256.45/79.51 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^2)).

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

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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 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳14 CdtProblem

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

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↳16 CdtProblem

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

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↳18 CdtProblem

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

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↳26 CdtProblem

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

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↳28 CdtProblem

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

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↳30 CdtProblem

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

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↳32 CdtProblem

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

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↳34 CdtProblem

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

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↳36 CdtProblem

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

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↳38 CdtProblem

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

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↳40 CdtProblem

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

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↳42 CdtProblem

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

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↳44 CdtProblem

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

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↳46 CdtProblem

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

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↳48 CdtProblem

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

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↳50 CdtProblem

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

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↳52 CdtProblem

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

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↳54 CdtProblem

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

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↳56 CdtProblem

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

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↳58 CdtProblem

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

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↳60 CdtProblem

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↳61 CdtUnreachableProof (⇔)

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↳62 CdtProblem

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

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↳64 CdtProblem

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

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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^2))))

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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(and(true, X)) → mark(X) 256.45/79.56
active(and(false, Y)) → mark(false) 256.45/79.56
active(if(true, X, Y)) → mark(X) 256.45/79.56
active(if(false, X, Y)) → mark(Y) 256.45/79.56
active(add(0, X)) → mark(X) 256.45/79.56
active(add(s(X), Y)) → mark(s(add(X, Y))) 256.45/79.56
active(first(0, X)) → mark(nil) 256.45/79.56
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z))) 256.45/79.56
active(from(X)) → mark(cons(X, from(s(X)))) 256.45/79.56
active(and(X1, X2)) → and(active(X1), X2) 256.45/79.56
active(if(X1, X2, X3)) → if(active(X1), X2, X3) 256.45/79.56
active(add(X1, X2)) → add(active(X1), X2) 256.45/79.56
active(first(X1, X2)) → first(active(X1), X2) 256.45/79.56
active(first(X1, X2)) → first(X1, active(X2)) 256.45/79.56
and(mark(X1), X2) → mark(and(X1, X2)) 256.45/79.56
if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) 256.45/79.56
add(mark(X1), X2) → mark(add(X1, X2)) 256.45/79.56
first(mark(X1), X2) → mark(first(X1, X2)) 256.45/79.56
first(X1, mark(X2)) → mark(first(X1, X2)) 256.45/79.56
proper(and(X1, X2)) → and(proper(X1), proper(X2)) 256.45/79.56
proper(true) → ok(true) 256.45/79.56
proper(false) → ok(false) 256.45/79.56
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3)) 256.45/79.56
proper(add(X1, X2)) → add(proper(X1), proper(X2)) 256.45/79.56
proper(0) → ok(0) 256.45/79.56
proper(s(X)) → s(proper(X)) 256.45/79.56
proper(first(X1, X2)) → first(proper(X1), proper(X2)) 256.45/79.56
proper(nil) → ok(nil) 256.45/79.56
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 256.45/79.56
proper(from(X)) → from(proper(X)) 256.45/79.56
and(ok(X1), ok(X2)) → ok(and(X1, X2)) 256.45/79.56
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) 256.45/79.56
add(ok(X1), ok(X2)) → ok(add(X1, X2)) 256.45/79.56
s(ok(X)) → ok(s(X)) 256.45/79.56
first(ok(X1), ok(X2)) → ok(first(X1, X2)) 256.45/79.56
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 256.45/79.56
from(ok(X)) → ok(from(X)) 256.45/79.56
top(mark(X)) → top(proper(X)) 256.45/79.56
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(and(true, z0)) → mark(z0) 256.45/79.56
active(and(false, z0)) → mark(false) 256.45/79.56
active(if(true, z0, z1)) → mark(z0) 256.45/79.56
active(if(false, z0, z1)) → mark(z1) 256.45/79.56
active(add(0, z0)) → mark(z0) 256.45/79.56
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.45/79.56
active(first(0, z0)) → mark(nil) 256.45/79.56
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.45/79.56
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.45/79.56
active(and(z0, z1)) → and(active(z0), z1) 256.45/79.56
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.45/79.56
active(add(z0, z1)) → add(active(z0), z1) 256.45/79.56
active(first(z0, z1)) → first(active(z0), z1) 256.45/79.56
active(first(z0, z1)) → first(z0, active(z1)) 256.45/79.56
and(mark(z0), z1) → mark(and(z0, z1)) 256.45/79.56
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.45/79.56
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.45/79.56
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.45/79.56
add(mark(z0), z1) → mark(add(z0, z1)) 256.45/79.56
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.45/79.56
first(mark(z0), z1) → mark(first(z0, z1)) 256.45/79.56
first(z0, mark(z1)) → mark(first(z0, z1)) 256.45/79.56
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.45/79.56
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.45/79.56
proper(true) → ok(true) 256.45/79.56
proper(false) → ok(false) 256.45/79.56
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.45/79.56
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.45/79.56
proper(0) → ok(0) 256.45/79.56
proper(s(z0)) → s(proper(z0)) 256.45/79.56
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.45/79.56
proper(nil) → ok(nil) 256.45/79.56
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.45/79.56
proper(from(z0)) → from(proper(z0)) 256.45/79.56
s(ok(z0)) → ok(s(z0)) 256.45/79.56
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.45/79.56
from(ok(z0)) → ok(from(z0)) 256.45/79.56
top(mark(z0)) → top(proper(z0)) 256.45/79.56
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(add(s(z0), z1)) → c5(S(add(z0, z1)), ADD(z0, z1)) 256.45/79.56
ACTIVE(first(s(z0), cons(z1, z2))) → c7(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 256.45/79.56
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 256.45/79.56
ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.56
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.56
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.56
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.56
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.56
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.56
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.56
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.56
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.56
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.45/79.56
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.45/79.56
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.45/79.56
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.45/79.56
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.45/79.56
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.56
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.45/79.56
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.56
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.45/79.56
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.56
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.56
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.45/79.58
S(ok(z0)) → c34(S(z0)) 256.45/79.58
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.45/79.58
FROM(ok(z0)) → c36(FROM(z0)) 256.45/79.58
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.45/79.58
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(add(s(z0), z1)) → c5(S(add(z0, z1)), ADD(z0, z1)) 256.45/79.58
ACTIVE(first(s(z0), cons(z1, z2))) → c7(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 256.45/79.58
ACTIVE(from(z0)) → c8(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 256.45/79.58
ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.58
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.58
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.58
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.58
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.58
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.58
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.58
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.58
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.58
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.45/79.58
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.45/79.58
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.45/79.58
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.45/79.58
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.45/79.58
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.58
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.45/79.58
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.58
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.45/79.58
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.58
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.58
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.45/79.58
S(ok(z0)) → c34(S(z0)) 256.45/79.58
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.45/79.58
FROM(ok(z0)) → c36(FROM(z0)) 256.45/79.58
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.45/79.58
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38

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

Removed 2 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.45/79.58
active(and(false, z0)) → mark(false) 256.45/79.58
active(if(true, z0, z1)) → mark(z0) 256.45/79.58
active(if(false, z0, z1)) → mark(z1) 256.45/79.58
active(add(0, z0)) → mark(z0) 256.45/79.58
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.45/79.58
active(first(0, z0)) → mark(nil) 256.45/79.58
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.45/79.58
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.45/79.58
active(and(z0, z1)) → and(active(z0), z1) 256.45/79.58
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.45/79.58
active(add(z0, z1)) → add(active(z0), z1) 256.45/79.58
active(first(z0, z1)) → first(active(z0), z1) 256.45/79.58
active(first(z0, z1)) → first(z0, active(z1)) 256.45/79.58
and(mark(z0), z1) → mark(and(z0, z1)) 256.45/79.58
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.45/79.58
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.45/79.58
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.45/79.58
add(mark(z0), z1) → mark(add(z0, z1)) 256.45/79.58
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.45/79.58
first(mark(z0), z1) → mark(first(z0, z1)) 256.45/79.58
first(z0, mark(z1)) → mark(first(z0, z1)) 256.45/79.58
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.45/79.58
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.45/79.58
proper(true) → ok(true) 256.45/79.58
proper(false) → ok(false) 256.45/79.58
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.45/79.58
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.45/79.58
proper(0) → ok(0) 256.45/79.58
proper(s(z0)) → s(proper(z0)) 256.45/79.58
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.45/79.58
proper(nil) → ok(nil) 256.45/79.58
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.45/79.58
proper(from(z0)) → from(proper(z0)) 256.45/79.58
s(ok(z0)) → ok(s(z0)) 256.45/79.58
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.45/79.58
from(ok(z0)) → ok(from(z0)) 256.45/79.58
top(mark(z0)) → top(proper(z0)) 256.45/79.58
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(add(s(z0), z1)) → c5(S(add(z0, z1)), ADD(z0, z1)) 256.45/79.58
ACTIVE(first(s(z0), cons(z1, z2))) → c7(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 256.45/79.58
ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.58
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.58
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.58
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.58
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.58
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.58
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.58
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.58
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.58
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.45/79.58
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.45/79.58
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.45/79.58
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.45/79.58
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.45/79.58
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.58
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.45/79.58
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.45/79.59
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.45/79.59
S(ok(z0)) → c34(S(z0)) 256.45/79.59
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.45/79.59
FROM(ok(z0)) → c36(FROM(z0)) 256.45/79.59
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.45/79.59
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.45/79.59
ACTIVE(from(z0)) → c8(S(z0))

S tuples:

ACTIVE(add(s(z0), z1)) → c5(S(add(z0, z1)), ADD(z0, z1)) 256.45/79.59
ACTIVE(first(s(z0), cons(z1, z2))) → c7(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 256.45/79.59
ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.59
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.59
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.59
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.59
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.59
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.59
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.45/79.59
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.45/79.59
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.45/79.59
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.45/79.59
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.45/79.59
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.45/79.59
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.45/79.59
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.45/79.59
S(ok(z0)) → c34(S(z0)) 256.45/79.59
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.45/79.59
FROM(ok(z0)) → c36(FROM(z0)) 256.45/79.59
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.45/79.59
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.45/79.59
ACTIVE(from(z0)) → c8(S(z0))

K tuples:none
Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c5, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8

256.45/79.59

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

Split RHS of tuples not part of any SCC

256.45/79.59

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.45/79.59
active(and(false, z0)) → mark(false) 256.45/79.59
active(if(true, z0, z1)) → mark(z0) 256.45/79.59
active(if(false, z0, z1)) → mark(z1) 256.45/79.59
active(add(0, z0)) → mark(z0) 256.45/79.59
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.45/79.59
active(first(0, z0)) → mark(nil) 256.45/79.59
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.45/79.59
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.45/79.59
active(and(z0, z1)) → and(active(z0), z1) 256.45/79.59
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.45/79.59
active(add(z0, z1)) → add(active(z0), z1) 256.45/79.59
active(first(z0, z1)) → first(active(z0), z1) 256.45/79.59
active(first(z0, z1)) → first(z0, active(z1)) 256.45/79.59
and(mark(z0), z1) → mark(and(z0, z1)) 256.45/79.59
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.45/79.59
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.45/79.59
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.45/79.59
add(mark(z0), z1) → mark(add(z0, z1)) 256.45/79.59
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.45/79.59
first(mark(z0), z1) → mark(first(z0, z1)) 256.45/79.59
first(z0, mark(z1)) → mark(first(z0, z1)) 256.45/79.59
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.45/79.59
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.45/79.59
proper(true) → ok(true) 256.45/79.59
proper(false) → ok(false) 256.45/79.59
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.45/79.59
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.45/79.59
proper(0) → ok(0) 256.45/79.59
proper(s(z0)) → s(proper(z0)) 256.45/79.59
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.45/79.59
proper(nil) → ok(nil) 256.45/79.59
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.45/79.59
proper(from(z0)) → from(proper(z0)) 256.45/79.59
s(ok(z0)) → ok(s(z0)) 256.45/79.59
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.45/79.59
from(ok(z0)) → ok(from(z0)) 256.45/79.59
top(mark(z0)) → top(proper(z0)) 256.45/79.59
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.59
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.59
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.59
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.59
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.59
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.59
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.45/79.59
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.45/79.59
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.45/79.59
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.45/79.59
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.45/79.59
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.45/79.59
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.45/79.59
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.45/79.59
S(ok(z0)) → c34(S(z0)) 256.45/79.59
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.45/79.59
FROM(ok(z0)) → c36(FROM(z0)) 256.45/79.59
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.45/79.59
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.45/79.59
ACTIVE(from(z0)) → c8(S(z0)) 256.45/79.59
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.45/79.59
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.45/79.59
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.45/79.59
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))

S tuples:

ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.59
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.59
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.59
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.59
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.59
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.59
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.45/79.59
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.45/79.59
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.45/79.59
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.45/79.59
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.45/79.59
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.45/79.59
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.45/79.59
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.45/79.59
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.45/79.59
S(ok(z0)) → c34(S(z0)) 256.45/79.59
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.45/79.59
FROM(ok(z0)) → c36(FROM(z0)) 256.45/79.59
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.45/79.59
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.45/79.59
ACTIVE(from(z0)) → c8(S(z0)) 256.45/79.59
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.45/79.59
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.45/79.59
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.45/79.59
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))

K tuples:none
Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c

256.45/79.59

(7) 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.

