YES(O(1), O(n^3)) 256.87/81.68 YES(O(1), O(n^3)) 257.06/81.78 257.06/81.78 257.06/81.78 257.06/81.78 257.06/81.78 257.06/81.78 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 257.06/81.78 257.06/81.78 257.06/81.78
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

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))
257.06/81.78 
↳4 CdtProblem
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↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳8 CdtProblem
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↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳10 CdtProblem
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↳11 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
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↳12 CdtProblem
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↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳14 CdtProblem
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↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳16 CdtProblem
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↳17 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
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↳18 CdtProblem
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↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳20 CdtProblem
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↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳22 CdtProblem
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↳23 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
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↳24 CdtProblem
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↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳26 CdtProblem
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↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳28 CdtProblem
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↳29 CdtGraphSplitRhsProof (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 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳60 CdtProblem
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↳61 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
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↳62 CdtProblem
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↳63 CdtUnreachableProof (⇔)
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↳64 CdtProblem
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↳65 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳66 CdtProblem
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↳67 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳68 CdtProblem
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↳69 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
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↳70 CdtProblem
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↳71 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳72 CdtProblem
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↳73 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳74 CdtProblem
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↳75 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳76 CdtProblem
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↳77 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳78 CdtProblem
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↳79 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳80 CdtProblem
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↳81 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳82 CdtProblem
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↳83 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳84 BOUNDS(O(1), O(1))
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(0) Obligation:

