0)) → zero(proper(z0)) 286.36/91.66
proper(s(z0)) → s(proper(z0)) 286.36/91.66
proper(0) → ok(0) 286.36/91.66
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.66
proper(p(z0)) → p(proper(z0)) 286.36/91.66
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.66
proper(true) → ok(true) 286.36/91.66
proper(false) → ok(false) 286.36/91.66
top(mark(z0)) → top(proper(z0)) 286.36/91.66
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.66
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.66
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.66
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.66
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.66
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.66
S(mark(z0)) → c25(S(z0)) 286.36/91.66
S(ok(z0)) → c26(S(z0)) 286.36/91.66
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.66
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.66
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.66
P(mark(z0)) → c30(P(z0)) 286.36/91.66
P(ok(z0)) → c31(P(z0)) 286.36/91.66
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.66
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.66
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.66
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.66
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.66
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.66
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.66
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.66
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.66
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.66
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.66
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.66
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.66
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.66
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.66
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.66
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.66
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.66
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.66
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.66
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.66
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.66
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.66
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.66
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.66
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.66
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.66
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.66
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.66
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.66
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.66
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.66
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.66
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.66
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.66
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
TOP(mark(0)) → c45(TOP(ok(0))) 286.36/91.66
TOP(mark(true)) → c45(TOP(ok(true))) 286.36/91.66
TOP(mark(false)) → c45(TOP(ok(false)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.66
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.66
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.66
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.66
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.66
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.66
S(mark(z0)) → c25(S(z0)) 286.36/91.66
S(ok(z0)) → c26(S(z0)) 286.36/91.66
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.66
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.66
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.66
P(mark(z0)) → c30(P(z0)) 286.36/91.66
P(ok(z0)) → c31(P(z0)) 286.36/91.66
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.66
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.66
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.66
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.66
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.66
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.66
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.66
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.66
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.66
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.66
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.66
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.66
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.66
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.66
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.66
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.66
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.66
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.66
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.66
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.66
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.66
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.66
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.66
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.66
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.66
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.66
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.66
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.66
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.66
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.66
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.66
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.66
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.66
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.66
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.66
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
TOP(mark(0)) → c45(TOP(ok(0))) 286.36/91.66
TOP(mark(true)) → c45(TOP(ok(true))) 286.36/91.66
TOP(mark(false)) → c45(TOP(ok(false)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, TOP, PROPER

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c46, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41, c41, c42, c42, c45, c45

286.36/91.66

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(fact(z0))) → c46(TOP(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
TOP(ok(prod(0, z0))) → c46(TOP(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
TOP(ok(prod(s(z0), z1))) → c46(TOP(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
TOP(ok(if(true, z0, z1))) → c46(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 286.36/91.66
TOP(ok(if(false, z0, z1))) → c46(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 286.36/91.66
TOP(ok(zero(0))) → c46(TOP(mark(true)), ACTIVE(zero(0))) 286.36/91.66
TOP(ok(zero(s(z0)))) → c46(TOP(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
TOP(ok(p(s(z0)))) → c46(TOP(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
TOP(ok(fact(z0))) → c46(TOP(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
TOP(ok(if(z0, z1, z2))) → c46(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
TOP(ok(zero(z0))) → c46(TOP(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
TOP(ok(s(z0))) → c46(TOP(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
TOP(ok(prod(z0, z1))) → c46(TOP(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
TOP(ok(prod(z0, z1))) → c46(TOP(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
TOP(ok(p(z0))) → c46(TOP(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))

286.36/91.66

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.66
active(add(0, z0)) → mark(z0) 286.36/91.66
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.66
active(prod(0, z0)) → mark(0) 286.36/91.66
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.66
active(if(true, z0, z1)) → mark(z0) 286.36/91.66
active(if(false, z0, z1)) → mark(z1) 286.36/91.66
active(zero(0)) → mark(true) 286.36/91.66
active(zero(s(z0))) → mark(false) 286.36/91.66
active(p(s(z0))) → mark(z0) 286.36/91.66
active(fact(z0)) → fact(active(z0)) 286.36/91.66
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.66
active(zero(z0)) → zero(active(z0)) 286.36/91.66
active(s(z0)) → s(active(z0)) 286.36/91.66
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.66
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.66
active(p(z0)) → p(active(z0)) 286.36/91.66
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.66
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.66
fact(mark(z0)) → mark(fact(z0)) 286.36/91.66
fact(ok(z0)) → ok(fact(z0)) 286.36/91.66
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.66
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.66
zero(mark(z0)) → mark(zero(z0)) 286.36/91.66
zero(ok(z0)) → ok(zero(z0)) 286.36/91.66
s(mark(z0)) → mark(s(z0)) 286.36/91.66
s(ok(z0)) → ok(s(z0)) 286.36/91.66
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.66
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.66
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.66
p(mark(z0)) → mark(p(z0)) 286.36/91.66
p(ok(z0)) → ok(p(z0)) 286.36/91.66
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.66
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.66
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.66
proper(fact(z0)) → fact(proper(z0)) 286.36/91.66
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.66
proper(zero(z0)) → zero(proper(z0)) 286.36/91.66
proper(s(z0)) → s(proper(z0)) 286.36/91.66
proper(0) → ok(0) 286.36/91.66
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.66
proper(p(z0)) → p(proper(z0)) 286.36/91.66
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.66
proper(true) → ok(true) 286.36/91.66
proper(false) → ok(false) 286.36/91.66
top(mark(z0)) → top(proper(z0)) 286.36/91.66
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.66
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.66
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.66
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.66
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.66
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.66
S(mark(z0)) → c25(S(z0)) 286.36/91.66
S(ok(z0)) → c26(S(z0)) 286.36/91.66
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.66
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.66
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.66
P(mark(z0)) → c30(P(z0)) 286.36/91.66
P(ok(z0)) → c31(P(z0)) 286.36/91.66
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.66
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.66
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.66
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.66
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.66
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.66
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.66
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.66
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.66
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.66
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.66
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.66
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.66
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.66
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.66
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.66
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.66
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.66
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.66
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.66
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.66
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.66
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.66
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.66
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.66
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.66
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.66
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.66
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.66
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.66
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.66
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.66
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.66
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.66
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.66
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.66
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.66
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.66
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.66
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.66
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.66
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.66
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.66
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.66
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.66
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.66
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.66
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.66
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.66
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.66
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.66
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.66
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.66
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.66
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.66
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.66
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.66
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.66
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.66
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.66
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.66
TOP(mark(0)) → c45(TOP(ok(0))) 286.36/91.66
TOP(mark(true)) → c45(TOP(ok(true))) 286.36/91.66
TOP(mark(false)) → c45(TOP(ok(false))) 286.36/91.66
TOP(ok(fact(z0))) → c46(TOP(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
TOP(ok(prod(0, z0))) → c46(TOP(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
TOP(ok(prod(s(z0), z1))) → c46(TOP(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
TOP(ok(if(true, z0, z1))) → c46(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 286.36/91.66
TOP(ok(if(false, z0, z1))) → c46(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 286.36/91.66
TOP(ok(zero(0))) → c46(TOP(mark(true)), ACTIVE(zero(0))) 286.36/91.66
TOP(ok(zero(s(z0)))) → c46(TOP(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
TOP(ok(p(s(z0)))) → c46(TOP(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
TOP(ok(fact(z0))) → c46(TOP(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
TOP(ok(if(z0, z1, z2))) → c46(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
TOP(ok(zero(z0))) → c46(TOP(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
TOP(ok(s(z0))) → c46(TOP(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
TOP(ok(prod(z0, z1))) → c46(TOP(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
TOP(ok(prod(z0, z1))) → c46(TOP(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
TOP(ok(p(z0))) → c46(TOP(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))