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

first(mark(z0), z1) → mark(first(z0, z1)) 256.45/79.59
first(z0, mark(z1)) → mark(first(z0, z1)) 256.45/79.59
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.45/79.59
add(mark(z0), z1) → mark(add(z0, z1)) 256.45/79.59
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.45/79.59
active(and(true, z0)) → mark(z0) 256.45/79.59
active(and(false, z0)) → mark(false) 256.45/79.59
active(if(true, z0, z1)) → mark(z0) 256.45/79.59
active(if(false, z0, z1)) → mark(z1) 256.45/79.59
active(add(0, z0)) → mark(z0) 256.45/79.59
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.45/79.59
active(first(0, z0)) → mark(nil) 256.45/79.59
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.45/79.59
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.45/79.59
active(and(z0, z1)) → and(active(z0), z1) 256.45/79.59
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.45/79.59
active(add(z0, z1)) → add(active(z0), z1) 256.45/79.59
active(first(z0, z1)) → first(active(z0), z1) 256.45/79.59
active(first(z0, z1)) → first(z0, active(z1)) 256.45/79.59
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.45/79.59
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.45/79.59
and(mark(z0), z1) → mark(and(z0, z1)) 256.45/79.59
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.45/79.59
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.45/79.59
s(ok(z0)) → ok(s(z0)) 256.45/79.59
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.45/79.59
proper(true) → ok(true) 256.45/79.59
proper(false) → ok(false) 256.45/79.59
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.45/79.59
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.45/79.59
proper(0) → ok(0) 256.45/79.59
proper(s(z0)) → s(proper(z0)) 256.45/79.59
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.45/79.59
proper(nil) → ok(nil) 256.45/79.59
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.45/79.59
proper(from(z0)) → from(proper(z0)) 256.45/79.59
from(ok(z0)) → ok(from(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.45/79.59
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.45/79.59
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.45/79.59
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.45/79.59
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.45/79.59
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.45/79.59
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.45/79.59
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.66
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.66
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.66
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.66
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.66
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.66
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.66
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.66
S(ok(z0)) → c34(S(z0)) 256.79/79.66
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.66
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.66
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.66
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.66
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))

The order we found is given by the following interpretation:
Polynomial interpretation : 256.79/79.66

POL(0) = 0 256.79/79.66
POL(ACTIVE(x₁)) = 0 256.79/79.66
POL(ADD(x₁, x₂)) = 0 256.79/79.66
POL(AND(x₁, x₂)) = 0 256.79/79.66
POL(CONS(x₁, x₂)) = 0 256.79/79.66
POL(FIRST(x₁, x₂)) = 0 256.79/79.66
POL(FROM(x₁)) = 0 256.79/79.66
POL(IF(x₁, x₂, x₃)) = 0 256.79/79.66
POL(PROPER(x₁)) = 0 256.79/79.66
POL(S(x₁)) = 0 256.79/79.66
POL(TOP(x₁)) = [2]x₁ 256.79/79.66
POL(active(x₁)) = x₁ 256.79/79.66
POL(add(x₁, x₂)) = [4] + [4]x₁ + x₂ 256.79/79.66
POL(and(x₁, x₂)) = [4]x₁ + x₂ 256.79/79.66
POL(c(x₁)) = x₁ 256.79/79.66
POL(c10(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c11(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c12(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c13(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c14(x₁)) = x₁ 256.79/79.66
POL(c15(x₁)) = x₁ 256.79/79.66
POL(c16(x₁)) = x₁ 256.79/79.66
POL(c17(x₁)) = x₁ 256.79/79.66
POL(c18(x₁)) = x₁ 256.79/79.66
POL(c19(x₁)) = x₁ 256.79/79.66
POL(c20(x₁)) = x₁ 256.79/79.66
POL(c21(x₁)) = x₁ 256.79/79.66
POL(c22(x₁)) = x₁ 256.79/79.66
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 256.79/79.66
POL(c26(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄ 256.79/79.66
POL(c27(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 256.79/79.66
POL(c29(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c30(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 256.79/79.66
POL(c32(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 256.79/79.66
POL(c33(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c34(x₁)) = x₁ 256.79/79.66
POL(c35(x₁)) = x₁ 256.79/79.66
POL(c36(x₁)) = x₁ 256.79/79.66
POL(c37(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c38(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(c8(x₁)) = x₁ 256.79/79.66
POL(c9(x₁, x₂)) = x₁ + x₂ 256.79/79.66
POL(cons(x₁, x₂)) = 0 256.79/79.66
POL(false) = [2] 256.79/79.66
POL(first(x₁, x₂)) = [4] + [4]x₁ + x₂ 256.79/79.66
POL(from(x₁)) = [4] 256.79/79.66
POL(if(x₁, x₂, x₃)) = [4]x₁ + x₂ + x₃ 256.79/79.66
POL(mark(x₁)) = [4] + x₁ 256.79/79.66
POL(nil) = 0 256.79/79.66
POL(ok(x₁)) = x₁ 256.79/79.66
POL(proper(x₁)) = x₁ 256.79/79.66
POL(s(x₁)) = 0 256.79/79.66
POL(true) = [1]

256.79/79.66

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.66
active(and(false, z0)) → mark(false) 256.79/79.66
active(if(true, z0, z1)) → mark(z0) 256.79/79.66
active(if(false, z0, z1)) → mark(z1) 256.79/79.66
active(add(0, z0)) → mark(z0) 256.79/79.66
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.66
active(first(0, z0)) → mark(nil) 256.79/79.66
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.66
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.66
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.66
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.66
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.66
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.66
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.66
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.66
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.66
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.66
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.66
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.66
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.66
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.66
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.66
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.66
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.66
proper(true) → ok(true) 256.79/79.66
proper(false) → ok(false) 256.79/79.66
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.66
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.66
proper(0) → ok(0) 256.79/79.66
proper(s(z0)) → s(proper(z0)) 256.79/79.66
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.66
proper(nil) → ok(nil) 256.79/79.66
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.66
proper(from(z0)) → from(proper(z0)) 256.79/79.66
s(ok(z0)) → ok(s(z0)) 256.79/79.66
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.66
from(ok(z0)) → ok(from(z0)) 256.79/79.66
top(mark(z0)) → top(proper(z0)) 256.79/79.66
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.66
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.66
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.66
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.66
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.66
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.66
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.66
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.66
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.66
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.66
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.66
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.66
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.66
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.66
S(ok(z0)) → c34(S(z0)) 256.79/79.66
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.66
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.66
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.66
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.66
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))

S tuples:

ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.66
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.66
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.66
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.66
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.66
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.66
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.66
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.66
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.66
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.66
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.66
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.66
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.66
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.66
S(ok(z0)) → c34(S(z0)) 256.79/79.66
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.66
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.66
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.66
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c

256.79/79.66

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c9(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(and(true, z0))) 256.79/79.66
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1), ACTIVE(and(false, z0))) 256.79/79.66
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.66
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.66
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.66
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.66
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.66
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.66
ACTIVE(and(from(z0), x1)) → c9(AND(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.66
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.66
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.66
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.66
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.66
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

256.79/79.66

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.66
active(and(false, z0)) → mark(false) 256.79/79.66
active(if(true, z0, z1)) → mark(z0) 256.79/79.66
active(if(false, z0, z1)) → mark(z1) 256.79/79.66
active(add(0, z0)) → mark(z0) 256.79/79.66
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.66
active(first(0, z0)) → mark(nil) 256.79/79.66
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.66
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.66
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.66
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.66
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.66
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.66
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.66
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.66
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.66
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.66
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.66
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.66
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.66
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.66
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.66
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.66
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.66
proper(true) → ok(true) 256.79/79.66
proper(false) → ok(false) 256.79/79.66
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.66
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.66
proper(0) → ok(0) 256.79/79.66
proper(s(z0)) → s(proper(z0)) 256.79/79.66
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.66
proper(nil) → ok(nil) 256.79/79.66
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.66
proper(from(z0)) → from(proper(z0)) 256.79/79.66
s(ok(z0)) → ok(s(z0)) 256.79/79.66
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.66
from(ok(z0)) → ok(from(z0)) 256.79/79.66
top(mark(z0)) → top(proper(z0)) 256.79/79.66
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.66
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.66
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.66
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.66
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.66
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.66
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.66
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.66
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.66
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.66
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.66
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.66
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.66
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.66
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.66
S(ok(z0)) → c34(S(z0)) 256.79/79.66
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.66
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.66
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.66
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.66
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.66
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.66
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.66
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(and(true, z0))) 256.79/79.66
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1), ACTIVE(and(false, z0))) 256.79/79.66
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.66
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.66
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.66
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.66
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.66
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.66
ACTIVE(and(from(z0), x1)) → c9(AND(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.66
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.66
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.66
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.66
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.66
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

S tuples:

ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.66
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.66
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.66
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.66
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.66
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.66
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.66
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.66
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.66
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.66
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.66
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c9(AND(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9

256.79/79.67

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

Removed 2 trailing tuple parts

256.79/79.67

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.67
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c9(AND(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1))

S tuples:

ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.67
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c9(AND(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9

256.79/79.67

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

Split RHS of tuples not part of any SCC

256.79/79.67

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.67
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0)))

S tuples:

ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) 256.79/79.67
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1

256.79/79.67

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

Use narrowing to replace ACTIVE(if(z0, z1, z2)) → c10(IF(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c10(IF(mark(cons(z0, from(s(z0)))), x1, x2), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1)))

256.79/79.67

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c10(IF(mark(cons(z0, from(s(z0)))), x1, x2), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1)))

S tuples:

ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c10(IF(mark(cons(z0, from(s(z0)))), x1, x2), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10

256.79/79.67

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

Removed 6 trailing tuple parts

256.79/79.67

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c10(IF(mark(cons(z0, from(s(z0)))), x1, x2), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2))

S tuples:

ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c10(IF(mark(cons(z0, from(s(z0)))), x1, x2), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10

256.79/79.67

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

Split RHS of tuples not part of any SCC

256.79/79.67

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0)))

S tuples:

ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2

256.79/79.67

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

Use narrowing to replace ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0)) by

ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c11(ADD(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c11(ADD(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

256.79/79.67

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c11(ADD(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c11(ADD(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1), ACTIVE(and(true, z0))) 256.79/79.67
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1), ACTIVE(and(false, z0))) 256.79/79.67
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1), ACTIVE(if(true, z0, z1))) 256.79/79.67
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1), ACTIVE(if(false, z0, z1))) 256.79/79.67
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1), ACTIVE(add(0, z0))) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c11(ADD(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c11(ADD(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11

256.79/79.67

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

Removed 7 trailing tuple parts

256.79/79.67

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c11(ADD(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c11(ADD(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.67
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.67
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c9(AND(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c10(IF(mark(s(add(z0, z1))), x1, x2), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c11(ADD(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c11(ADD(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.67
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.67
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11

256.79/79.67

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

Split RHS of tuples not part of any SCC

256.79/79.67

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.67
active(and(false, z0)) → mark(false) 256.79/79.67
active(if(true, z0, z1)) → mark(z0) 256.79/79.67
active(if(false, z0, z1)) → mark(z1) 256.79/79.67
active(add(0, z0)) → mark(z0) 256.79/79.67
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.67
active(first(0, z0)) → mark(nil) 256.79/79.67
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.67
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.67
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.67
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.67
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.67
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.67
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.67
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.67
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.67
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.67
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.67
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.67
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.67
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.67
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.67
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.67
proper(true) → ok(true) 256.79/79.67
proper(false) → ok(false) 256.79/79.67
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.67
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.67
proper(0) → ok(0) 256.79/79.67
proper(s(z0)) → s(proper(z0)) 256.79/79.67
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.67
proper(nil) → ok(nil) 256.79/79.67
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.67
proper(from(z0)) → from(proper(z0)) 256.79/79.67
s(ok(z0)) → ok(s(z0)) 256.79/79.67
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.67
from(ok(z0)) → ok(from(z0)) 256.79/79.67
top(mark(z0)) → top(proper(z0)) 256.79/79.67
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.67
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.67
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.67
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.67
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.67
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.67
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.67
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.67
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.67
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.67
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.67
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.67
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.67
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.67
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.67
S(ok(z0)) → c34(S(z0)) 256.79/79.67
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.67
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.67
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.67
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.67
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.67
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.67
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.67
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.67
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.67
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.67
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.67
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.67
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.67
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.67
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.67
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.67
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.67
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.67
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.67
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.67
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.67
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.67
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.67
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.67
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.68
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.68
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.68
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.68
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.68
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.68
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.68
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.68
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.68
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.68
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.68
PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.68
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.68
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.68
S(ok(z0)) → c34(S(z0)) 256.79/79.68
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.68
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.68
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.68
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.68
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.68
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.68
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.68
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.68
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.68
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.68
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.68
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.68
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.68
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.68
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.68
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.68
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.68
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.68
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.68
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.68
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.68
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.68
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.68
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.68
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.68
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.68
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.68
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.68
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.68
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.68
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.68
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.68
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.68
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.68
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.68
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.68
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.68
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.68
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.68
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.68
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.68
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.68
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.68
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.68
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.68
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.68
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.68
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.68
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.68
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.68
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.68
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.68
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.68
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.68
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.68
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3