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

active(U11(tt, N)) → mark(N) 257.06/81.78
active(U21(tt, M, N)) → mark(s(plus(N, M))) 257.06/81.78
active(and(tt, X)) → mark(X) 257.06/81.78
active(isNat(0)) → mark(tt) 257.06/81.78
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2))) 257.06/81.78
active(isNat(s(V1))) → mark(isNat(V1)) 257.06/81.78
active(plus(N, 0)) → mark(U11(isNat(N), N)) 257.06/81.78
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N)) 257.06/81.78
active(U11(X1, X2)) → U11(active(X1), X2) 257.06/81.78
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3) 257.06/81.78
active(s(X)) → s(active(X)) 257.06/81.78
active(plus(X1, X2)) → plus(active(X1), X2) 257.06/81.78
active(plus(X1, X2)) → plus(X1, active(X2)) 257.06/81.78
active(and(X1, X2)) → and(active(X1), X2) 257.06/81.78
U11(mark(X1), X2) → mark(U11(X1, X2)) 257.06/81.78
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3)) 257.06/81.78
s(mark(X)) → mark(s(X)) 257.06/81.78
plus(mark(X1), X2) → mark(plus(X1, X2)) 257.06/81.78
plus(X1, mark(X2)) → mark(plus(X1, X2)) 257.06/81.78
and(mark(X1), X2) → mark(and(X1, X2)) 257.06/81.78
proper(U11(X1, X2)) → U11(proper(X1), proper(X2)) 257.06/81.78
proper(tt) → ok(tt) 257.06/81.78
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3)) 257.06/81.78
proper(s(X)) → s(proper(X)) 257.06/81.78
proper(plus(X1, X2)) → plus(proper(X1), proper(X2)) 257.06/81.78
proper(and(X1, X2)) → and(proper(X1), proper(X2)) 257.06/81.78
proper(isNat(X)) → isNat(proper(X)) 257.06/81.78
proper(0) → ok(0) 257.06/81.78
U11(ok(X1), ok(X2)) → ok(U11(X1, X2)) 257.06/81.78
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3)) 257.06/81.78
s(ok(X)) → ok(s(X)) 257.06/81.78
plus(ok(X1), ok(X2)) → ok(plus(X1, X2)) 257.06/81.78
and(ok(X1), ok(X2)) → ok(and(X1, X2)) 257.06/81.78
isNat(ok(X)) → ok(isNat(X)) 257.06/81.78
top(mark(X)) → top(proper(X)) 257.06/81.78
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(U11(tt, z0)) → mark(z0) 257.06/81.78
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.06/81.78
active(and(tt, z0)) → mark(z0) 257.06/81.78
active(isNat(0)) → mark(tt) 257.06/81.78
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.06/81.78
active(isNat(s(z0))) → mark(isNat(z0)) 257.06/81.78
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.06/81.78
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.06/81.78
active(U11(z0, z1)) → U11(active(z0), z1) 257.06/81.78
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.06/81.78
active(s(z0)) → s(active(z0)) 257.06/81.78
active(plus(z0, z1)) → plus(active(z0), z1) 257.06/81.78
active(plus(z0, z1)) → plus(z0, active(z1)) 257.06/81.78
active(and(z0, z1)) → and(active(z0), z1) 257.06/81.78
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.06/81.78
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.06/81.78
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.06/81.78
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.06/81.78
s(mark(z0)) → mark(s(z0)) 257.06/81.78
s(ok(z0)) → ok(s(z0)) 257.06/81.78
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.06/81.78
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.06/81.78
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.06/81.78
and(mark(z0), z1) → mark(and(z0, z1)) 257.06/81.78
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.06/81.78
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.06/81.78
proper(tt) → ok(tt) 257.06/81.78
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.06/81.78
proper(s(z0)) → s(proper(z0)) 257.06/81.78
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.06/81.78
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.06/81.78
proper(isNat(z0)) → isNat(proper(z0)) 257.06/81.78
proper(0) → ok(0) 257.06/81.78
isNat(ok(z0)) → ok(isNat(z0)) 257.06/81.78
top(mark(z0)) → top(proper(z0)) 257.06/81.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0)) 257.06/81.78
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1)) 257.06/81.78
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.06/81.78
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0)) 257.06/81.78
ACTIVE(plus(z0, s(z1))) → c7(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0)) 257.06/81.78
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) 257.06/81.78
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.06/81.78
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.06/81.78
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.06/81.78
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.06/81.78
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.06/81.78
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.06/81.78
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.06/81.78
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.06/81.78
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.06/81.78
S(mark(z0)) → c18(S(z0)) 257.06/81.78
S(ok(z0)) → c19(S(z0)) 257.06/81.78
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.06/81.78
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.06/81.78
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.06/81.78
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.06/81.78
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.06/81.78
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.06/81.78
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.06/81.78
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.06/81.78
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.06/81.78
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.06/81.78
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.45/81.87
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.45/81.87
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.45/81.87
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0)) 257.45/81.87
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1)) 257.45/81.87
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.45/81.87
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0)) 257.45/81.87
ACTIVE(plus(z0, s(z1))) → c7(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0)) 257.45/81.87
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.45/81.87
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.45/81.87
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.45/81.87
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.45/81.87
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.45/81.87
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.45/81.87
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.45/81.87
S(mark(z0)) → c18(S(z0)) 257.45/81.87
S(ok(z0)) → c19(S(z0)) 257.45/81.87
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.45/81.87
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.45/81.87
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.45/81.87
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.45/81.87
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.45/81.87
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.45/81.87
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.45/81.87
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.45/81.87
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.45/81.87
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.45/81.87
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35