S tuples:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.66
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.66
FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.66
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.66
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.66
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.66
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.66
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.66
S(mark(z0)) → c25(S(z0)) 286.36/91.66
S(ok(z0)) → c26(S(z0)) 286.36/91.66
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.66
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.66
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.66
P(mark(z0)) → c30(P(z0)) 286.36/91.66
P(ok(z0)) → c31(P(z0)) 286.36/91.66
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.66
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.66
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.66
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.66
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.66
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.66
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.66
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.66
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.66
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.66
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.66
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.66
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.66
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.66
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.66
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.66
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.66
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.66
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.66
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.66
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.67
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.67
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.67
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.67
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.67
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.67
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.67
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.67
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.67
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.67
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.67
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.67
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.67
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.67
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.67
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.67
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.67
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.67
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.67
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.67
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.67
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.67
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.67
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.67
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.67
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.67
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.67
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.67
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.67
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.67
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.67
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.67
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.67
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.67
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.67
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.67
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.67
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.67
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.67
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.67
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.67
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.67
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.67
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.67
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.67
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.67
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.67
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.67
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.67
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.67
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.67
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.67
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.67
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.67
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
TOP(mark(0)) → c45(TOP(ok(0))) 286.36/91.67
TOP(mark(true)) → c45(TOP(ok(true))) 286.36/91.67
TOP(mark(false)) → c45(TOP(ok(false))) 286.36/91.67
TOP(ok(fact(z0))) → c46(TOP(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
TOP(ok(prod(0, z0))) → c46(TOP(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
TOP(ok(prod(s(z0), z1))) → c46(TOP(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
TOP(ok(if(true, z0, z1))) → c46(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 286.36/91.67
TOP(ok(if(false, z0, z1))) → c46(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 286.36/91.67
TOP(ok(zero(0))) → c46(TOP(mark(true)), ACTIVE(zero(0))) 286.36/91.67
TOP(ok(zero(s(z0)))) → c46(TOP(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
TOP(ok(p(s(z0)))) → c46(TOP(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
TOP(ok(fact(z0))) → c46(TOP(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
TOP(ok(if(z0, z1, z2))) → c46(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
TOP(ok(zero(z0))) → c46(TOP(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
TOP(ok(s(z0))) → c46(TOP(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
TOP(ok(prod(z0, z1))) → c46(TOP(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
TOP(ok(prod(z0, z1))) → c46(TOP(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
TOP(ok(p(z0))) → c46(TOP(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

ACTIVE, FACT, IF, ZERO, S, PROD, P, ADD, PROPER, TOP

Compound Symbols:

c14, c15, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c2, c4, c1, c10, c11, c10, c11, c12, c12, c13, c13, c16, c16, c35, c35, c36, c36, c37, c37, c38, c38, c40, c40, c41, c41, c42, c42, c45, c45, c46