256.79/79.68

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

Use narrowing to replace PROPER(and(z0, z1)) → c23(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.68
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 256.79/79.68
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 256.79/79.68
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.68
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.68
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 256.79/79.68
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.68
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.68
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 256.79/79.68
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.68
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.68
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.68
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 256.79/79.68
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 256.79/79.68
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.68
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.68
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 256.79/79.68
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.68
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.68
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 256.79/79.68
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.68
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

256.79/79.68

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.68
active(and(false, z0)) → mark(false) 256.79/79.68
active(if(true, z0, z1)) → mark(z0) 256.79/79.68
active(if(false, z0, z1)) → mark(z1) 256.79/79.68
active(add(0, z0)) → mark(z0) 256.79/79.68
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.68
active(first(0, z0)) → mark(nil) 256.79/79.68
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.68
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.68
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.68
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.68
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.68
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.68
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.68
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.68
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.68
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.68
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.68
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.68
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.68
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.68
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.68
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.68
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.68
proper(true) → ok(true) 256.79/79.68
proper(false) → ok(false) 256.79/79.68
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.68
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.68
proper(0) → ok(0) 256.79/79.68
proper(s(z0)) → s(proper(z0)) 256.79/79.68
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.68
proper(nil) → ok(nil) 256.79/79.68
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.68
proper(from(z0)) → from(proper(z0)) 256.79/79.68
s(ok(z0)) → ok(s(z0)) 256.79/79.68
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.68
from(ok(z0)) → ok(from(z0)) 256.79/79.68
top(mark(z0)) → top(proper(z0)) 256.79/79.68
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.68
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.68
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.68
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.68
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.68
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.68
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.68
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.68
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.68
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.68
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.68
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.68
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.68
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.68
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.68
S(ok(z0)) → c34(S(z0)) 256.79/79.68
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.68
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.68
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.68
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.68
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.68
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.68
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.68
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.68
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.68
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.68
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.69
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.69
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.69
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.69
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 256.79/79.69
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.69
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 256.79/79.69
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 256.79/79.69
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.69
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 256.79/79.69
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.69
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.69
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 256.79/79.69
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.69
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 256.79/79.69
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 256.79/79.69
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.69
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 256.79/79.69
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 256.79/79.69
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.69
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.69
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.69
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.69
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.69
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.69
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.69
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.69
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.69
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.69
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.69
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.69
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.69
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.69
S(ok(z0)) → c34(S(z0)) 256.79/79.69
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.69
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.69
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.69
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.69
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.69
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.69
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.69
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.69
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 256.79/79.69
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.69
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 256.79/79.69
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 256.79/79.69
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.69
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 256.79/79.69
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.69
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.69
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 256.79/79.69
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.69
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 256.79/79.69
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 256.79/79.69
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.69
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 256.79/79.69
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 256.79/79.69
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23

256.79/79.69

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

Removed 8 trailing tuple parts

256.79/79.69

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.69
active(and(false, z0)) → mark(false) 256.79/79.69
active(if(true, z0, z1)) → mark(z0) 256.79/79.69
active(if(false, z0, z1)) → mark(z1) 256.79/79.69
active(add(0, z0)) → mark(z0) 256.79/79.69
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.69
active(first(0, z0)) → mark(nil) 256.79/79.69
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.69
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.69
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.69
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.69
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.69
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.69
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.69
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.69
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.69
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.69
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.69
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.69
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.69
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.69
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.69
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.69
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.69
proper(true) → ok(true) 256.79/79.69
proper(false) → ok(false) 256.79/79.69
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.69
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.69
proper(0) → ok(0) 256.79/79.69
proper(s(z0)) → s(proper(z0)) 256.79/79.69
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.69
proper(nil) → ok(nil) 256.79/79.69
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.69
proper(from(z0)) → from(proper(z0)) 256.79/79.69
s(ok(z0)) → ok(s(z0)) 256.79/79.69
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.69
from(ok(z0)) → ok(from(z0)) 256.79/79.69
top(mark(z0)) → top(proper(z0)) 256.79/79.69
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.69
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.69
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.69
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.69
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.69
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.69
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.69
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.69
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.69
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.69
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.69
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.69
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.69
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.69
S(ok(z0)) → c34(S(z0)) 256.79/79.69
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.69
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.69
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.69
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.69
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.69
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.69
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.69
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.69
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.69
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 256.79/79.69
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.69
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.69
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.69
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.69
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.69
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.69
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 256.79/79.69
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.69
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.69
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.69
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.69
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.69
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.69
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.69
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.69
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.69
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.69
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.69
PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 256.79/79.69
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.69
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.69
S(ok(z0)) → c34(S(z0)) 256.79/79.69
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.69
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.69
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.69
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.69
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.69
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.69
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.69
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.69
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 256.79/79.69
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.69
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.69
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.69
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.69
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.69
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.69
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 256.79/79.69
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c26, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23

256.79/79.69

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

Use narrowing to replace PROPER(if(z0, z1, z2)) → c26(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 256.79/79.69
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 256.79/79.69
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 256.79/79.69
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 256.79/79.69
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1), PROPER(nil)) 256.79/79.69
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 256.79/79.69
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 256.79/79.69
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 256.79/79.69
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 256.79/79.69
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(nil), PROPER(x2)) 256.79/79.69
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 256.79/79.69
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(nil), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2))

256.79/79.69

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 256.79/79.69
active(and(false, z0)) → mark(false) 256.79/79.69
active(if(true, z0, z1)) → mark(z0) 256.79/79.69
active(if(false, z0, z1)) → mark(z1) 256.79/79.69
active(add(0, z0)) → mark(z0) 256.79/79.69
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 256.79/79.69
active(first(0, z0)) → mark(nil) 256.79/79.69
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 256.79/79.69
active(from(z0)) → mark(cons(z0, from(s(z0)))) 256.79/79.69
active(and(z0, z1)) → and(active(z0), z1) 256.79/79.69
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 256.79/79.69
active(add(z0, z1)) → add(active(z0), z1) 256.79/79.69
active(first(z0, z1)) → first(active(z0), z1) 256.79/79.69
active(first(z0, z1)) → first(z0, active(z1)) 256.79/79.69
and(mark(z0), z1) → mark(and(z0, z1)) 256.79/79.69
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 256.79/79.69
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 256.79/79.69
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 256.79/79.69
add(mark(z0), z1) → mark(add(z0, z1)) 256.79/79.69
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 256.79/79.69
first(mark(z0), z1) → mark(first(z0, z1)) 256.79/79.69
first(z0, mark(z1)) → mark(first(z0, z1)) 256.79/79.69
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 256.79/79.69
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 256.79/79.69
proper(true) → ok(true) 256.79/79.69
proper(false) → ok(false) 256.79/79.69
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 256.79/79.69
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 256.79/79.69
proper(0) → ok(0) 256.79/79.69
proper(s(z0)) → s(proper(z0)) 256.79/79.69
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 256.79/79.69
proper(nil) → ok(nil) 256.79/79.69
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 256.79/79.69
proper(from(z0)) → from(proper(z0)) 256.79/79.69
s(ok(z0)) → ok(s(z0)) 256.79/79.69
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 256.79/79.69
from(ok(z0)) → ok(from(z0)) 256.79/79.69
top(mark(z0)) → top(proper(z0)) 256.79/79.69
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.69
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.69
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.69
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.69
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.69
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.69
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.69
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.69
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.69
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.69
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.69
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.69
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.69
S(ok(z0)) → c34(S(z0)) 256.79/79.69
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.69
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.69
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 256.79/79.69
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.69
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.69
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.69
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.69
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.69
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.69
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 256.79/79.69
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.69
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.69
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.69
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.69
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.69
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.69
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 256.79/79.69
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 256.79/79.69
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 256.79/79.69
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 256.79/79.69
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 256.79/79.69
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1), PROPER(nil)) 256.79/79.69
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 256.79/79.69
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 256.79/79.69
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 256.79/79.69
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 256.79/79.69
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(nil), PROPER(x2)) 256.79/79.69
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 256.79/79.69
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(nil), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 256.79/79.69
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 256.79/79.69
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 256.79/79.69
AND(mark(z0), z1) → c14(AND(z0, z1)) 256.79/79.69
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 256.79/79.69
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 256.79/79.69
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 256.79/79.69
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 256.79/79.69
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 256.79/79.69
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 256.79/79.69
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 256.79/79.69
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 256.79/79.69
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 256.79/79.69
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 256.79/79.69
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 256.79/79.69
S(ok(z0)) → c34(S(z0)) 256.79/79.69
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 256.79/79.69
FROM(ok(z0)) → c36(FROM(z0)) 256.79/79.69
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 256.79/79.69
ACTIVE(from(z0)) → c8(S(z0)) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 256.79/79.69
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 256.79/79.69
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 256.79/79.69
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 256.79/79.69
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 256.79/79.69
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 256.79/79.69
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 256.79/79.69
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 256.79/79.69
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 256.79/79.69
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 256.79/79.69
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 256.79/79.69
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 256.79/79.69
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 256.79/79.69
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 256.79/79.69
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 256.79/79.69
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 256.79/79.69
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 256.79/79.69
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 256.79/79.69
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 256.79/79.69
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 256.79/79.69
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 256.79/79.69
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 256.79/79.69
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 256.79/79.69
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 256.79/79.69
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 256.79/79.69
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 256.79/79.69
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 256.79/79.69
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 256.79/79.69
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 256.79/79.69
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 256.79/79.69
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 256.79/79.69
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 256.79/79.69
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 256.79/79.69
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 256.79/79.69
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 256.79/79.69
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 256.79/79.69
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1), PROPER(nil)) 256.79/79.69
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 256.79/79.69
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 256.79/79.69
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 256.79/79.69
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 256.79/79.69
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 256.79/79.69
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(nil), PROPER(x2)) 256.79/79.69
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 256.79/79.69
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.77
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(nil), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26

257.15/79.77

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

Removed 12 trailing tuple parts

257.15/79.77

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.77
active(and(false, z0)) → mark(false) 257.15/79.77
active(if(true, z0, z1)) → mark(z0) 257.15/79.77
active(if(false, z0, z1)) → mark(z1) 257.15/79.77
active(add(0, z0)) → mark(z0) 257.15/79.77
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.77
active(first(0, z0)) → mark(nil) 257.15/79.77
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.77
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.77
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.77
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.77
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.77
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.77
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.77
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.77
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.77
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.77
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.77
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.77
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.77
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.77
proper(true) → ok(true) 257.15/79.77
proper(false) → ok(false) 257.15/79.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.77
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.77
proper(0) → ok(0) 257.15/79.77
proper(s(z0)) → s(proper(z0)) 257.15/79.77
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.77
proper(nil) → ok(nil) 257.15/79.77
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.77
proper(from(z0)) → from(proper(z0)) 257.15/79.77
s(ok(z0)) → ok(s(z0)) 257.15/79.77
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.77
from(ok(z0)) → ok(from(z0)) 257.15/79.77
top(mark(z0)) → top(proper(z0)) 257.15/79.77
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.77
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.77
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.77
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.77
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.77
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.77
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.77
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.77
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.77
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.77
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.77
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 257.15/79.77
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.77
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.77
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.77
S(ok(z0)) → c34(S(z0)) 257.15/79.77
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.77
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.77
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.77
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.77
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.77
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.77
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.77
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.77
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.77
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.77
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.77
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.77
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.77
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.77
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.77
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.77
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.77
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.77
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.77
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.77
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.77
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.77
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.77
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.77
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.77
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.77
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.77
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.77
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.77
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.77
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.77
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.77
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.77
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.77
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.77
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.77
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.77
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.77
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.77
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.77
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.77
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.77
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.77
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.77
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.77
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.77
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.77
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.77
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.77
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.77
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.77
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.77
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.77
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.77
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.77
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.77
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.77
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.77
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.77
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.77
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.77
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.77
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.77
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.77
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.77
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.77
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.77
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.77
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.77
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.77
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.77
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.77
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.77
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.77
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.77
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.77
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.77
PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 257.15/79.77
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.77
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.77
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.77
S(ok(z0)) → c34(S(z0)) 257.15/79.77
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.77
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.77
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.77
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.77
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.77
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.77
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.77
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.77
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.77
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.77
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.77
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.77
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.77
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.77
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.77
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.77
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.77
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.77
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.77
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.77
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.77
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.77
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.77
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.77
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.77
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.77
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.77
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.77
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.77
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.77
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.77
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.77
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.77
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.77
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.77
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.77
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.77
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.77
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.77
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.77
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.77
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.77
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.77
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.77
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.77
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.77
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.77
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.77
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.77
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.77
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.77
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.77
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.77
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.77
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.77
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.77
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.77
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.77
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.77
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.77
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.77
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.77
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.77
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.77
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.77
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.77
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.77
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.77
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.77
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.77
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.77
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.77
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c27, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26

257.15/79.78

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

Use narrowing to replace PROPER(add(z0, z1)) → c27(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.78
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.15/79.78
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.15/79.78
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.78
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.15/79.78
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.78
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.78
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.15/79.78
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.78
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.15/79.78
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.15/79.78
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.78
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.15/79.78
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.78
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.15/79.78
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