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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(U11(tt, z0)) → mark(z0) 257.45/81.87
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.45/81.87
active(and(tt, z0)) → mark(z0) 257.45/81.87
active(isNat(0)) → mark(tt) 257.45/81.87
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.45/81.87
active(isNat(s(z0))) → mark(isNat(z0)) 257.45/81.87
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.45/81.87
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.45/81.87
active(U11(z0, z1)) → U11(active(z0), z1) 257.45/81.87
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.45/81.87
active(s(z0)) → s(active(z0)) 257.45/81.87
active(plus(z0, z1)) → plus(active(z0), z1) 257.45/81.87
active(plus(z0, z1)) → plus(z0, active(z1)) 257.45/81.87
active(and(z0, z1)) → and(active(z0), z1) 257.45/81.87
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.45/81.87
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.45/81.87
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.45/81.87
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.45/81.87
s(mark(z0)) → mark(s(z0)) 257.45/81.87
s(ok(z0)) → ok(s(z0)) 257.45/81.87
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.45/81.87
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.45/81.87
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.45/81.87
and(mark(z0), z1) → mark(and(z0, z1)) 257.45/81.87
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.45/81.87
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.45/81.87
proper(tt) → ok(tt) 257.45/81.87
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.45/81.87
proper(s(z0)) → s(proper(z0)) 257.45/81.87
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.45/81.87
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.45/81.87
proper(isNat(z0)) → isNat(proper(z0)) 257.45/81.87
proper(0) → ok(0) 257.45/81.87
isNat(ok(z0)) → ok(isNat(z0)) 257.45/81.87
top(mark(z0)) → top(proper(z0)) 257.45/81.87
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0)) 257.45/81.87
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1)) 257.45/81.87
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.45/81.87
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0)) 257.45/81.87
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.45/81.87
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.45/81.87
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.45/81.87
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.45/81.87
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.45/81.87
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.45/81.87
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.45/81.87
S(mark(z0)) → c18(S(z0)) 257.45/81.87
S(ok(z0)) → c19(S(z0)) 257.45/81.87
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.45/81.87
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.45/81.87
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.45/81.87
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.45/81.87
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.45/81.87
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.45/81.87
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.45/81.87
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.45/81.87
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.45/81.87
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.45/81.87
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, s(z1))) → c7(ISNAT(z1), ISNAT(z0))
S tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0)) 257.45/81.87
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1)) 257.45/81.87
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.45/81.87
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0)) 257.45/81.87
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.45/81.87
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.45/81.87
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.45/81.87
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.45/81.87
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.45/81.87
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.45/81.87
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.45/81.87
S(mark(z0)) → c18(S(z0)) 257.45/81.87
S(ok(z0)) → c19(S(z0)) 257.45/81.87
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.45/81.87
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.45/81.87
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.45/81.87
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.45/81.87
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.45/81.87
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.45/81.87
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.45/81.87
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.45/81.87
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.45/81.87
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.45/81.87
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, s(z1))) → c7(ISNAT(z1), ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c1, c4, c5, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c7

257.45/81.87
257.45/81.87

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

Split RHS of tuples not part of any SCC
257.45/81.87
257.45/81.87

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.45/81.87
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.45/81.87
active(and(tt, z0)) → mark(z0) 257.45/81.87
active(isNat(0)) → mark(tt) 257.45/81.87
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.45/81.87
active(isNat(s(z0))) → mark(isNat(z0)) 257.45/81.87
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.45/81.87
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.45/81.87
active(U11(z0, z1)) → U11(active(z0), z1) 257.45/81.87
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.45/81.87
active(s(z0)) → s(active(z0)) 257.45/81.87
active(plus(z0, z1)) → plus(active(z0), z1) 257.45/81.87
active(plus(z0, z1)) → plus(z0, active(z1)) 257.45/81.87
active(and(z0, z1)) → and(active(z0), z1) 257.45/81.87
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.45/81.87
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.45/81.87
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.45/81.87
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.45/81.87
s(mark(z0)) → mark(s(z0)) 257.45/81.87
s(ok(z0)) → ok(s(z0)) 257.45/81.87
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.45/81.87
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.45/81.87
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.45/81.87
and(mark(z0), z1) → mark(and(z0, z1)) 257.45/81.87
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.45/81.87
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.45/81.87
proper(tt) → ok(tt) 257.45/81.87
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.45/81.87
proper(s(z0)) → s(proper(z0)) 257.45/81.87
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.45/81.87
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.45/81.87
proper(isNat(z0)) → isNat(proper(z0)) 257.45/81.87
proper(0) → ok(0) 257.45/81.87
isNat(ok(z0)) → ok(isNat(z0)) 257.45/81.87
top(mark(z0)) → top(proper(z0)) 257.45/81.87
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.45/81.87
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.45/81.87
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.45/81.87
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.45/81.87
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.45/81.87
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.45/81.87
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.45/81.87
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.45/81.87
S(mark(z0)) → c18(S(z0)) 257.45/81.87
S(ok(z0)) → c19(S(z0)) 257.45/81.87
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.45/81.87
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.45/81.87
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.45/81.87
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.45/81.87
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.45/81.87
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.45/81.87
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.45/81.87
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.45/81.87
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.45/81.87
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.45/81.87
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.45/81.87
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.45/81.87
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.45/81.87
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.45/81.87
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.45/81.87
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.45/81.87
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.45/81.87
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.45/81.87
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.45/81.87
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.45/81.87
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.45/81.87
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.45/81.87
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c