286.36/91.67

(59) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(prod(z0, z1)) → c14(PROD(active(z0), z1), ACTIVE(z0)) 286.36/91.67
ACTIVE(prod(z0, z1)) → c15(PROD(z0, active(z1)), ACTIVE(z1)) 286.36/91.67
ACTIVE(add(z0, z1)) → c17(ADD(active(z0), z1), ACTIVE(z0)) 286.36/91.67
ACTIVE(add(z0, z1)) → c18(ADD(z0, active(z1)), ACTIVE(z1)) 286.36/91.67
ACTIVE(add(s(z0), z1)) → c2(ADD(z0, z1)) 286.36/91.67
ACTIVE(prod(s(z0), z1)) → c4(PROD(z0, z1)) 286.36/91.67
ACTIVE(fact(z0)) → c1(ZERO(z0)) 286.36/91.67
ACTIVE(fact(z0)) → c1(P(z0)) 286.36/91.67
ACTIVE(fact(fact(z0))) → c10(FACT(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(fact(add(0, z0))) → c10(FACT(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(fact(add(s(z0), z1))) → c10(FACT(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(fact(prod(0, z0))) → c10(FACT(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(fact(prod(s(z0), z1))) → c10(FACT(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(fact(zero(s(z0)))) → c10(FACT(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(fact(p(s(z0)))) → c10(FACT(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(fact(fact(z0))) → c10(FACT(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(fact(if(z0, z1, z2))) → c10(FACT(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(fact(zero(z0))) → c10(FACT(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(fact(s(z0))) → c10(FACT(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(fact(prod(z0, z1))) → c10(FACT(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(fact(p(z0))) → c10(FACT(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(fact(add(z0, z1))) → c10(FACT(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))), x1, x2), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(if(add(0, z0), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c11(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(if(prod(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(if(prod(s(z0), z1), x1, x2)) → c11(IF(mark(add(z1, prod(z0, z1))), x1, x2), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(if(zero(s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(if(p(s(z0)), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(if(fact(z0), x1, x2)) → c11(IF(fact(active(z0)), x1, x2), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(if(zero(z0), x1, x2)) → c11(IF(zero(active(z0)), x1, x2), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(active(z0), z1), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(if(prod(z0, z1), x1, x2)) → c11(IF(prod(z0, active(z1)), x1, x2), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(if(p(z0), x1, x2)) → c11(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(if(add(z0, z1), x1, x2)) → c11(IF(add(z0, active(z1)), x1, x2), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(fact(if(true, z0, z1))) → c10(FACT(mark(z0))) 286.36/91.67
ACTIVE(fact(if(false, z0, z1))) → c10(FACT(mark(z1))) 286.36/91.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 286.36/91.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 286.36/91.67
ACTIVE(zero(fact(z0))) → c12(ZERO(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(zero(add(0, z0))) → c12(ZERO(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(zero(add(s(z0), z1))) → c12(ZERO(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(zero(prod(0, z0))) → c12(ZERO(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(zero(prod(s(z0), z1))) → c12(ZERO(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(zero(zero(s(z0)))) → c12(ZERO(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(zero(p(s(z0)))) → c12(ZERO(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(zero(fact(z0))) → c12(ZERO(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(zero(if(z0, z1, z2))) → c12(ZERO(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(zero(zero(z0))) → c12(ZERO(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(zero(s(z0))) → c12(ZERO(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(zero(prod(z0, z1))) → c12(ZERO(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(zero(p(z0))) → c12(ZERO(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(zero(add(z0, z1))) → c12(ZERO(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(fact(zero(0))) → c10(FACT(mark(true))) 286.36/91.67
ACTIVE(if(zero(0), x1, x2)) → c11(IF(mark(true), x1, x2)) 286.36/91.67
ACTIVE(zero(if(true, z0, z1))) → c12(ZERO(mark(z0))) 286.36/91.67
ACTIVE(zero(if(false, z0, z1))) → c12(ZERO(mark(z1))) 286.36/91.67
ACTIVE(zero(zero(0))) → c12(ZERO(mark(true))) 286.36/91.67
ACTIVE(s(fact(z0))) → c13(S(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(s(add(0, z0))) → c13(S(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(s(add(s(z0), z1))) → c13(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(s(prod(0, z0))) → c13(S(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(s(prod(s(z0), z1))) → c13(S(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(s(zero(s(z0)))) → c13(S(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(s(p(s(z0)))) → c13(S(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(s(fact(z0))) → c13(S(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(s(if(z0, z1, z2))) → c13(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(s(zero(z0))) → c13(S(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(s(s(z0))) → c13(S(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(s(prod(z0, z1))) → c13(S(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(s(prod(z0, z1))) → c13(S(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(s(p(z0))) → c13(S(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(s(add(z0, z1))) → c13(S(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(s(add(z0, z1))) → c13(S(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(s(if(true, z0, z1))) → c13(S(mark(z0))) 286.36/91.67
ACTIVE(s(if(false, z0, z1))) → c13(S(mark(z1))) 286.36/91.67
ACTIVE(s(zero(0))) → c13(S(mark(true))) 286.36/91.67
ACTIVE(p(fact(z0))) → c16(P(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(p(add(0, z0))) → c16(P(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
ACTIVE(p(add(s(z0), z1))) → c16(P(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
ACTIVE(p(prod(0, z0))) → c16(P(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
ACTIVE(p(prod(s(z0), z1))) → c16(P(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