257.15/79.78

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.78
active(and(false, z0)) → mark(false) 257.15/79.78
active(if(true, z0, z1)) → mark(z0) 257.15/79.78
active(if(false, z0, z1)) → mark(z1) 257.15/79.78
active(add(0, z0)) → mark(z0) 257.15/79.78
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.78
active(first(0, z0)) → mark(nil) 257.15/79.78
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.78
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.78
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.78
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.78
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.78
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.78
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.78
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.78
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.78
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.78
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.78
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.78
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.78
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.78
proper(true) → ok(true) 257.15/79.78
proper(false) → ok(false) 257.15/79.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.78
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.78
proper(0) → ok(0) 257.15/79.78
proper(s(z0)) → s(proper(z0)) 257.15/79.78
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.78
proper(nil) → ok(nil) 257.15/79.78
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.78
proper(from(z0)) → from(proper(z0)) 257.15/79.78
s(ok(z0)) → ok(s(z0)) 257.15/79.78
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.78
from(ok(z0)) → ok(from(z0)) 257.15/79.78
top(mark(z0)) → top(proper(z0)) 257.15/79.78
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.78
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.78
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.78
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.78
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.78
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.78
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.78
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.78
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.78
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.78
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.78
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 257.15/79.78
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.78
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.78
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.78
S(ok(z0)) → c34(S(z0)) 257.15/79.78
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.78
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.78
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.78
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.78
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.78
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.78
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.78
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.78
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.78
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.78
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.78
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.78
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.78
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.78
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.78
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.78
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.78
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.78
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.78
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.78
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.78
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.78
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.78
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.78
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.78
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.78
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.78
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.78
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.78
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.78
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.78
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.78
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.78
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.78
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.78
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.78
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.15/79.78
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.15/79.78
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.78
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.15/79.78
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.78
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.78
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.15/79.78
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.78
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.15/79.78
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.15/79.78
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.78
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.15/79.78
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.78
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.15/79.78
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.78
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.78
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.78
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.78
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.78
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.78
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.78
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.78
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.78
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.78
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.78
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 257.15/79.78
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.78
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.78
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.78
S(ok(z0)) → c34(S(z0)) 257.15/79.78
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.78
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.78
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.78
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.78
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.78
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.78
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.78
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.78
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.78
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.78
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.78
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.78
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.78
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.78
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.78
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.78
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.78
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.78
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.78
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.78
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.78
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.78
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.78
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.78
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.78
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.78
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.78
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.78
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.78
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.78
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.78
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.78
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.78
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.78
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.78
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.78
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.15/79.78
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.15/79.78
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.78
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.15/79.78
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.78
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.78
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.15/79.78
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.78
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.15/79.78
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.15/79.78
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.78
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.15/79.78
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.78
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.15/79.78
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27

257.15/79.78

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

Removed 8 trailing tuple parts

257.15/79.78

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.78
active(and(false, z0)) → mark(false) 257.15/79.78
active(if(true, z0, z1)) → mark(z0) 257.15/79.78
active(if(false, z0, z1)) → mark(z1) 257.15/79.78
active(add(0, z0)) → mark(z0) 257.15/79.78
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.78
active(first(0, z0)) → mark(nil) 257.15/79.78
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.78
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.78
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.78
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.78
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.78
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.78
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.78
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.78
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.78
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.78
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.78
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.78
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.78
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.78
proper(true) → ok(true) 257.15/79.78
proper(false) → ok(false) 257.15/79.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.78
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.78
proper(0) → ok(0) 257.15/79.78
proper(s(z0)) → s(proper(z0)) 257.15/79.78
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.78
proper(nil) → ok(nil) 257.15/79.78
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.78
proper(from(z0)) → from(proper(z0)) 257.15/79.78
s(ok(z0)) → ok(s(z0)) 257.15/79.78
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.78
from(ok(z0)) → ok(from(z0)) 257.15/79.78
top(mark(z0)) → top(proper(z0)) 257.15/79.78
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.78
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.78
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.78
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.78
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.78
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.78
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.78
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.78
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.78
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.78
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.78
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 257.15/79.78
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.78
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.78
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.78
S(ok(z0)) → c34(S(z0)) 257.15/79.78
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.78
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.78
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.78
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.78
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.78
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.78
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.78
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.78
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.78
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.78
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.78
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.78
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.78
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.78
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.78
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.78
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.78
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.78
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.78
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.78
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.78
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.78
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.78
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.78
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.78
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.78
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.78
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.78
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.78
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.78
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.78
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.78
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.78
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.78
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.78
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.78
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.78
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.78
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.78
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.78
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.78
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.78
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.78
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.78
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.78
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.78
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.78
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.78
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.78
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.78
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.78
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.78
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 257.15/79.79
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c29, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27

257.15/79.79

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

Use narrowing to replace PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) by

PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true)), PROPER(true)) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false)), PROPER(false)) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0)), PROPER(0)) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil)), PROPER(nil)) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0)))

257.15/79.79

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.79
active(and(false, z0)) → mark(false) 257.15/79.79
active(if(true, z0, z1)) → mark(z0) 257.15/79.79
active(if(false, z0, z1)) → mark(z1) 257.15/79.79
active(add(0, z0)) → mark(z0) 257.15/79.79
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.79
active(first(0, z0)) → mark(nil) 257.15/79.79
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.79
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.79
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.79
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.79
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.79
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.79
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.79
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.79
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.79
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.79
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.79
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.79
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.79
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.79
proper(true) → ok(true) 257.15/79.79
proper(false) → ok(false) 257.15/79.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.79
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.79
proper(0) → ok(0) 257.15/79.79
proper(s(z0)) → s(proper(z0)) 257.15/79.79
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.79
proper(nil) → ok(nil) 257.15/79.79
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.79
proper(from(z0)) → from(proper(z0)) 257.15/79.79
s(ok(z0)) → ok(s(z0)) 257.15/79.79
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.79
from(ok(z0)) → ok(from(z0)) 257.15/79.79
top(mark(z0)) → top(proper(z0)) 257.15/79.79
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true)), PROPER(true)) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false)), PROPER(false)) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0)), PROPER(0)) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil)), PROPER(nil)) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true)), PROPER(true)) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false)), PROPER(false)) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0)), PROPER(0)) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil)), PROPER(nil)) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29

257.15/79.79

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

Removed 4 trailing tuple parts

257.15/79.79

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.79
active(and(false, z0)) → mark(false) 257.15/79.79
active(if(true, z0, z1)) → mark(z0) 257.15/79.79
active(if(false, z0, z1)) → mark(z1) 257.15/79.79
active(add(0, z0)) → mark(z0) 257.15/79.79
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.79
active(first(0, z0)) → mark(nil) 257.15/79.79
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.79
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.79
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.79
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.79
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.79
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.79
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.79
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.79
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.79
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.79
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.79
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.79
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.79
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.79
proper(true) → ok(true) 257.15/79.79
proper(false) → ok(false) 257.15/79.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.79
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.79
proper(0) → ok(0) 257.15/79.79
proper(s(z0)) → s(proper(z0)) 257.15/79.79
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.79
proper(nil) → ok(nil) 257.15/79.79
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.79
proper(from(z0)) → from(proper(z0)) 257.15/79.79
s(ok(z0)) → ok(s(z0)) 257.15/79.79
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.79
from(ok(z0)) → ok(from(z0)) 257.15/79.79
top(mark(z0)) → top(proper(z0)) 257.15/79.79
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true))) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true))) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c30, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29

257.15/79.79

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

Use narrowing to replace PROPER(first(z0, z1)) → c30(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.15/79.79
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.15/79.79
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.15/79.79
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.15/79.79
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.15/79.79
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.15/79.79
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.15/79.79
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.15/79.79
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

257.15/79.79

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.79
active(and(false, z0)) → mark(false) 257.15/79.79
active(if(true, z0, z1)) → mark(z0) 257.15/79.79
active(if(false, z0, z1)) → mark(z1) 257.15/79.79
active(add(0, z0)) → mark(z0) 257.15/79.79
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.79
active(first(0, z0)) → mark(nil) 257.15/79.79
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.79
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.79
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.79
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.79
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.79
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.79
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.79
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.79
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.79
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.79
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.79
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.79
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.79
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.79
proper(true) → ok(true) 257.15/79.79
proper(false) → ok(false) 257.15/79.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.79
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.79
proper(0) → ok(0) 257.15/79.79
proper(s(z0)) → s(proper(z0)) 257.15/79.79
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.79
proper(nil) → ok(nil) 257.15/79.79
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.79
proper(from(z0)) → from(proper(z0)) 257.15/79.79
s(ok(z0)) → ok(s(z0)) 257.15/79.79
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.79
from(ok(z0)) → ok(from(z0)) 257.15/79.79
top(mark(z0)) → top(proper(z0)) 257.15/79.79
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true))) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil))) 257.15/79.79
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.15/79.79
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.15/79.79
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.15/79.79
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.15/79.79
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.15/79.79
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.15/79.79
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.15/79.79
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.15/79.79
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.79
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.79
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.79
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.79
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.79
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.79
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.79
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.79
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.79
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.79
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.79
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.79
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.79
S(ok(z0)) → c34(S(z0)) 257.15/79.79
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.79
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.79
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.79
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.79
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.79
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.79
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.79
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.79
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.79
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.79
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.79
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.79
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.79
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.79
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.79
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.79
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.79
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.79
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.79
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.79
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.79
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.79
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.79
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.79
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.79
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.79
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.79
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.79
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.79
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.79
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.79
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.79
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.79
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.79
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.79
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.15/79.79
PROPER(s(true)) → c29(S(ok(true))) 257.15/79.79
PROPER(s(false)) → c29(S(ok(false))) 257.15/79.79
PROPER(s(0)) → c29(S(ok(0))) 257.15/79.79
PROPER(s(nil)) → c29(S(ok(nil))) 257.15/79.79
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.79
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.15/79.79
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.15/79.79
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.79
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.79
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.15/79.79
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.79
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.79
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.15/79.79
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.79
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.79
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.15/79.79
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.15/79.79
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.79
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.79
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.15/79.79
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.79
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.15/79.80
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30

257.15/79.80

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

Removed 8 trailing tuple parts

257.15/79.80

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.15/79.80
active(and(false, z0)) → mark(false) 257.15/79.80
active(if(true, z0, z1)) → mark(z0) 257.15/79.80
active(if(false, z0, z1)) → mark(z1) 257.15/79.80
active(add(0, z0)) → mark(z0) 257.15/79.80
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.15/79.80
active(first(0, z0)) → mark(nil) 257.15/79.80
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.15/79.80
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.15/79.80
active(and(z0, z1)) → and(active(z0), z1) 257.15/79.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.15/79.80
active(add(z0, z1)) → add(active(z0), z1) 257.15/79.80
active(first(z0, z1)) → first(active(z0), z1) 257.15/79.80
active(first(z0, z1)) → first(z0, active(z1)) 257.15/79.80
and(mark(z0), z1) → mark(and(z0, z1)) 257.15/79.80
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.15/79.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.15/79.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.15/79.80
add(mark(z0), z1) → mark(add(z0, z1)) 257.15/79.80
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.15/79.80
first(mark(z0), z1) → mark(first(z0, z1)) 257.15/79.80
first(z0, mark(z1)) → mark(first(z0, z1)) 257.15/79.80
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.15/79.80
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.15/79.80
proper(true) → ok(true) 257.15/79.80
proper(false) → ok(false) 257.15/79.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.15/79.80
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.15/79.80
proper(0) → ok(0) 257.15/79.80
proper(s(z0)) → s(proper(z0)) 257.15/79.80
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.15/79.80
proper(nil) → ok(nil) 257.15/79.80
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.15/79.80
proper(from(z0)) → from(proper(z0)) 257.15/79.80
s(ok(z0)) → ok(s(z0)) 257.15/79.80
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.15/79.80
from(ok(z0)) → ok(from(z0)) 257.15/79.80
top(mark(z0)) → top(proper(z0)) 257.15/79.80
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.80
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.80
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.80
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.80
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.80
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.80
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.80
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.80
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.80
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.80
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.80
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.80
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.80
S(ok(z0)) → c34(S(z0)) 257.15/79.80
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.80
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.80
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.15/79.80
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.80
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.80
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.80
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.80
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.80
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.80
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.80
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.80
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.80
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.80
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.15/79.80
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.80
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.15/79.80
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.15/79.80
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.80
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.15/79.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.15/79.80
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.15/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.15/79.80
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.15/79.80
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.15/79.80
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.80
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.15/79.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.15/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.15/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.15/79.80
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.80
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.80
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.15/79.80
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.15/79.80
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.15/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.15/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.15/79.80
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.15/79.80
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.15/79.80
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.80
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.15/79.80
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.80
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.15/79.80
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.15/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.15/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.15/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.15/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.15/79.80
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.15/79.80
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.15/79.80
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.80
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.80
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.80
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.80
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.80
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.80
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.80
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.80
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.80
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.80
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.15/79.80
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.15/79.80
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.15/79.80
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.80
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.15/79.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.15/79.80
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.15/79.80
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.15/79.80
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.15/79.80
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.15/79.80
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.15/79.80
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.15/79.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.15/79.80
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.15/79.80
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.15/79.80
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.15/79.80
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.15/79.80
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.15/79.80
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.15/79.80
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.15/79.80
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.15/79.80
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.15/79.80
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.80
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.80
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.80
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.15/79.80
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.15/79.80
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.80
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.80
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.80
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.80
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.80
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.80
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.80
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.80
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.80
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.80
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.15/79.80
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.15/79.80
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.15/79.80
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.80
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.15/79.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.15/79.80
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.15/79.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.15/79.80
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.15/79.80
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.15/79.80
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.15/79.80
PROPER(s(true)) → c29(S(ok(true))) 257.15/79.80
PROPER(s(false)) → c29(S(ok(false))) 257.15/79.80
PROPER(s(0)) → c29(S(ok(0))) 257.15/79.80
PROPER(s(nil)) → c29(S(ok(nil))) 257.15/79.80
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.15/79.80
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.15/79.80
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.15/79.80
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.15/79.80
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.15/79.80
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.15/79.80
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.15/79.80
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.15/79.80
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.15/79.80
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.15/79.80
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.15/79.80
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.15/79.80
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.15/79.80
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.15/79.80
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.15/79.80
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.15/79.80
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.15/79.80
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.15/79.80
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.15/79.80
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.15/79.80
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.15/79.80
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.15/79.80
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.15/79.80
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.15/79.80
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.15/79.80
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.15/79.80
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.15/79.80
PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.15/79.80
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.15/79.80
S(ok(z0)) → c34(S(z0)) 257.15/79.80
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.15/79.80
FROM(ok(z0)) → c36(FROM(z0)) 257.15/79.80
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.15/79.80
ACTIVE(from(z0)) → c8(S(z0)) 257.15/79.80
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.15/79.80
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.15/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.15/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.15/79.80
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.15/79.80
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.15/79.80
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.80
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.80
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.80
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.80
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.80
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.80
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.80
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.80
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.80
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.80
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.80
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.80
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.80
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c32, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30