257.87/81.94
257.87/81.94

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

Use narrowing to replace ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) by

ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
257.87/81.94
257.87/81.94

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8

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257.87/81.94

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

Removed 2 trailing tuple parts
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257.87/81.94

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8

257.87/81.94
257.87/81.94

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

Split RHS of tuples not part of any SCC
257.87/81.94
257.87/81.94

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1

257.87/81.94
257.87/81.94

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

Use narrowing to replace ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
257.87/81.94
257.87/81.94

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9

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257.87/81.94

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

Removed 2 trailing tuple parts
257.87/81.94
257.87/81.94

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9

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257.87/81.94

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

Split RHS of tuples not part of any SCC
257.87/81.94
257.87/81.94

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.94
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.94
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.94
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.94
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.94
S(mark(z0)) → c18(S(z0)) 257.87/81.94
S(ok(z0)) → c19(S(z0)) 257.87/81.94
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.94
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.94
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.94
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.94
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.94
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.94
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.94
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.94
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.94
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.94
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.94
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.94
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.94
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.94
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.94
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.94
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.94
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.94
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.87/81.94
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 257.87/81.94
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 257.87/81.94
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 257.87/81.94
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2

257.87/81.94
257.87/81.94

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

Use narrowing to replace ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)), ACTIVE(U11(tt, z0))) 257.87/81.94
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 257.87/81.94
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 257.87/81.94
ACTIVE(s(isNat(0))) → c10(S(mark(tt)), ACTIVE(isNat(0))) 257.87/81.94
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.94
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 257.87/81.94
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 257.87/81.94
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 257.87/81.94
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 257.87/81.94
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 257.87/81.94
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 257.87/81.94
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 257.87/81.94
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
257.87/81.94
257.87/81.94