ACTIVE(p(zero(s(z0)))) → c16(P(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
ACTIVE(p(p(s(z0)))) → c16(P(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
ACTIVE(p(fact(z0))) → c16(P(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
ACTIVE(p(if(z0, z1, z2))) → c16(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
ACTIVE(p(zero(z0))) → c16(P(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
ACTIVE(p(s(z0))) → c16(P(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
ACTIVE(p(prod(z0, z1))) → c16(P(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(p(prod(z0, z1))) → c16(P(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
ACTIVE(p(p(z0))) → c16(P(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
ACTIVE(p(add(z0, z1))) → c16(P(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(p(add(z0, z1))) → c16(P(add(z0, active(z1))), ACTIVE(add(z0, z1))) 286.36/91.67
ACTIVE(p(if(true, z0, z1))) → c16(P(mark(z0))) 286.36/91.67
ACTIVE(p(if(false, z0, z1))) → c16(P(mark(z1))) 286.36/91.67
ACTIVE(p(zero(0))) → c16(P(mark(true))) 286.36/91.67
PROPER(fact(fact(z0))) → c35(FACT(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(fact(if(z0, z1, z2))) → c35(FACT(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(fact(zero(z0))) → c35(FACT(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(fact(s(z0))) → c35(FACT(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(fact(prod(z0, z1))) → c35(FACT(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(fact(p(z0))) → c35(FACT(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(fact(add(z0, z1))) → c35(FACT(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(fact(0)) → c35(FACT(ok(0))) 286.36/91.67
PROPER(fact(true)) → c35(FACT(ok(true))) 286.36/91.67
PROPER(fact(false)) → c35(FACT(ok(false))) 286.36/91.67
PROPER(if(x0, x1, fact(z0))) → c36(IF(proper(x0), proper(x1), fact(proper(z0))), PROPER(x0), PROPER(x1), PROPER(fact(z0))) 286.36/91.67
PROPER(if(x0, x1, if(z0, z1, z2))) → c36(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(if(x0, x1, zero(z0))) → c36(IF(proper(x0), proper(x1), zero(proper(z0))), PROPER(x0), PROPER(x1), PROPER(zero(z0))) 286.36/91.67
PROPER(if(x0, x1, s(z0))) → c36(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 286.36/91.67
PROPER(if(x0, x1, prod(z0, z1))) → c36(IF(proper(x0), proper(x1), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(if(x0, x1, p(z0))) → c36(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 286.36/91.67
PROPER(if(x0, x1, add(z0, z1))) → c36(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 286.36/91.67
PROPER(if(x0, fact(z0), x2)) → c36(IF(proper(x0), fact(proper(z0)), proper(x2)), PROPER(x0), PROPER(fact(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, if(z0, z1, z2), x2)) → c36(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, zero(z0), x2)) → c36(IF(proper(x0), zero(proper(z0)), proper(x2)), PROPER(x0), PROPER(zero(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, s(z0), x2)) → c36(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, prod(z0, z1), x2)) → c36(IF(proper(x0), prod(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(prod(z0, z1)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, p(z0), x2)) → c36(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 286.36/91.67
PROPER(if(x0, add(z0, z1), x2)) → c36(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 286.36/91.67
PROPER(if(fact(z0), x1, x2)) → c36(IF(fact(proper(z0)), proper(x1), proper(x2)), PROPER(fact(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(if(z0, z1, z2), x1, x2)) → c36(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(zero(z0), x1, x2)) → c36(IF(zero(proper(z0)), proper(x1), proper(x2)), PROPER(zero(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(s(z0), x1, x2)) → c36(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(prod(z0, z1), x1, x2)) → c36(IF(prod(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(prod(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(p(z0), x1, x2)) → c36(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(add(z0, z1), x1, x2)) → c36(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(x0, x1, 0)) → c36(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 286.36/91.67
PROPER(if(x0, x1, true)) → c36(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 286.36/91.67
PROPER(if(x0, x1, false)) → c36(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 286.36/91.67
PROPER(if(x0, 0, x2)) → c36(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.67
PROPER(if(x0, true, x2)) → c36(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.67
PROPER(if(x0, false, x2)) → c36(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 286.36/91.67
PROPER(if(0, x1, x2)) → c36(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(true, x1, x2)) → c36(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(if(false, x1, x2)) → c36(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 286.36/91.67
PROPER(zero(fact(z0))) → c37(ZERO(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(zero(if(z0, z1, z2))) → c37(ZERO(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(zero(zero(z0))) → c37(ZERO(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(zero(s(z0))) → c37(ZERO(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(zero(prod(z0, z1))) → c37(ZERO(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(zero(p(z0))) → c37(ZERO(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(zero(add(z0, z1))) → c37(ZERO(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(zero(0)) → c37(ZERO(ok(0))) 286.36/91.67
PROPER(zero(true)) → c37(ZERO(ok(true))) 286.36/91.67
PROPER(zero(false)) → c37(ZERO(ok(false))) 286.36/91.67
PROPER(s(fact(z0))) → c38(S(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(s(if(z0, z1, z2))) → c38(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(s(zero(z0))) → c38(S(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(s(s(z0))) → c38(S(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(s(prod(z0, z1))) → c38(S(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(s(p(z0))) → c38(S(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(s(add(z0, z1))) → c38(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(s(0)) → c38(S(ok(0))) 286.36/91.67
PROPER(s(true)) → c38(S(ok(true))) 286.36/91.67
PROPER(s(false)) → c38(S(ok(false))) 286.36/91.67
PROPER(prod(x0, fact(z0))) → c40(PROD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.67