257.52/79.80

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

Use narrowing to replace PROPER(cons(z0, z1)) → c32(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.52/79.80
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.52/79.80
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.52/79.80
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.52/79.80
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.52/79.80
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.52/79.80
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.52/79.80
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.52/79.80
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

257.52/79.80

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.80
active(and(false, z0)) → mark(false) 257.52/79.80
active(if(true, z0, z1)) → mark(z0) 257.52/79.80
active(if(false, z0, z1)) → mark(z1) 257.52/79.80
active(add(0, z0)) → mark(z0) 257.52/79.80
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.80
active(first(0, z0)) → mark(nil) 257.52/79.80
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.80
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.80
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.80
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.80
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.80
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.80
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.80
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.80
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.80
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.80
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.80
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.80
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.80
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.80
proper(true) → ok(true) 257.52/79.80
proper(false) → ok(false) 257.52/79.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.80
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.80
proper(0) → ok(0) 257.52/79.80
proper(s(z0)) → s(proper(z0)) 257.52/79.80
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.80
proper(nil) → ok(nil) 257.52/79.80
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.80
proper(from(z0)) → from(proper(z0)) 257.52/79.80
s(ok(z0)) → ok(s(z0)) 257.52/79.80
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.80
from(ok(z0)) → ok(from(z0)) 257.52/79.80
top(mark(z0)) → top(proper(z0)) 257.52/79.80
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.80
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.80
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.80
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.80
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.80
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.80
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.80
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.80
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.80
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.80
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.80
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.52/79.80
S(ok(z0)) → c34(S(z0)) 257.52/79.80
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.80
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.80
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.52/79.80
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.80
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.80
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.80
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.80
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.80
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.80
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.80
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.80
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.80
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.80
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.80
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.80
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.80
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.80
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.80
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.52/79.80
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.52/79.80
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.52/79.80
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.52/79.80
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.52/79.80
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.52/79.80
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.52/79.80
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.52/79.80
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.80
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.80
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.80
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.80
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.80
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.80
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.80
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.80
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.80
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.80
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.80
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.52/79.80
S(ok(z0)) → c34(S(z0)) 257.52/79.80
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.80
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.80
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.80
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.80
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.80
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.80
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.80
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.80
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.80
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.80
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.80
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.80
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.80
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.80
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.80
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.80
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.80
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 257.52/79.80
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 257.52/79.80
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 257.52/79.80
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 257.52/79.80
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 257.52/79.80
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 257.52/79.80
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 257.52/79.80
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 257.52/79.80
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32

257.52/79.80

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

Removed 8 trailing tuple parts

257.52/79.80

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.80
active(and(false, z0)) → mark(false) 257.52/79.80
active(if(true, z0, z1)) → mark(z0) 257.52/79.80
active(if(false, z0, z1)) → mark(z1) 257.52/79.80
active(add(0, z0)) → mark(z0) 257.52/79.80
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.80
active(first(0, z0)) → mark(nil) 257.52/79.80
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.80
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.80
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.80
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.80
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.80
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.80
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.80
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.80
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.80
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.80
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.80
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.80
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.80
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.80
proper(true) → ok(true) 257.52/79.80
proper(false) → ok(false) 257.52/79.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.80
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.80
proper(0) → ok(0) 257.52/79.80
proper(s(z0)) → s(proper(z0)) 257.52/79.80
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.80
proper(nil) → ok(nil) 257.52/79.80
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.80
proper(from(z0)) → from(proper(z0)) 257.52/79.80
s(ok(z0)) → ok(s(z0)) 257.52/79.80
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.80
from(ok(z0)) → ok(from(z0)) 257.52/79.80
top(mark(z0)) → top(proper(z0)) 257.52/79.80
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.80
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.80
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.80
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.80
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.80
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.80
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.80
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.80
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.80
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.80
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.80
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.52/79.80
S(ok(z0)) → c34(S(z0)) 257.52/79.80
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.80
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.80
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.52/79.80
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.80
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.80
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.80
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.80
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.80
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.80
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.80
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.80
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.80
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.80
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.80
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.80
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.80
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.80
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.80
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.80
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.80
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.80
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.80
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.80
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.80
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.80
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.80
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.80
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.80
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.80
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.80
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.80
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.80
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.80
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.80
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.80
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.80
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.80
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.80
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.80
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.80
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.80
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.80
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.80
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.80
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.80
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.80
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.80
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.80
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.80
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.80
PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) 257.52/79.80
S(ok(z0)) → c34(S(z0)) 257.52/79.80
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.80
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.80
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.80
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.80
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.80
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.80
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.80
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.80
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.80
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.80
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.80
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.80
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.80
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.80
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.80
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.80
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.80
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.80
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.80
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.80
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.80
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.80
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.81
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.81
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.81
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.81
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.81
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.81
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, PROPER, S, CONS, FROM, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c33, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32, c32

257.52/79.81

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

Use narrowing to replace PROPER(from(z0)) → c33(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(from(true)) → c33(FROM(ok(true)), PROPER(true)) 257.52/79.81
PROPER(from(false)) → c33(FROM(ok(false)), PROPER(false)) 257.52/79.81
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(from(0)) → c33(FROM(ok(0)), PROPER(0)) 257.52/79.81
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(from(nil)) → c33(FROM(ok(nil)), PROPER(nil)) 257.52/79.81
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0)))

257.52/79.81

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.81
active(and(false, z0)) → mark(false) 257.52/79.81
active(if(true, z0, z1)) → mark(z0) 257.52/79.81
active(if(false, z0, z1)) → mark(z1) 257.52/79.81
active(add(0, z0)) → mark(z0) 257.52/79.81
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.81
active(first(0, z0)) → mark(nil) 257.52/79.81
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.81
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.81
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.81
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.81
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.81
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.81
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.81
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.81
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.81
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.81
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.81
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.81
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.81
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.81
proper(true) → ok(true) 257.52/79.81
proper(false) → ok(false) 257.52/79.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.81
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.81
proper(0) → ok(0) 257.52/79.81
proper(s(z0)) → s(proper(z0)) 257.52/79.81
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.81
proper(nil) → ok(nil) 257.52/79.81
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.81
proper(from(z0)) → from(proper(z0)) 257.52/79.81
s(ok(z0)) → ok(s(z0)) 257.52/79.81
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.81
from(ok(z0)) → ok(from(z0)) 257.52/79.81
top(mark(z0)) → top(proper(z0)) 257.52/79.81
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.81
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.81
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.81
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.81
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.81
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.81
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.81
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.81
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.81
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.81
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.81
S(ok(z0)) → c34(S(z0)) 257.52/79.81
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.81
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.81
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.52/79.81
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.81
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.81
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.81
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.81
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.81
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.81
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.81
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.81
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.81
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.81
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.81
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.81
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.81
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(from(true)) → c33(FROM(ok(true)), PROPER(true)) 257.52/79.81
PROPER(from(false)) → c33(FROM(ok(false)), PROPER(false)) 257.52/79.81
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(from(0)) → c33(FROM(ok(0)), PROPER(0)) 257.52/79.81
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(from(nil)) → c33(FROM(ok(nil)), PROPER(nil)) 257.52/79.81
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.81
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.81
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.81
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.81
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.81
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.81
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.81
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.81
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.81
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.81
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.81
S(ok(z0)) → c34(S(z0)) 257.52/79.81
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.81
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.81
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.81
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.81
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.81
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.81
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.81
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.81
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.81
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.81
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.81
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.81
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.81
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.81
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.81
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(from(true)) → c33(FROM(ok(true)), PROPER(true)) 257.52/79.81
PROPER(from(false)) → c33(FROM(ok(false)), PROPER(false)) 257.52/79.81
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(from(0)) → c33(FROM(ok(0)), PROPER(0)) 257.52/79.81
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(from(nil)) → c33(FROM(ok(nil)), PROPER(nil)) 257.52/79.81
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32, c32, c33

257.52/79.81

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

Removed 4 trailing tuple parts

257.52/79.81

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.81
active(and(false, z0)) → mark(false) 257.52/79.81
active(if(true, z0, z1)) → mark(z0) 257.52/79.81
active(if(false, z0, z1)) → mark(z1) 257.52/79.81
active(add(0, z0)) → mark(z0) 257.52/79.81
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.81
active(first(0, z0)) → mark(nil) 257.52/79.81
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.81
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.81
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.81
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.81
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.81
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.81
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.81
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.81
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.81
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.81
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.81
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.81
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.81
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.81
proper(true) → ok(true) 257.52/79.81
proper(false) → ok(false) 257.52/79.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.81
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.81
proper(0) → ok(0) 257.52/79.81
proper(s(z0)) → s(proper(z0)) 257.52/79.81
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.81
proper(nil) → ok(nil) 257.52/79.81
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.81
proper(from(z0)) → from(proper(z0)) 257.52/79.81
s(ok(z0)) → ok(s(z0)) 257.52/79.81
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.81
from(ok(z0)) → ok(from(z0)) 257.52/79.81
top(mark(z0)) → top(proper(z0)) 257.52/79.81
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.81
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.81
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.81
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.81
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.81
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.81
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.81
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.81
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.81
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.81
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.81
S(ok(z0)) → c34(S(z0)) 257.52/79.81
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.81
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.81
TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) 257.52/79.81
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.81
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.81
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.81
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.81
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.81
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.81
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.81
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.81
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.81
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.81
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.81
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.81
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.81
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.81
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.81
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.81
PROPER(from(nil)) → c33(FROM(ok(nil)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.81
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.81
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.81
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.81
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.81
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.81
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.81
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.81
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.81
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.81
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.81
S(ok(z0)) → c34(S(z0)) 257.52/79.81
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.81
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.81
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.81
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.81
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.81
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.81
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.81
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.81
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.81
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.81
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.81
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.81
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.81
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.81
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.81
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.81
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.81
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.81
PROPER(from(nil)) → c33(FROM(ok(nil)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c37, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32, c32, c33, c33

257.52/79.81

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

Use narrowing to replace TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(and(z0, z1))) → c37(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
TOP(mark(true)) → c37(TOP(ok(true)), PROPER(true)) 257.52/79.81
TOP(mark(false)) → c37(TOP(ok(false)), PROPER(false)) 257.52/79.81
TOP(mark(if(z0, z1, z2))) → c37(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
TOP(mark(0)) → c37(TOP(ok(0)), PROPER(0)) 257.52/79.81
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
TOP(mark(first(z0, z1))) → c37(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
TOP(mark(nil)) → c37(TOP(ok(nil)), PROPER(nil)) 257.52/79.81
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
TOP(mark(from(z0))) → c37(TOP(from(proper(z0))), PROPER(from(z0)))