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 257.87/81.94
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 257.87/81.94
active(and(tt, z0)) → mark(z0) 257.87/81.94
active(isNat(0)) → mark(tt) 257.87/81.94
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 257.87/81.94
active(isNat(s(z0))) → mark(isNat(z0)) 257.87/81.94
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 257.87/81.94
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 257.87/81.94
active(U11(z0, z1)) → U11(active(z0), z1) 257.87/81.94
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 257.87/81.94
active(s(z0)) → s(active(z0)) 257.87/81.94
active(plus(z0, z1)) → plus(active(z0), z1) 257.87/81.94
active(plus(z0, z1)) → plus(z0, active(z1)) 257.87/81.94
active(and(z0, z1)) → and(active(z0), z1) 257.87/81.94
U11(mark(z0), z1) → mark(U11(z0, z1)) 257.87/81.94
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 257.87/81.94
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 257.87/81.94
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 257.87/81.94
s(mark(z0)) → mark(s(z0)) 257.87/81.94
s(ok(z0)) → ok(s(z0)) 257.87/81.94
plus(mark(z0), z1) → mark(plus(z0, z1)) 257.87/81.94
plus(z0, mark(z1)) → mark(plus(z0, z1)) 257.87/81.94
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 257.87/81.94
and(mark(z0), z1) → mark(and(z0, z1)) 257.87/81.94
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 257.87/81.94
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 257.87/81.94
proper(tt) → ok(tt) 257.87/81.94
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 257.87/81.94
proper(s(z0)) → s(proper(z0)) 257.87/81.94
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 257.87/81.94
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 257.87/81.94
proper(isNat(z0)) → isNat(proper(z0)) 257.87/81.94
proper(0) → ok(0) 257.87/81.94
isNat(ok(z0)) → ok(isNat(z0)) 257.87/81.94
top(mark(z0)) → top(proper(z0)) 257.87/81.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.94
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.94
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.95
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.95
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.95
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.95
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.95
S(mark(z0)) → c18(S(z0)) 257.87/81.95
S(ok(z0)) → c19(S(z0)) 257.87/81.95
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.95
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.95
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.95
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.95
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.95
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.95
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.95
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.95
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.95
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.95
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.95
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.95
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.95
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.95
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.95
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.95
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.95
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.95
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.95
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.95
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.95
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.95
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.95
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.95
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.95
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.95
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.95
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.95
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.95
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.95
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.95
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.95
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.95
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.95
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.95
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.95
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.95
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.95
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.95
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.95
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.95
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.95
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.95
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.95
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.95
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.95
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.87/81.95
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 257.87/81.95
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 257.87/81.95
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 257.87/81.95
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.95
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 257.87/81.95
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.95
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 257.87/81.95
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.95
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 257.87/81.95
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 257.87/81.95
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)), ACTIVE(U11(tt, z0))) 257.87/81.95
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 257.87/81.95
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 257.87/81.95
ACTIVE(s(isNat(0))) → c10(S(mark(tt)), ACTIVE(isNat(0))) 257.87/81.95
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 257.87/81.95
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 257.87/81.95
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 257.87/81.95
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 257.87/81.95
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 257.87/81.95
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 257.87/81.95
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 257.87/81.95
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 257.87/81.95
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 257.87/81.95
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 257.87/81.95
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 257.87/81.95
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 257.87/81.95
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 257.87/81.95
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 257.87/81.95
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 257.87/81.95
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 257.87/81.95
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 257.87/81.95
S(mark(z0)) → c18(S(z0)) 257.87/81.95
S(ok(z0)) → c19(S(z0)) 257.87/81.95
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 257.87/81.95
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 257.87/81.95
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 257.87/81.95
AND(mark(z0), z1) → c23(AND(z0, z1)) 257.87/81.95
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 257.87/81.95
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.95
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 257.87/81.95
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 257.87/81.95
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.95
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 257.87/81.95
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 257.87/81.95
ISNAT(ok(z0)) → c33(ISNAT(z0)) 257.87/81.95
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 257.87/81.95
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 257.87/81.95
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 257.87/81.95
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 257.87/81.95
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 257.87/81.95
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 257.87/81.95
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 257.87/81.95
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 257.87/81.95
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 257.87/81.95
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 257.87/81.95
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 257.87/81.95
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 257.87/81.95
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 257.87/81.96
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 257.87/81.96
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 257.87/81.96
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 257.87/81.96
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 257.87/81.96
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 257.87/81.96
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 257.87/81.96
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 257.87/81.96
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 257.87/81.96
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 257.87/81.96
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 257.87/81.96
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.96
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 257.87/81.96
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 257.87/81.96
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 257.87/81.96
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 257.87/81.96
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 257.87/81.96
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 257.87/81.96
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 257.87/81.96
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 257.87/81.96
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.96
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 257.87/81.96
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 257.87/81.96
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 257.87/81.96
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 257.87/81.96
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 257.87/81.96
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.96
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 257.87/81.96
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 257.87/81.96
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 257.87/81.96
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 257.87/81.96
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 257.87/81.96
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 257.87/81.96
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)), ACTIVE(U11(tt, z0))) 257.87/81.96
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 257.87/81.96
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(s(isNat(0))) → c10(S(mark(tt)), ACTIVE(isNat(0))) 258.20/82.00
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10