PROPER(prod(x0, if(z0, z1, z2))) → c40(PROD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(prod(x0, zero(z0))) → c40(PROD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.67
PROPER(prod(x0, s(z0))) → c40(PROD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.67
PROPER(prod(x0, prod(z0, z1))) → c40(PROD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(prod(x0, p(z0))) → c40(PROD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.67
PROPER(prod(x0, add(z0, z1))) → c40(PROD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.67
PROPER(prod(fact(z0), x1)) → c40(PROD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(if(z0, z1, z2), x1)) → c40(PROD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.67
PROPER(prod(zero(z0), x1)) → c40(PROD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(s(z0), x1)) → c40(PROD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(prod(z0, z1), x1)) → c40(PROD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(prod(p(z0), x1)) → c40(PROD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.67
PROPER(prod(add(z0, z1), x1)) → c40(PROD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(prod(x0, 0)) → c40(PROD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.67
PROPER(prod(x0, true)) → c40(PROD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.67
PROPER(prod(x0, false)) → c40(PROD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.67
PROPER(prod(0, x1)) → c40(PROD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(prod(true, x1)) → c40(PROD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(prod(false, x1)) → c40(PROD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(p(fact(z0))) → c41(P(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
PROPER(p(if(z0, z1, z2))) → c41(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(p(zero(z0))) → c41(P(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
PROPER(p(s(z0))) → c41(P(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
PROPER(p(prod(z0, z1))) → c41(P(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(p(p(z0))) → c41(P(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
PROPER(p(add(z0, z1))) → c41(P(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
PROPER(p(0)) → c41(P(ok(0))) 286.36/91.67
PROPER(p(true)) → c41(P(ok(true))) 286.36/91.67
PROPER(p(false)) → c41(P(ok(false))) 286.36/91.67
PROPER(add(x0, fact(z0))) → c42(ADD(proper(x0), fact(proper(z0))), PROPER(x0), PROPER(fact(z0))) 286.36/91.67
PROPER(add(x0, if(z0, z1, z2))) → c42(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 286.36/91.67
PROPER(add(x0, zero(z0))) → c42(ADD(proper(x0), zero(proper(z0))), PROPER(x0), PROPER(zero(z0))) 286.36/91.67
PROPER(add(x0, s(z0))) → c42(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 286.36/91.67
PROPER(add(x0, prod(z0, z1))) → c42(ADD(proper(x0), prod(proper(z0), proper(z1))), PROPER(x0), PROPER(prod(z0, z1))) 286.36/91.67
PROPER(add(x0, p(z0))) → c42(ADD(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 286.36/91.67
PROPER(add(x0, add(z0, z1))) → c42(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 286.36/91.67
PROPER(add(fact(z0), x1)) → c42(ADD(fact(proper(z0)), proper(x1)), PROPER(fact(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(if(z0, z1, z2), x1)) → c42(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 286.36/91.67
PROPER(add(zero(z0), x1)) → c42(ADD(zero(proper(z0)), proper(x1)), PROPER(zero(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(s(z0), x1)) → c42(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(prod(z0, z1), x1)) → c42(ADD(prod(proper(z0), proper(z1)), proper(x1)), PROPER(prod(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(add(p(z0), x1)) → c42(ADD(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 286.36/91.67
PROPER(add(add(z0, z1), x1)) → c42(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 286.36/91.67
PROPER(add(x0, 0)) → c42(ADD(proper(x0), ok(0)), PROPER(x0)) 286.36/91.67
PROPER(add(x0, true)) → c42(ADD(proper(x0), ok(true)), PROPER(x0)) 286.36/91.67
PROPER(add(x0, false)) → c42(ADD(proper(x0), ok(false)), PROPER(x0)) 286.36/91.67
PROPER(add(0, x1)) → c42(ADD(ok(0), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(add(true, x1)) → c42(ADD(ok(true), proper(x1)), PROPER(x1)) 286.36/91.67
PROPER(add(false, x1)) → c42(ADD(ok(false), proper(x1)), PROPER(x1)) 286.36/91.67
TOP(mark(fact(z0))) → c45(TOP(fact(proper(z0))), PROPER(fact(z0))) 286.36/91.67
TOP(mark(if(z0, z1, z2))) → c45(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 286.36/91.67
TOP(mark(zero(z0))) → c45(TOP(zero(proper(z0))), PROPER(zero(z0))) 286.36/91.67
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0))) 286.36/91.67
TOP(mark(prod(z0, z1))) → c45(TOP(prod(proper(z0), proper(z1))), PROPER(prod(z0, z1))) 286.36/91.67
TOP(mark(p(z0))) → c45(TOP(p(proper(z0))), PROPER(p(z0))) 286.36/91.67
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 286.36/91.67
TOP(ok(fact(z0))) → c46(TOP(mark(if(zero(z0), s(0), prod(z0, fact(p(z0)))))), ACTIVE(fact(z0))) 286.36/91.67
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0))) 286.36/91.67
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 286.36/91.67
TOP(ok(prod(0, z0))) → c46(TOP(mark(0)), ACTIVE(prod(0, z0))) 286.36/91.67
TOP(ok(prod(s(z0), z1))) → c46(TOP(mark(add(z1, prod(z0, z1)))), ACTIVE(prod(s(z0), z1))) 286.36/91.67
TOP(ok(if(true, z0, z1))) → c46(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 286.36/91.67
TOP(ok(if(false, z0, z1))) → c46(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 286.36/91.67
TOP(ok(zero(0))) → c46(TOP(mark(true)), ACTIVE(zero(0))) 286.36/91.67
TOP(ok(zero(s(z0)))) → c46(TOP(mark(false)), ACTIVE(zero(s(z0)))) 286.36/91.67
TOP(ok(p(s(z0)))) → c46(TOP(mark(z0)), ACTIVE(p(s(z0)))) 286.36/91.67
TOP(ok(fact(z0))) → c46(TOP(fact(active(z0))), ACTIVE(fact(z0))) 286.36/91.67
TOP(ok(if(z0, z1, z2))) → c46(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 286.36/91.67
TOP(ok(zero(z0))) → c46(TOP(zero(active(z0))), ACTIVE(zero(z0))) 286.36/91.67
TOP(ok(s(z0))) → c46(TOP(s(active(z0))), ACTIVE(s(z0))) 286.36/91.67
TOP(ok(prod(z0, z1))) → c46(TOP(prod(active(z0), z1)), ACTIVE(prod(z0, z1))) 286.36/91.67
TOP(ok(prod(z0, z1))) → c46(TOP(prod(z0, active(z1))), ACTIVE(prod(z0, z1))) 286.36/91.67
TOP(ok(p(z0))) → c46(TOP(p(active(z0))), ACTIVE(p(z0))) 286.36/91.67
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 286.36/91.67
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))