257.52/79.81

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.81
active(and(false, z0)) → mark(false) 257.52/79.81
active(if(true, z0, z1)) → mark(z0) 257.52/79.81
active(if(false, z0, z1)) → mark(z1) 257.52/79.81
active(add(0, z0)) → mark(z0) 257.52/79.81
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.81
active(first(0, z0)) → mark(nil) 257.52/79.81
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.81
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.81
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.81
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.81
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.81
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.81
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.81
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.81
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.81
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.81
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.81
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.81
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.81
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.81
proper(true) → ok(true) 257.52/79.81
proper(false) → ok(false) 257.52/79.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.81
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.81
proper(0) → ok(0) 257.52/79.81
proper(s(z0)) → s(proper(z0)) 257.52/79.81
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.81
proper(nil) → ok(nil) 257.52/79.81
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.81
proper(from(z0)) → from(proper(z0)) 257.52/79.81
s(ok(z0)) → ok(s(z0)) 257.52/79.81
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.81
from(ok(z0)) → ok(from(z0)) 257.52/79.81
top(mark(z0)) → top(proper(z0)) 257.52/79.81
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.81
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.81
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.81
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.81
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.81
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.81
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.81
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.81
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.81
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.81
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.81
S(ok(z0)) → c34(S(z0)) 257.52/79.81
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.81
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.81
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.81
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.81
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.81
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.81
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.81
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.81
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.81
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.81
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.81
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.81
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.81
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.81
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.81
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.81
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.81
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.81
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.81
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.81
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.81
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.81
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.81
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.81
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.81
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.81
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.81
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.81
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.81
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.81
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.81
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.81
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.81
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.81
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.81
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.81
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.81
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.81
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.81
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.81
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.81
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.81
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.81
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.81
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.81
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.81
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.81
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.81
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.81
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.81
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.81
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.81
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.82
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.82
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.82
PROPER(from(nil)) → c33(FROM(ok(nil))) 257.52/79.82
TOP(mark(and(z0, z1))) → c37(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
TOP(mark(true)) → c37(TOP(ok(true)), PROPER(true)) 257.52/79.82
TOP(mark(false)) → c37(TOP(ok(false)), PROPER(false)) 257.52/79.82
TOP(mark(if(z0, z1, z2))) → c37(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
TOP(mark(0)) → c37(TOP(ok(0)), PROPER(0)) 257.52/79.82
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
TOP(mark(first(z0, z1))) → c37(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
TOP(mark(nil)) → c37(TOP(ok(nil)), PROPER(nil)) 257.52/79.82
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
TOP(mark(from(z0))) → c37(TOP(from(proper(z0))), PROPER(from(z0)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.82
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.82
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.82
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.82
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.82
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.82
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.82
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.82
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.82
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.82
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.82
S(ok(z0)) → c34(S(z0)) 257.52/79.82
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.82
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.82
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.82
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.82
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.82
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.82
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.82
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.82
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.82
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.82
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.82
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.82
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.82
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.82
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.82
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.82
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.82
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.82
PROPER(from(nil)) → c33(FROM(ok(nil)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32, c32, c33, c33, c37

257.52/79.82

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

Removed 4 trailing tuple parts

257.52/79.82

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.82
active(and(false, z0)) → mark(false) 257.52/79.82
active(if(true, z0, z1)) → mark(z0) 257.52/79.82
active(if(false, z0, z1)) → mark(z1) 257.52/79.82
active(add(0, z0)) → mark(z0) 257.52/79.82
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.82
active(first(0, z0)) → mark(nil) 257.52/79.82
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.82
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.82
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.82
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.82
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.82
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.82
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.82
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.82
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.82
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.82
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.82
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.82
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.82
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.82
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.82
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.82
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.82
proper(true) → ok(true) 257.52/79.82
proper(false) → ok(false) 257.52/79.82
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.82
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.82
proper(0) → ok(0) 257.52/79.82
proper(s(z0)) → s(proper(z0)) 257.52/79.82
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.82
proper(nil) → ok(nil) 257.52/79.82
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.82
proper(from(z0)) → from(proper(z0)) 257.52/79.82
s(ok(z0)) → ok(s(z0)) 257.52/79.82
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.82
from(ok(z0)) → ok(from(z0)) 257.52/79.82
top(mark(z0)) → top(proper(z0)) 257.52/79.82
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.82
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.82
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.82
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.82
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.82
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.82
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.82
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.82
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.82
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.82
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.82
S(ok(z0)) → c34(S(z0)) 257.52/79.82
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.82
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.82
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.82
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.82
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.82
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.82
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.82
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.82
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.82
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.82
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.82
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.82
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.82
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.82
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.82
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.82
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.82
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.82
PROPER(from(nil)) → c33(FROM(ok(nil))) 257.52/79.82
TOP(mark(and(z0, z1))) → c37(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
TOP(mark(if(z0, z1, z2))) → c37(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
TOP(mark(first(z0, z1))) → c37(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
TOP(mark(from(z0))) → c37(TOP(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
TOP(mark(true)) → c37(TOP(ok(true))) 257.52/79.82
TOP(mark(false)) → c37(TOP(ok(false))) 257.52/79.82
TOP(mark(0)) → c37(TOP(ok(0))) 257.52/79.82
TOP(mark(nil)) → c37(TOP(ok(nil)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.82
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.82
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.82
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.82
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.82
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.82
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.82
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.82
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.82
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.82
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.82
S(ok(z0)) → c34(S(z0)) 257.52/79.82
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.82
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.82
TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) 257.52/79.82
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.82
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.82
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.82
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.82
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.82
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.82
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.82
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.82
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.82
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.82
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.82
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.82
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.82
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.82
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.82
PROPER(from(nil)) → c33(FROM(ok(nil)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c38, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32, c32, c33, c33, c37, c37

257.52/79.82

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

Use narrowing to replace TOP(ok(z0)) → c38(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(and(true, z0))) → c38(TOP(mark(z0)), ACTIVE(and(true, z0))) 257.52/79.82
TOP(ok(and(false, z0))) → c38(TOP(mark(false)), ACTIVE(and(false, z0))) 257.52/79.82
TOP(ok(if(true, z0, z1))) → c38(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 257.52/79.82
TOP(ok(if(false, z0, z1))) → c38(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 257.52/79.82
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0))) 257.52/79.82
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 257.52/79.82
TOP(ok(first(0, z0))) → c38(TOP(mark(nil)), ACTIVE(first(0, z0))) 257.52/79.82
TOP(ok(first(s(z0), cons(z1, z2)))) → c38(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
TOP(ok(from(z0))) → c38(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 257.52/79.82
TOP(ok(and(z0, z1))) → c38(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 257.52/79.82
TOP(ok(if(z0, z1, z2))) → c38(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 257.52/79.82
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 257.52/79.82
TOP(ok(first(z0, z1))) → c38(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 257.52/79.82
TOP(ok(first(z0, z1))) → c38(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))

257.52/79.82

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.82
active(and(false, z0)) → mark(false) 257.52/79.82
active(if(true, z0, z1)) → mark(z0) 257.52/79.82
active(if(false, z0, z1)) → mark(z1) 257.52/79.82
active(add(0, z0)) → mark(z0) 257.52/79.82
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.82
active(first(0, z0)) → mark(nil) 257.52/79.82
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.82
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.82
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.82
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.82
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.82
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.82
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.82
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.82
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.82
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.82
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.82
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.82
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.82
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.82
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.82
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.82
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.82
proper(true) → ok(true) 257.52/79.82
proper(false) → ok(false) 257.52/79.82
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.82
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.82
proper(0) → ok(0) 257.52/79.82
proper(s(z0)) → s(proper(z0)) 257.52/79.82
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.82
proper(nil) → ok(nil) 257.52/79.82
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.82
proper(from(z0)) → from(proper(z0)) 257.52/79.82
s(ok(z0)) → ok(s(z0)) 257.52/79.82
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.82
from(ok(z0)) → ok(from(z0)) 257.52/79.82
top(mark(z0)) → top(proper(z0)) 257.52/79.82
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.82
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.82
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.82
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.82
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.82
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.82
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.82
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.82
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.82
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.82
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.82
S(ok(z0)) → c34(S(z0)) 257.52/79.82
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.82
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.82
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.82
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.82
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.82
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.82
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.82
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.82
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.82
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.82
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.82
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.82
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.82
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.82
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.82
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.82
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.82
PROPER(from(nil)) → c33(FROM(ok(nil))) 257.52/79.82
TOP(mark(and(z0, z1))) → c37(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
TOP(mark(if(z0, z1, z2))) → c37(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
TOP(mark(first(z0, z1))) → c37(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
TOP(mark(from(z0))) → c37(TOP(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
TOP(mark(true)) → c37(TOP(ok(true))) 257.52/79.82
TOP(mark(false)) → c37(TOP(ok(false))) 257.52/79.82
TOP(mark(0)) → c37(TOP(ok(0))) 257.52/79.82
TOP(mark(nil)) → c37(TOP(ok(nil))) 257.52/79.82
TOP(ok(and(true, z0))) → c38(TOP(mark(z0)), ACTIVE(and(true, z0))) 257.52/79.82
TOP(ok(and(false, z0))) → c38(TOP(mark(false)), ACTIVE(and(false, z0))) 257.52/79.82
TOP(ok(if(true, z0, z1))) → c38(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 257.52/79.82
TOP(ok(if(false, z0, z1))) → c38(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 257.52/79.82
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0))) 257.52/79.82
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 257.52/79.82
TOP(ok(first(0, z0))) → c38(TOP(mark(nil)), ACTIVE(first(0, z0))) 257.52/79.82
TOP(ok(first(s(z0), cons(z1, z2)))) → c38(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
TOP(ok(from(z0))) → c38(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 257.52/79.82
TOP(ok(and(z0, z1))) → c38(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 257.52/79.82
TOP(ok(if(z0, z1, z2))) → c38(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 257.52/79.82
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 257.52/79.82
TOP(ok(first(z0, z1))) → c38(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 257.52/79.82
TOP(ok(first(z0, z1))) → c38(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))

S tuples:

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.82
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.82
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.82
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.82
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.82
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.82
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.82
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.82
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.82
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.82
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.82
S(ok(z0)) → c34(S(z0)) 257.52/79.82
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.82
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.82
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.82
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.82
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.82
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.82
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.82
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.82
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.82
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.82
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.82
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.82
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.82
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.82
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.82
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.82
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.82
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.82
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.82
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.82
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.82
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.82
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.82
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.82
PROPER(from(nil)) → c33(FROM(ok(nil))) 257.52/79.82
TOP(ok(and(true, z0))) → c38(TOP(mark(z0)), ACTIVE(and(true, z0))) 257.52/79.82
TOP(ok(and(false, z0))) → c38(TOP(mark(false)), ACTIVE(and(false, z0))) 257.52/79.82
TOP(ok(if(true, z0, z1))) → c38(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 257.52/79.82
TOP(ok(if(false, z0, z1))) → c38(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 257.52/79.82
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0))) 257.52/79.82
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 257.52/79.82
TOP(ok(first(0, z0))) → c38(TOP(mark(nil)), ACTIVE(first(0, z0))) 257.52/79.82
TOP(ok(first(s(z0), cons(z1, z2)))) → c38(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
TOP(ok(from(z0))) → c38(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 257.52/79.82
TOP(ok(and(z0, z1))) → c38(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 257.52/79.82
TOP(ok(if(z0, z1, z2))) → c38(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 257.52/79.82
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 257.52/79.82
TOP(ok(first(z0, z1))) → c38(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 257.52/79.82
TOP(ok(first(z0, z1))) → c38(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))

K tuples:

TOP(mark(z0)) → c37(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

ACTIVE, AND, IF, ADD, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c8, c, c9, c9, c1, c10, c10, c2, c11, c11, c3, c23, c23, c26, c26, c27, c27, c29, c29, c30, c30, c32, c32, c33, c33, c37, c37, c38