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

Removed 2 trailing tuple parts
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258.20/82.00

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.00
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.00
active(and(tt, z0)) → mark(z0) 258.20/82.00
active(isNat(0)) → mark(tt) 258.20/82.00
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.00
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.00
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.00
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.00
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.00
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.00
active(s(z0)) → s(active(z0)) 258.20/82.00
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.00
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.00
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.00
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.00
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.00
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.00
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.00
s(mark(z0)) → mark(s(z0)) 258.20/82.00
s(ok(z0)) → ok(s(z0)) 258.20/82.00
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.00
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.00
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.00
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.00
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.00
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.00
proper(tt) → ok(tt) 258.20/82.00
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.00
proper(s(z0)) → s(proper(z0)) 258.20/82.00
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.00
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.00
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.00
proper(0) → ok(0) 258.20/82.00
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.00
top(mark(z0)) → top(proper(z0)) 258.20/82.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.00
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.00
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.00
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 258.20/82.00
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.00
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.00
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.00
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.00
S(mark(z0)) → c18(S(z0)) 258.20/82.00
S(ok(z0)) → c19(S(z0)) 258.20/82.00
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.00
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.00
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.00
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.00
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.00
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.00
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.00
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.00
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.00
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.00
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.00
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.00
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.00
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.00
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.00
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.00
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.00
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.00
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.00
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.00
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.00
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.00
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.00
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.00
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.00
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.00
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 258.20/82.00
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.00
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.00
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.00
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.00
S(mark(z0)) → c18(S(z0)) 258.20/82.00
S(ok(z0)) → c19(S(z0)) 258.20/82.00
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.00
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.00
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.00
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.00
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.00
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.00
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.00
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.00
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.00
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.00
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.00
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.00
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.00
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.00
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.00
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.00
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.00
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.00
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.00
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.00
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.00
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.00
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.00
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10

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

Split RHS of tuples not part of any SCC
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(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.00
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.00
active(and(tt, z0)) → mark(z0) 258.20/82.00
active(isNat(0)) → mark(tt) 258.20/82.00
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.00
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.00
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.00
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.00
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.00
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.00
active(s(z0)) → s(active(z0)) 258.20/82.00
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.00
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.00
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.00
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.00
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.00
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.00
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.00
s(mark(z0)) → mark(s(z0)) 258.20/82.00
s(ok(z0)) → ok(s(z0)) 258.20/82.00
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.00
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.00
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.00
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.00
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.00
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.00
proper(tt) → ok(tt) 258.20/82.00
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.00
proper(s(z0)) → s(proper(z0)) 258.20/82.00
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.00
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.00
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.00
proper(0) → ok(0) 258.20/82.00
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.00
top(mark(z0)) → top(proper(z0)) 258.20/82.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.00
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.00
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.00
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 258.20/82.00
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.00
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.00
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.00
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.00
S(mark(z0)) → c18(S(z0)) 258.20/82.00
S(ok(z0)) → c19(S(z0)) 258.20/82.00
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.00
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.00
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.00
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.00
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.00
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.00
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.00
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.00
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.00
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.00
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.00
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.00
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.00
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.00
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.00
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.00
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.00
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.00
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.00
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.00
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.00
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.00
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.00
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.00
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.00
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.00
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.00
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.00
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.00
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.00
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.00
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.00
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.00
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.00
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.00
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.00
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3

258.20/82.01
258.20/82.01

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(U11(tt, z0))) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1), ACTIVE(isNat(0))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
258.20/82.01
258.20/82.01