286.36/91.67

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
TOP(mark(0)) → c45(TOP(ok(0))) 286.36/91.67
TOP(mark(true)) → c45(TOP(ok(true))) 286.36/91.67
TOP(mark(false)) → c45(TOP(ok(false)))

S tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
TOP(mark(0)) → c45(TOP(ok(0))) 286.36/91.67
TOP(mark(true)) → c45(TOP(ok(true))) 286.36/91.67
TOP(mark(false)) → c45(TOP(ok(false)))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD, TOP

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c45

286.36/91.67

(61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

286.36/91.67

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
TOP(mark(0)) → c45 286.36/91.67
TOP(mark(true)) → c45 286.36/91.67
TOP(mark(false)) → c45

S tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
TOP(mark(0)) → c45 286.36/91.67
TOP(mark(true)) → c45 286.36/91.67
TOP(mark(false)) → c45

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD, TOP

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c45

286.36/91.67

(63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

TOP(mark(true)) → c45 286.36/91.67
TOP(mark(0)) → c45 286.36/91.67
TOP(mark(false)) → c45

286.36/91.67

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

K tuples:none
Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(65) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = 0 286.36/91.67
POL(FACT(x₁)) = [2]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = 0 286.36/91.67
POL(P(x₁)) = 0 286.36/91.67
POL(PROD(x₁, x₂)) = 0 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = 0 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [1] + x₁ 286.36/91.67
POL(ok(x₁)) = [1] + x₁

286.36/91.67

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(67) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = 0 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = 0 286.36/91.67
POL(P(x₁)) = 0 286.36/91.67
POL(PROD(x₁, x₂)) = 0 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [1] + x₁ 286.36/91.67
POL(ok(x₁)) = [1] + x₁

286.36/91.67

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(69) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [2]x₁ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = 0 286.36/91.67
POL(P(x₁)) = x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = 0 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = [5]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [2] + x₁ 286.36/91.67
POL(ok(x₁)) = [1] + x₁

286.36/91.67

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(71) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [5]x₁ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = [2]x₁ 286.36/91.67
POL(P(x₁)) = [3]x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = 0 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = [5]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [1] + x₁ 286.36/91.67
POL(ok(x₁)) = x₁

286.36/91.67

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(73) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [5]x₁ + x₂ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = x₃ 286.36/91.67
POL(P(x₁)) = [3]x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = 0 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = [3]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = x₁ 286.36/91.67
POL(ok(x₁)) = [1] + x₁

286.36/91.67

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(75) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PROD(mark(z0), z1) → c27(PROD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [5]x₁ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = [3]x₁ + [3]x₂ + [5]x₃ 286.36/91.67
POL(P(x₁)) = [3]x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = x₁ 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = [3]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [2] + x₁ 286.36/91.67
POL(ok(x₁)) = x₁

286.36/91.67

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(77) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [5]x₁ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = [5]x₁ + [5]x₂ + [5]x₃ 286.36/91.67
POL(P(x₁)) = [3]x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = x₁ 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = [3]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = x₁ 286.36/91.67
POL(ok(x₁)) = [1] + x₁

286.36/91.67

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(79) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(mark(z0)) → c25(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [3]x₁ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = [5]x₁ + [5]x₂ + [5]x₃ 286.36/91.67
POL(P(x₁)) = [3]x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = [3]x₁ 286.36/91.67
POL(S(x₁)) = [2]x₁ 286.36/91.67
POL(ZERO(x₁)) = [3]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [1] + x₁ 286.36/91.67
POL(ok(x₁)) = x₁

286.36/91.67

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.67
active(add(0, z0)) → mark(z0) 286.36/91.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.67
active(prod(0, z0)) → mark(0) 286.36/91.67
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.67
active(if(true, z0, z1)) → mark(z0) 286.36/91.67
active(if(false, z0, z1)) → mark(z1) 286.36/91.67
active(zero(0)) → mark(true) 286.36/91.67
active(zero(s(z0))) → mark(false) 286.36/91.67
active(p(s(z0))) → mark(z0) 286.36/91.67
active(fact(z0)) → fact(active(z0)) 286.36/91.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.67
active(zero(z0)) → zero(active(z0)) 286.36/91.67
active(s(z0)) → s(active(z0)) 286.36/91.67
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.67
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.67
active(p(z0)) → p(active(z0)) 286.36/91.67
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.67
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.67
fact(mark(z0)) → mark(fact(z0)) 286.36/91.67
fact(ok(z0)) → ok(fact(z0)) 286.36/91.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.67
zero(mark(z0)) → mark(zero(z0)) 286.36/91.67
zero(ok(z0)) → ok(zero(z0)) 286.36/91.67
s(mark(z0)) → mark(s(z0)) 286.36/91.67
s(ok(z0)) → ok(s(z0)) 286.36/91.67
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.67
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.67
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.67
p(mark(z0)) → mark(p(z0)) 286.36/91.67
p(ok(z0)) → ok(p(z0)) 286.36/91.67
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.67
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.67
proper(fact(z0)) → fact(proper(z0)) 286.36/91.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.67
proper(zero(z0)) → zero(proper(z0)) 286.36/91.67
proper(s(z0)) → s(proper(z0)) 286.36/91.67
proper(0) → ok(0) 286.36/91.67
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.67
proper(p(z0)) → p(proper(z0)) 286.36/91.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.67
proper(true) → ok(true) 286.36/91.67
proper(false) → ok(false) 286.36/91.67
top(mark(z0)) → top(proper(z0)) 286.36/91.67
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
S(mark(z0)) → c25(S(z0))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.67

(81) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PROD(z0, mark(z1)) → c28(PROD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.67
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.67
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.67
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.67
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.67
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.67
S(mark(z0)) → c25(S(z0)) 286.36/91.67
S(ok(z0)) → c26(S(z0)) 286.36/91.67
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.67
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.67
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.67
P(mark(z0)) → c30(P(z0)) 286.36/91.67
P(ok(z0)) → c31(P(z0)) 286.36/91.67
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.67
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.67
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.67