257.52/79.82

(61) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(first(z0, z1)) → c12(FIRST(active(z0), z1), ACTIVE(z0)) 257.52/79.82
ACTIVE(first(z0, z1)) → c13(FIRST(z0, active(z1)), ACTIVE(z1)) 257.52/79.82
ACTIVE(from(z0)) → c8(S(z0)) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(S(add(z0, z1))) 257.52/79.82
ACTIVE(add(s(z0), z1)) → c(ADD(z0, z1)) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 257.52/79.82
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 257.52/79.82
ACTIVE(and(first(0, z0), x1)) → c9(AND(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(and(first(s(z0), cons(z1, z2)), x1)) → c9(AND(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(and(and(z0, z1), x1)) → c9(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(and(if(z0, z1, z2), x1)) → c9(AND(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(and(add(z0, z1), x1)) → c9(AND(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(first(z0, z1), x1)) → c9(AND(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(and(true, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(and(false, z0), x1)) → c9(AND(mark(false), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(AND(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(and(from(z0), x1)) → c1(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(if(first(0, z0), x1, x2)) → c10(IF(mark(nil), x1, x2), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(if(first(s(z0), cons(z1, z2)), x1, x2)) → c10(IF(mark(cons(z1, first(z0, z2))), x1, x2), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(if(and(z0, z1), x1, x2)) → c10(IF(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c10(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(if(add(z0, z1), x1, x2)) → c10(IF(add(active(z0), z1), x1, x2), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(active(z0), z1), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(if(first(z0, z1), x1, x2)) → c10(IF(first(z0, active(z1)), x1, x2), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(if(true, z0, z1), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(and(if(false, z0, z1), x1)) → c9(AND(mark(z1), x1)) 257.52/79.82
ACTIVE(if(and(true, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(and(false, z0), x1, x2)) → c10(IF(mark(false), x1, x2)) 257.52/79.82
ACTIVE(if(if(true, z0, z1), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(if(if(false, z0, z1), x1, x2)) → c10(IF(mark(z1), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(IF(mark(cons(z0, from(s(z0)))), x1, x2)) 257.52/79.82
ACTIVE(if(from(z0), x1, x2)) → c2(ACTIVE(from(z0))) 257.52/79.82
ACTIVE(add(first(0, z0), x1)) → c11(ADD(mark(nil), x1), ACTIVE(first(0, z0))) 257.52/79.82
ACTIVE(add(first(s(z0), cons(z1, z2)), x1)) → c11(ADD(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.82
ACTIVE(add(and(z0, z1), x1)) → c11(ADD(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.52/79.82
ACTIVE(add(if(z0, z1, z2), x1)) → c11(ADD(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 257.52/79.82
ACTIVE(add(add(z0, z1), x1)) → c11(ADD(add(active(z0), z1), x1), ACTIVE(add(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(add(first(z0, z1), x1)) → c11(ADD(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 257.52/79.82
ACTIVE(and(add(0, z0), x1)) → c9(AND(mark(z0), x1)) 257.52/79.82
ACTIVE(if(add(0, z0), x1, x2)) → c10(IF(mark(z0), x1, x2)) 257.52/79.82
ACTIVE(add(and(true, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(and(false, z0), x1)) → c11(ADD(mark(false), x1)) 257.52/79.82
ACTIVE(add(if(true, z0, z1), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(add(if(false, z0, z1), x1)) → c11(ADD(mark(z1), x1)) 257.52/79.82
ACTIVE(add(add(0, z0), x1)) → c11(ADD(mark(z0), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(AND(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(and(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(IF(mark(s(add(z0, z1))), x1, x2)) 257.52/79.82
ACTIVE(if(add(s(z0), z1), x1, x2)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ADD(mark(s(add(z0, z1))), x1)) 257.52/79.82
ACTIVE(add(add(s(z0), z1), x1)) → c3(ACTIVE(add(s(z0), z1))) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ADD(mark(cons(z0, from(s(z0)))), x1)) 257.52/79.82
ACTIVE(add(from(z0), x1)) → c3(ACTIVE(from(z0))) 257.52/79.82
PROPER(and(x0, and(z0, z1))) → c23(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.82
PROPER(and(x0, if(z0, z1, z2))) → c23(AND(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(and(x0, add(z0, z1))) → c23(AND(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.82
PROPER(and(x0, s(z0))) → c23(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.82
PROPER(and(x0, first(z0, z1))) → c23(AND(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.82
PROPER(and(x0, cons(z0, z1))) → c23(AND(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(and(x0, from(z0))) → c23(AND(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.82
PROPER(and(and(z0, z1), x1)) → c23(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(if(z0, z1, z2), x1)) → c23(AND(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.82
PROPER(and(add(z0, z1), x1)) → c23(AND(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(s(z0), x1)) → c23(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(first(z0, z1), x1)) → c23(AND(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(cons(z0, z1), x1)) → c23(AND(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.82
PROPER(and(from(z0), x1)) → c23(AND(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.82
PROPER(and(x0, true)) → c23(AND(proper(x0), ok(true)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, false)) → c23(AND(proper(x0), ok(false)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, 0)) → c23(AND(proper(x0), ok(0)), PROPER(x0)) 257.52/79.82
PROPER(and(x0, nil)) → c23(AND(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.82
PROPER(and(true, x1)) → c23(AND(ok(true), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(false, x1)) → c23(AND(ok(false), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(0, x1)) → c23(AND(ok(0), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(and(nil, x1)) → c23(AND(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.82
PROPER(if(x0, x1, and(z0, z1))) → c26(IF(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, if(z0, z1, z2))) → c26(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 257.52/79.82
PROPER(if(x0, x1, add(z0, z1))) → c26(IF(proper(x0), proper(x1), add(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(add(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, s(z0))) → c26(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 257.52/79.82
PROPER(if(x0, x1, first(z0, z1))) → c26(IF(proper(x0), proper(x1), first(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(first(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, cons(z0, z1))) → c26(IF(proper(x0), proper(x1), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(cons(z0, z1))) 257.52/79.82
PROPER(if(x0, x1, from(z0))) → c26(IF(proper(x0), proper(x1), from(proper(z0))), PROPER(x0), PROPER(x1), PROPER(from(z0))) 257.52/79.82
PROPER(if(x0, and(z0, z1), x2)) → c26(IF(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, if(z0, z1, z2), x2)) → c26(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, add(z0, z1), x2)) → c26(IF(proper(x0), add(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(add(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, s(z0), x2)) → c26(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, first(z0, z1), x2)) → c26(IF(proper(x0), first(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(first(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, cons(z0, z1), x2)) → c26(IF(proper(x0), cons(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(cons(z0, z1)), PROPER(x2)) 257.52/79.82
PROPER(if(x0, from(z0), x2)) → c26(IF(proper(x0), from(proper(z0)), proper(x2)), PROPER(x0), PROPER(from(z0)), PROPER(x2)) 257.52/79.82
PROPER(if(and(z0, z1), x1, x2)) → c26(IF(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(if(z0, z1, z2), x1, x2)) → c26(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(add(z0, z1), x1, x2)) → c26(IF(add(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(add(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.82
PROPER(if(s(z0), x1, x2)) → c26(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(first(z0, z1), x1, x2)) → c26(IF(first(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(first(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(cons(z0, z1), x1, x2)) → c26(IF(cons(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(cons(z0, z1)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(from(z0), x1, x2)) → c26(IF(from(proper(z0)), proper(x1), proper(x2)), PROPER(from(z0)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(x0, x1, true)) → c26(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 257.52/79.83
PROPER(if(x0, x1, false)) → c26(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 257.52/79.83
PROPER(if(x0, x1, 0)) → c26(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 257.52/79.83
PROPER(if(x0, x1, nil)) → c26(IF(proper(x0), proper(x1), ok(nil)), PROPER(x0), PROPER(x1)) 257.52/79.83
PROPER(if(x0, true, x2)) → c26(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.83
PROPER(if(x0, false, x2)) → c26(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.83
PROPER(if(x0, 0, x2)) → c26(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.83
PROPER(if(x0, nil, x2)) → c26(IF(proper(x0), ok(nil), proper(x2)), PROPER(x0), PROPER(x2)) 257.52/79.83
PROPER(if(true, x1, x2)) → c26(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(false, x1, x2)) → c26(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(0, x1, x2)) → c26(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(if(nil, x1, x2)) → c26(IF(ok(nil), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 257.52/79.83
PROPER(add(x0, and(z0, z1))) → c27(ADD(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.83
PROPER(add(x0, if(z0, z1, z2))) → c27(ADD(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.83
PROPER(add(x0, add(z0, z1))) → c27(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.83
PROPER(add(x0, s(z0))) → c27(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.83
PROPER(add(x0, first(z0, z1))) → c27(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.83
PROPER(add(x0, cons(z0, z1))) → c27(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.83
PROPER(add(x0, from(z0))) → c27(ADD(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.83
PROPER(add(and(z0, z1), x1)) → c27(ADD(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(add(if(z0, z1, z2), x1)) → c27(ADD(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.83
PROPER(add(add(z0, z1), x1)) → c27(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(add(s(z0), x1)) → c27(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.83
PROPER(add(first(z0, z1), x1)) → c27(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(add(cons(z0, z1), x1)) → c27(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(add(from(z0), x1)) → c27(ADD(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.83
PROPER(add(x0, true)) → c27(ADD(proper(x0), ok(true)), PROPER(x0)) 257.52/79.83
PROPER(add(x0, false)) → c27(ADD(proper(x0), ok(false)), PROPER(x0)) 257.52/79.83
PROPER(add(x0, 0)) → c27(ADD(proper(x0), ok(0)), PROPER(x0)) 257.52/79.83
PROPER(add(x0, nil)) → c27(ADD(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.83
PROPER(add(true, x1)) → c27(ADD(ok(true), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(add(false, x1)) → c27(ADD(ok(false), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(add(0, x1)) → c27(ADD(ok(0), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(add(nil, x1)) → c27(ADD(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(s(and(z0, z1))) → c29(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.83
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.83
PROPER(s(add(z0, z1))) → c29(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.83
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 257.52/79.83
PROPER(s(first(z0, z1))) → c29(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.83
PROPER(s(cons(z0, z1))) → c29(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.83
PROPER(s(from(z0))) → c29(S(from(proper(z0))), PROPER(from(z0))) 257.52/79.83
PROPER(s(true)) → c29(S(ok(true))) 257.52/79.83
PROPER(s(false)) → c29(S(ok(false))) 257.52/79.83
PROPER(s(0)) → c29(S(ok(0))) 257.52/79.83
PROPER(s(nil)) → c29(S(ok(nil))) 257.52/79.83
PROPER(first(x0, and(z0, z1))) → c30(FIRST(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.83
PROPER(first(x0, if(z0, z1, z2))) → c30(FIRST(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.83
PROPER(first(x0, add(z0, z1))) → c30(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.83
PROPER(first(x0, s(z0))) → c30(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.83
PROPER(first(x0, first(z0, z1))) → c30(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.83
PROPER(first(x0, cons(z0, z1))) → c30(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.83
PROPER(first(x0, from(z0))) → c30(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.83
PROPER(first(and(z0, z1), x1)) → c30(FIRST(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(first(if(z0, z1, z2), x1)) → c30(FIRST(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.83
PROPER(first(add(z0, z1), x1)) → c30(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(first(s(z0), x1)) → c30(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.83
PROPER(first(first(z0, z1), x1)) → c30(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(first(cons(z0, z1), x1)) → c30(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(first(from(z0), x1)) → c30(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.83
PROPER(first(x0, true)) → c30(FIRST(proper(x0), ok(true)), PROPER(x0)) 257.52/79.83
PROPER(first(x0, false)) → c30(FIRST(proper(x0), ok(false)), PROPER(x0)) 257.52/79.83
PROPER(first(x0, 0)) → c30(FIRST(proper(x0), ok(0)), PROPER(x0)) 257.52/79.83
PROPER(first(x0, nil)) → c30(FIRST(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.83
PROPER(first(true, x1)) → c30(FIRST(ok(true), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(first(false, x1)) → c30(FIRST(ok(false), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(first(0, x1)) → c30(FIRST(ok(0), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(first(nil, x1)) → c30(FIRST(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(cons(x0, and(z0, z1))) → c32(CONS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 257.52/79.83
PROPER(cons(x0, if(z0, z1, z2))) → c32(CONS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 257.52/79.83
PROPER(cons(x0, add(z0, z1))) → c32(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1))) 257.52/79.83
PROPER(cons(x0, s(z0))) → c32(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 257.52/79.83
PROPER(cons(x0, first(z0, z1))) → c32(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 257.52/79.83
PROPER(cons(x0, cons(z0, z1))) → c32(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 257.52/79.83
PROPER(cons(x0, from(z0))) → c32(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 257.52/79.83
PROPER(cons(and(z0, z1), x1)) → c32(CONS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(cons(if(z0, z1, z2), x1)) → c32(CONS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 257.52/79.83
PROPER(cons(add(z0, z1), x1)) → c32(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(cons(s(z0), x1)) → c32(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 257.52/79.83
PROPER(cons(first(z0, z1), x1)) → c32(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(cons(cons(z0, z1), x1)) → c32(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 257.52/79.83
PROPER(cons(from(z0), x1)) → c32(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 257.52/79.83
PROPER(cons(x0, true)) → c32(CONS(proper(x0), ok(true)), PROPER(x0)) 257.52/79.83
PROPER(cons(x0, false)) → c32(CONS(proper(x0), ok(false)), PROPER(x0)) 257.52/79.83
PROPER(cons(x0, 0)) → c32(CONS(proper(x0), ok(0)), PROPER(x0)) 257.52/79.83
PROPER(cons(x0, nil)) → c32(CONS(proper(x0), ok(nil)), PROPER(x0)) 257.52/79.83
PROPER(cons(true, x1)) → c32(CONS(ok(true), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(cons(false, x1)) → c32(CONS(ok(false), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(cons(0, x1)) → c32(CONS(ok(0), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(cons(nil, x1)) → c32(CONS(ok(nil), proper(x1)), PROPER(x1)) 257.52/79.83
PROPER(from(and(z0, z1))) → c33(FROM(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.83
PROPER(from(if(z0, z1, z2))) → c33(FROM(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.83
PROPER(from(add(z0, z1))) → c33(FROM(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.83
PROPER(from(s(z0))) → c33(FROM(s(proper(z0))), PROPER(s(z0))) 257.52/79.83
PROPER(from(first(z0, z1))) → c33(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.83
PROPER(from(cons(z0, z1))) → c33(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.83
PROPER(from(from(z0))) → c33(FROM(from(proper(z0))), PROPER(from(z0))) 257.52/79.83
PROPER(from(true)) → c33(FROM(ok(true))) 257.52/79.83
PROPER(from(false)) → c33(FROM(ok(false))) 257.52/79.83
PROPER(from(0)) → c33(FROM(ok(0))) 257.52/79.83
PROPER(from(nil)) → c33(FROM(ok(nil))) 257.52/79.83
TOP(mark(and(z0, z1))) → c37(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 257.52/79.83
TOP(mark(if(z0, z1, z2))) → c37(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 257.52/79.83
TOP(mark(add(z0, z1))) → c37(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1))) 257.52/79.83
TOP(mark(s(z0))) → c37(TOP(s(proper(z0))), PROPER(s(z0))) 257.52/79.83
TOP(mark(first(z0, z1))) → c37(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 257.52/79.83
TOP(mark(cons(z0, z1))) → c37(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 257.52/79.83
TOP(mark(from(z0))) → c37(TOP(from(proper(z0))), PROPER(from(z0))) 257.52/79.83
TOP(ok(and(true, z0))) → c38(TOP(mark(z0)), ACTIVE(and(true, z0))) 257.52/79.83
TOP(ok(and(false, z0))) → c38(TOP(mark(false)), ACTIVE(and(false, z0))) 257.52/79.83
TOP(ok(if(true, z0, z1))) → c38(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 257.52/79.83
TOP(ok(if(false, z0, z1))) → c38(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 257.52/79.83
TOP(ok(add(0, z0))) → c38(TOP(mark(z0)), ACTIVE(add(0, z0))) 257.52/79.83
TOP(ok(add(s(z0), z1))) → c38(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1))) 257.52/79.83
TOP(ok(first(0, z0))) → c38(TOP(mark(nil)), ACTIVE(first(0, z0))) 257.52/79.83
TOP(ok(first(s(z0), cons(z1, z2)))) → c38(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 257.52/79.83
TOP(ok(from(z0))) → c38(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 257.52/79.83
TOP(ok(and(z0, z1))) → c38(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 257.52/79.83
TOP(ok(if(z0, z1, z2))) → c38(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 257.52/79.83
TOP(ok(add(z0, z1))) → c38(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1))) 257.52/79.83
TOP(ok(first(z0, z1))) → c38(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 257.52/79.83
TOP(ok(first(z0, z1))) → c38(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))