POL(ADD(x₁, x₂)) = [3]x₁ 286.36/91.67
POL(FACT(x₁)) = [3]x₁ 286.36/91.67
POL(IF(x₁, x₂, x₃)) = [3]x₁ + [5]x₂ + [5]x₃ 286.36/91.67
POL(P(x₁)) = [3]x₁ 286.36/91.67
POL(PROD(x₁, x₂)) = [5]x₁ + x₂ 286.36/91.67
POL(S(x₁)) = 0 286.36/91.67
POL(ZERO(x₁)) = [3]x₁ 286.36/91.67
POL(c19(x₁)) = x₁ 286.36/91.67
POL(c20(x₁)) = x₁ 286.36/91.67
POL(c21(x₁)) = x₁ 286.36/91.67
POL(c22(x₁)) = x₁ 286.36/91.67
POL(c23(x₁)) = x₁ 286.36/91.67
POL(c24(x₁)) = x₁ 286.36/91.67
POL(c25(x₁)) = x₁ 286.36/91.67
POL(c26(x₁)) = x₁ 286.36/91.67
POL(c27(x₁)) = x₁ 286.36/91.67
POL(c28(x₁)) = x₁ 286.36/91.67
POL(c29(x₁)) = x₁ 286.36/91.67
POL(c30(x₁)) = x₁ 286.36/91.67
POL(c31(x₁)) = x₁ 286.36/91.67
POL(c32(x₁)) = x₁ 286.36/91.67
POL(c33(x₁)) = x₁ 286.36/91.67
POL(c34(x₁)) = x₁ 286.36/91.67
POL(mark(x₁)) = [1] + x₁ 286.36/91.67
POL(ok(x₁)) = [1] + x₁

286.36/91.68

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.68
active(add(0, z0)) → mark(z0) 286.36/91.68
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.68
active(prod(0, z0)) → mark(0) 286.36/91.68
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.68
active(if(true, z0, z1)) → mark(z0) 286.36/91.68
active(if(false, z0, z1)) → mark(z1) 286.36/91.68
active(zero(0)) → mark(true) 286.36/91.68
active(zero(s(z0))) → mark(false) 286.36/91.68
active(p(s(z0))) → mark(z0) 286.36/91.68
active(fact(z0)) → fact(active(z0)) 286.36/91.68
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.68
active(zero(z0)) → zero(active(z0)) 286.36/91.68
active(s(z0)) → s(active(z0)) 286.36/91.68
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.68
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.68
active(p(z0)) → p(active(z0)) 286.36/91.68
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.68
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.68
fact(mark(z0)) → mark(fact(z0)) 286.36/91.68
fact(ok(z0)) → ok(fact(z0)) 286.36/91.68
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.68
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.68
zero(mark(z0)) → mark(zero(z0)) 286.36/91.68
zero(ok(z0)) → ok(zero(z0)) 286.36/91.68
s(mark(z0)) → mark(s(z0)) 286.36/91.68
s(ok(z0)) → ok(s(z0)) 286.36/91.68
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.68
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.68
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.68
p(mark(z0)) → mark(p(z0)) 286.36/91.68
p(ok(z0)) → ok(p(z0)) 286.36/91.68
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.68
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.68
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.68
proper(fact(z0)) → fact(proper(z0)) 286.36/91.68
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.68
proper(zero(z0)) → zero(proper(z0)) 286.36/91.68
proper(s(z0)) → s(proper(z0)) 286.36/91.68
proper(0) → ok(0) 286.36/91.68
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.68
proper(p(z0)) → p(proper(z0)) 286.36/91.68
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.68
proper(true) → ok(true) 286.36/91.68
proper(false) → ok(false) 286.36/91.68
top(mark(z0)) → top(proper(z0)) 286.36/91.68
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.68
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.68
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.68
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.68
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.68
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.68
S(mark(z0)) → c25(S(z0)) 286.36/91.68
S(ok(z0)) → c26(S(z0)) 286.36/91.68
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.68
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.68
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.68
P(mark(z0)) → c30(P(z0)) 286.36/91.68
P(ok(z0)) → c31(P(z0)) 286.36/91.68
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.68
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.68
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:

S(ok(z0)) → c26(S(z0)) 286.36/91.68
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.68
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.68
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.68
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.68
P(mark(z0)) → c30(P(z0)) 286.36/91.68
P(ok(z0)) → c31(P(z0)) 286.36/91.68
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.68
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.68
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.68
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.68
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.68
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.68
S(mark(z0)) → c25(S(z0)) 286.36/91.68
PROD(z0, mark(z1)) → c28(PROD(z0, z1))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.68

(83) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(ok(z0)) → c26(S(z0)) 286.36/91.68
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.68
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.68
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.68
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.68
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.68
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.68
S(mark(z0)) → c25(S(z0)) 286.36/91.68
S(ok(z0)) → c26(S(z0)) 286.36/91.68
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.68
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.68
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.68
P(mark(z0)) → c30(P(z0)) 286.36/91.68
P(ok(z0)) → c31(P(z0)) 286.36/91.68
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.68
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.68
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 286.36/91.68

POL(ADD(x₁, x₂)) = [5]x₁ + [3]x₂ 286.36/91.68
POL(FACT(x₁)) = [3]x₁ 286.36/91.68
POL(IF(x₁, x₂, x₃)) = [3]x₁ + [5]x₂ + [5]x₃ 286.36/91.68
POL(P(x₁)) = [3]x₁ 286.36/91.68
POL(PROD(x₁, x₂)) = [3]x₁ + [3]x₂ 286.36/91.68
POL(S(x₁)) = [3]x₁ 286.36/91.68
POL(ZERO(x₁)) = [3]x₁ 286.36/91.68
POL(c19(x₁)) = x₁ 286.36/91.68
POL(c20(x₁)) = x₁ 286.36/91.68
POL(c21(x₁)) = x₁ 286.36/91.68
POL(c22(x₁)) = x₁ 286.36/91.68
POL(c23(x₁)) = x₁ 286.36/91.68
POL(c24(x₁)) = x₁ 286.36/91.68
POL(c25(x₁)) = x₁ 286.36/91.68
POL(c26(x₁)) = x₁ 286.36/91.68
POL(c27(x₁)) = x₁ 286.36/91.68
POL(c28(x₁)) = x₁ 286.36/91.68
POL(c29(x₁)) = x₁ 286.36/91.68
POL(c30(x₁)) = x₁ 286.36/91.68
POL(c31(x₁)) = x₁ 286.36/91.68
POL(c32(x₁)) = x₁ 286.36/91.68
POL(c33(x₁)) = x₁ 286.36/91.68
POL(c34(x₁)) = x₁ 286.36/91.68
POL(mark(x₁)) = [3] + x₁ 286.36/91.68
POL(ok(x₁)) = [3] + x₁

286.36/91.68

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(fact(z0)) → mark(if(zero(z0), s(0), prod(z0, fact(p(z0))))) 286.36/91.68
active(add(0, z0)) → mark(z0) 286.36/91.68
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 286.36/91.68
active(prod(0, z0)) → mark(0) 286.36/91.68
active(prod(s(z0), z1)) → mark(add(z1, prod(z0, z1))) 286.36/91.68
active(if(true, z0, z1)) → mark(z0) 286.36/91.68
active(if(false, z0, z1)) → mark(z1) 286.36/91.68
active(zero(0)) → mark(true) 286.36/91.68
active(zero(s(z0))) → mark(false) 286.36/91.68
active(p(s(z0))) → mark(z0) 286.36/91.68
active(fact(z0)) → fact(active(z0)) 286.36/91.68
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 286.36/91.68
active(zero(z0)) → zero(active(z0)) 286.36/91.68
active(s(z0)) → s(active(z0)) 286.36/91.68
active(prod(z0, z1)) → prod(active(z0), z1) 286.36/91.68
active(prod(z0, z1)) → prod(z0, active(z1)) 286.36/91.68
active(p(z0)) → p(active(z0)) 286.36/91.68
active(add(z0, z1)) → add(active(z0), z1) 286.36/91.68
active(add(z0, z1)) → add(z0, active(z1)) 286.36/91.68
fact(mark(z0)) → mark(fact(z0)) 286.36/91.68
fact(ok(z0)) → ok(fact(z0)) 286.36/91.68
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 286.36/91.68
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 286.36/91.68
zero(mark(z0)) → mark(zero(z0)) 286.36/91.68
zero(ok(z0)) → ok(zero(z0)) 286.36/91.68
s(mark(z0)) → mark(s(z0)) 286.36/91.68
s(ok(z0)) → ok(s(z0)) 286.36/91.68
prod(mark(z0), z1) → mark(prod(z0, z1)) 286.36/91.68
prod(z0, mark(z1)) → mark(prod(z0, z1)) 286.36/91.68
prod(ok(z0), ok(z1)) → ok(prod(z0, z1)) 286.36/91.68
p(mark(z0)) → mark(p(z0)) 286.36/91.68
p(ok(z0)) → ok(p(z0)) 286.36/91.68
add(mark(z0), z1) → mark(add(z0, z1)) 286.36/91.68
add(z0, mark(z1)) → mark(add(z0, z1)) 286.36/91.68
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 286.36/91.68
proper(fact(z0)) → fact(proper(z0)) 286.36/91.68
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 286.36/91.68
proper(zero(z0)) → zero(proper(z0)) 286.36/91.68
proper(s(z0)) → s(proper(z0)) 286.36/91.68
proper(0) → ok(0) 286.36/91.68
proper(prod(z0, z1)) → prod(proper(z0), proper(z1)) 286.36/91.68
proper(p(z0)) → p(proper(z0)) 286.36/91.68
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 286.36/91.68
proper(true) → ok(true) 286.36/91.68
proper(false) → ok(false) 286.36/91.68
top(mark(z0)) → top(proper(z0)) 286.36/91.68
top(ok(z0)) → top(active(z0))

Tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.68
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.68
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.68
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.68
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.68
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.68
S(mark(z0)) → c25(S(z0)) 286.36/91.68
S(ok(z0)) → c26(S(z0)) 286.36/91.68
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.68
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.68
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.68
P(mark(z0)) → c30(P(z0)) 286.36/91.68
P(ok(z0)) → c31(P(z0)) 286.36/91.68
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.68
ADD(z0, mark(z1)) → c33(ADD(z0, z1)) 286.36/91.68
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1))

S tuples:none
K tuples:

FACT(mark(z0)) → c19(FACT(z0)) 286.36/91.68
FACT(ok(z0)) → c20(FACT(z0)) 286.36/91.68
ZERO(mark(z0)) → c23(ZERO(z0)) 286.36/91.68
ZERO(ok(z0)) → c24(ZERO(z0)) 286.36/91.68
P(mark(z0)) → c30(P(z0)) 286.36/91.68
P(ok(z0)) → c31(P(z0)) 286.36/91.68
ADD(mark(z0), z1) → c32(ADD(z0, z1)) 286.36/91.68
ADD(ok(z0), ok(z1)) → c34(ADD(z0, z1)) 286.36/91.68
IF(mark(z0), z1, z2) → c21(IF(z0, z1, z2)) 286.36/91.68
IF(ok(z0), ok(z1), ok(z2)) → c22(IF(z0, z1, z2)) 286.36/91.68
PROD(mark(z0), z1) → c27(PROD(z0, z1)) 286.36/91.68
PROD(ok(z0), ok(z1)) → c29(PROD(z0, z1)) 286.36/91.68
S(mark(z0)) → c25(S(z0)) 286.36/91.68
PROD(z0, mark(z1)) → c28(PROD(z0, z1)) 286.36/91.68
S(ok(z0)) → c26(S(z0)) 286.36/91.68
ADD(z0, mark(z1)) → c33(ADD(z0, z1))

Defined Rule Symbols:

active, fact, if, zero, s, prod, p, add, proper, top

Defined Pair Symbols:

FACT, IF, ZERO, S, PROD, P, ADD

Compound Symbols:

c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

286.36/91.68

(85) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

286.36/91.68

(86) BOUNDS(O(1), O(1))

286.36/91.68