257.52/79.83

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.83
TOP(mark(true)) → c37(TOP(ok(true))) 257.52/79.83
TOP(mark(false)) → c37(TOP(ok(false))) 257.52/79.83
TOP(mark(0)) → c37(TOP(ok(0))) 257.52/79.83
TOP(mark(nil)) → c37(TOP(ok(nil)))

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:none
Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c37

257.52/79.83

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

Removed 4 trailing tuple parts

257.52/79.83

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.83
TOP(mark(true)) → c37 257.52/79.83
TOP(mark(false)) → c37 257.52/79.83
TOP(mark(0)) → c37 257.52/79.83
TOP(mark(nil)) → c37

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:none
Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36, c37

257.52/79.83

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

Removed 4 trailing nodes:

TOP(mark(true)) → c37 257.52/79.83
TOP(mark(0)) → c37 257.52/79.83
TOP(mark(false)) → c37 257.52/79.83
TOP(mark(nil)) → c37

257.52/79.83

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:none
Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = 0 257.52/79.83
POL(AND(x₁, x₂)) = 0 257.52/79.83
POL(CONS(x₁, x₂)) = 0 257.52/79.83
POL(FIRST(x₁, x₂)) = x₁ 257.52/79.83
POL(FROM(x₁)) = 0 257.52/79.83
POL(IF(x₁, x₂, x₃)) = 0 257.52/79.83
POL(S(x₁)) = 0 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = x₁ 257.52/79.83
POL(ok(x₁)) = [2] + x₁

257.52/79.83

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

ADD(mark(z0), z1) → c18(ADD(z0, z1))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = x₁ + x₂ 257.52/79.83
POL(AND(x₁, x₂)) = x₂ 257.52/79.83
POL(CONS(x₁, x₂)) = x₁ + x₂ 257.52/79.83
POL(FIRST(x₁, x₂)) = 0 257.52/79.83
POL(FROM(x₁)) = x₁ 257.52/79.83
POL(IF(x₁, x₂, x₃)) = x₂ 257.52/79.83
POL(S(x₁)) = 0 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = [1] + x₁ 257.52/79.83
POL(ok(x₁)) = x₁

257.52/79.83

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

FIRST(z0, mark(z1)) → c21(FIRST(z0, z1))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = 0 257.52/79.83
POL(AND(x₁, x₂)) = 0 257.52/79.83
POL(CONS(x₁, x₂)) = 0 257.52/79.83
POL(FIRST(x₁, x₂)) = x₂ 257.52/79.83
POL(FROM(x₁)) = 0 257.52/79.83
POL(IF(x₁, x₂, x₃)) = 0 257.52/79.83
POL(S(x₁)) = 0 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = [1] + x₁ 257.52/79.83
POL(ok(x₁)) = x₁

257.52/79.83

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

S(ok(z0)) → c34(S(z0))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = 0 257.52/79.83
POL(AND(x₁, x₂)) = 0 257.52/79.83
POL(CONS(x₁, x₂)) = 0 257.52/79.83
POL(FIRST(x₁, x₂)) = x₁ + [3]x₂ 257.52/79.83
POL(FROM(x₁)) = 0 257.52/79.83
POL(IF(x₁, x₂, x₃)) = 0 257.52/79.83
POL(S(x₁)) = x₁ 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = x₁ 257.52/79.83
POL(ok(x₁)) = [1] + x₁

257.52/79.83

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = 0 257.52/79.83
POL(AND(x₁, x₂)) = [2]x₁ 257.52/79.83
POL(CONS(x₁, x₂)) = x₁ 257.52/79.83
POL(FIRST(x₁, x₂)) = [3]x₂ 257.52/79.83
POL(FROM(x₁)) = x₁ 257.52/79.83
POL(IF(x₁, x₂, x₃)) = 0 257.52/79.83
POL(S(x₁)) = [5]x₁ 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = [2] + x₁ 257.52/79.83
POL(ok(x₁)) = [1] + x₁

257.52/79.83

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = x₁ + x₂ 257.52/79.83
POL(AND(x₁, x₂)) = [5]x₁ + [5]x₂ 257.52/79.83
POL(CONS(x₁, x₂)) = [5]x₁ + [5]x₂ 257.52/79.83
POL(FIRST(x₁, x₂)) = [3]x₂ 257.52/79.83
POL(FROM(x₁)) = [5]x₁ 257.52/79.83
POL(IF(x₁, x₂, x₃)) = [2]x₁ + x₂ 257.52/79.83
POL(S(x₁)) = [5]x₁ 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = [1] + x₁ 257.52/79.83
POL(ok(x₁)) = x₁

257.52/79.83

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(79) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) 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)) → c17(IF(z0, z1, z2))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = 0 257.52/79.83
POL(AND(x₁, x₂)) = x₂ 257.52/79.83
POL(CONS(x₁, x₂)) = x₁ + x₂ 257.52/79.83
POL(FIRST(x₁, x₂)) = 0 257.52/79.83
POL(FROM(x₁)) = [3]x₁ + [3]x₁² 257.52/79.83
POL(IF(x₁, x₂, x₃)) = x₃² 257.52/79.83
POL(S(x₁)) = [3]x₁ + [3]x₁² 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = 0 257.52/79.83
POL(ok(x₁)) = [1] + x₁

257.52/79.83

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = x₂ 257.52/79.83
POL(AND(x₁, x₂)) = [5]x₁ + [5]x₂ 257.52/79.83
POL(CONS(x₁, x₂)) = [5]x₁ + [3]x₂ 257.52/79.83
POL(FIRST(x₁, x₂)) = [3]x₂ 257.52/79.83
POL(FROM(x₁)) = [5]x₁ 257.52/79.83
POL(IF(x₁, x₂, x₃)) = [3]x₁ + [5]x₂ + [5]x₃ 257.52/79.83
POL(S(x₁)) = [5]x₁ 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = [1] + x₁ 257.52/79.83
POL(ok(x₁)) = [1] + x₁

257.52/79.83

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:

FIRST(mark(z0), z1) → c20(FIRST(z0, z1))

K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

(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.

FIRST(mark(z0), z1) → c20(FIRST(z0, z1))

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

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 257.52/79.83

POL(ADD(x₁, x₂)) = [3]x₁ + [5]x₂ 257.52/79.83
POL(AND(x₁, x₂)) = [3]x₁ + [5]x₂ 257.52/79.83
POL(CONS(x₁, x₂)) = [5]x₁ + [5]x₂ 257.52/79.83
POL(FIRST(x₁, x₂)) = [3]x₁ + [5]x₂ 257.52/79.83
POL(FROM(x₁)) = [3]x₁ 257.52/79.83
POL(IF(x₁, x₂, x₃)) = [3]x₁ + [3]x₂ + [5]x₃ 257.52/79.83
POL(S(x₁)) = [5]x₁ 257.52/79.83
POL(c14(x₁)) = x₁ 257.52/79.83
POL(c15(x₁)) = x₁ 257.52/79.83
POL(c16(x₁)) = x₁ 257.52/79.83
POL(c17(x₁)) = x₁ 257.52/79.83
POL(c18(x₁)) = x₁ 257.52/79.83
POL(c19(x₁)) = x₁ 257.52/79.83
POL(c20(x₁)) = x₁ 257.52/79.83
POL(c21(x₁)) = x₁ 257.52/79.83
POL(c22(x₁)) = x₁ 257.52/79.83
POL(c34(x₁)) = x₁ 257.52/79.83
POL(c35(x₁)) = x₁ 257.52/79.83
POL(c36(x₁)) = x₁ 257.52/79.83
POL(mark(x₁)) = [1] + x₁ 257.52/79.83
POL(ok(x₁)) = x₁

257.52/79.83

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(true, z0)) → mark(z0) 257.52/79.83
active(and(false, z0)) → mark(false) 257.52/79.83
active(if(true, z0, z1)) → mark(z0) 257.52/79.83
active(if(false, z0, z1)) → mark(z1) 257.52/79.83
active(add(0, z0)) → mark(z0) 257.52/79.83
active(add(s(z0), z1)) → mark(s(add(z0, z1))) 257.52/79.83
active(first(0, z0)) → mark(nil) 257.52/79.83
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 257.52/79.83
active(from(z0)) → mark(cons(z0, from(s(z0)))) 257.52/79.83
active(and(z0, z1)) → and(active(z0), z1) 257.52/79.83
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 257.52/79.83
active(add(z0, z1)) → add(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(active(z0), z1) 257.52/79.83
active(first(z0, z1)) → first(z0, active(z1)) 257.52/79.83
and(mark(z0), z1) → mark(and(z0, z1)) 257.52/79.83
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.52/79.83
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 257.52/79.83
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 257.52/79.83
add(mark(z0), z1) → mark(add(z0, z1)) 257.52/79.83
add(ok(z0), ok(z1)) → ok(add(z0, z1)) 257.52/79.83
first(mark(z0), z1) → mark(first(z0, z1)) 257.52/79.83
first(z0, mark(z1)) → mark(first(z0, z1)) 257.52/79.83
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 257.52/79.83
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.52/79.83
proper(true) → ok(true) 257.52/79.83
proper(false) → ok(false) 257.52/79.83
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 257.52/79.83
proper(add(z0, z1)) → add(proper(z0), proper(z1)) 257.52/79.83
proper(0) → ok(0) 257.52/79.83
proper(s(z0)) → s(proper(z0)) 257.52/79.83
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 257.52/79.83
proper(nil) → ok(nil) 257.52/79.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 257.52/79.83
proper(from(z0)) → from(proper(z0)) 257.52/79.83
s(ok(z0)) → ok(s(z0)) 257.52/79.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 257.52/79.83
from(ok(z0)) → ok(from(z0)) 257.52/79.83
top(mark(z0)) → top(proper(z0)) 257.52/79.83
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0))

S tuples:none
K tuples:

FIRST(ok(z0), ok(z1)) → c22(FIRST(z0, z1)) 257.52/79.83
ADD(mark(z0), z1) → c18(ADD(z0, z1)) 257.52/79.83
FIRST(z0, mark(z1)) → c21(FIRST(z0, z1)) 257.52/79.83
S(ok(z0)) → c34(S(z0)) 257.52/79.83
AND(mark(z0), z1) → c14(AND(z0, z1)) 257.52/79.83
AND(ok(z0), ok(z1)) → c15(AND(z0, z1)) 257.52/79.83
CONS(ok(z0), ok(z1)) → c35(CONS(z0, z1)) 257.52/79.83
FROM(ok(z0)) → c36(FROM(z0)) 257.52/79.83
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 257.52/79.83
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 257.52/79.83
ADD(ok(z0), ok(z1)) → c19(ADD(z0, z1)) 257.52/79.83
FIRST(mark(z0), z1) → c20(FIRST(z0, z1))

Defined Rule Symbols:

active, and, if, add, first, proper, s, cons, from, top

Defined Pair Symbols:

AND, IF, ADD, FIRST, S, CONS, FROM

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c34, c35, c36

257.52/79.83

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

The set S is empty

257.52/79.83

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

257.52/79.83