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.01
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.01
active(and(tt, z0)) → mark(z0) 258.20/82.01
active(isNat(0)) → mark(tt) 258.20/82.01
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.01
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.01
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.01
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.01
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.01
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.01
active(s(z0)) → s(active(z0)) 258.20/82.01
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.01
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.01
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.01
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.01
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.01
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.01
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.01
s(mark(z0)) → mark(s(z0)) 258.20/82.01
s(ok(z0)) → ok(s(z0)) 258.20/82.01
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.01
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.01
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.01
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.01
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.01
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.01
proper(tt) → ok(tt) 258.20/82.01
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.01
proper(s(z0)) → s(proper(z0)) 258.20/82.01
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.01
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.01
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.01
proper(0) → ok(0) 258.20/82.01
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.01
top(mark(z0)) → top(proper(z0)) 258.20/82.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(U11(tt, z0))) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1), ACTIVE(isNat(0))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(U11(tt, z0))) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(and(tt, z0))) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1), ACTIVE(isNat(0))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13

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258.20/82.01

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

Removed 6 trailing tuple parts
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258.20/82.01

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.01
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.01
active(and(tt, z0)) → mark(z0) 258.20/82.01
active(isNat(0)) → mark(tt) 258.20/82.01
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.01
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.01
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.01
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.01
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.01
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.01
active(s(z0)) → s(active(z0)) 258.20/82.01
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.01
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.01
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.01
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.01
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.01
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.01
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.01
s(mark(z0)) → mark(s(z0)) 258.20/82.01
s(ok(z0)) → ok(s(z0)) 258.20/82.01
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.01
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.01
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.01
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.01
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.01
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.01
proper(tt) → ok(tt) 258.20/82.01
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.01
proper(s(z0)) → s(proper(z0)) 258.20/82.01
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.01
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.01
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.01
proper(0) → ok(0) 258.20/82.01
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.01
top(mark(z0)) → top(proper(z0)) 258.20/82.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13

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

Split RHS of tuples not part of any SCC
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(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.01
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.01
active(and(tt, z0)) → mark(z0) 258.20/82.01
active(isNat(0)) → mark(tt) 258.20/82.01
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.01
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.01
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.01
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.01
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.01
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.01
active(s(z0)) → s(active(z0)) 258.20/82.01
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.01
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.01
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.01
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.01
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.01
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.01
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.01
s(mark(z0)) → mark(s(z0)) 258.20/82.01
s(ok(z0)) → ok(s(z0)) 258.20/82.01
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.01
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.01
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.01
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.01
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.01
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.01
proper(tt) → ok(tt) 258.20/82.01
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.01
proper(s(z0)) → s(proper(z0)) 258.20/82.01
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.01
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.01
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.01
proper(0) → ok(0) 258.20/82.01
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.01
top(mark(z0)) → top(proper(z0)) 258.20/82.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4

258.20/82.01
258.20/82.01

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

Use narrowing to replace PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.01
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.01
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.01
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.01
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.01
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.01
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.01
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.01
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.01
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.01
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.01
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.01
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
258.20/82.01
258.20/82.01

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.01
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.01
active(and(tt, z0)) → mark(z0) 258.20/82.01
active(isNat(0)) → mark(tt) 258.20/82.01
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.01
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.01
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.01
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.01
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.01
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.01
active(s(z0)) → s(active(z0)) 258.20/82.01
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.01
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.01
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.01
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.01
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.01
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.01
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.01
s(mark(z0)) → mark(s(z0)) 258.20/82.01
s(ok(z0)) → ok(s(z0)) 258.20/82.01
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.01
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.01
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.01
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.01
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.01
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.01
proper(tt) → ok(tt) 258.20/82.01
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.01
proper(s(z0)) → s(proper(z0)) 258.20/82.01
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.01
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.01
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.01
proper(0) → ok(0) 258.20/82.01
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.01
top(mark(z0)) → top(proper(z0)) 258.20/82.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.01
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.01
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.01
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.01
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.01
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.01
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.01
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.01
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.01
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.01
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.01
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.01
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.01
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.01
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.01
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.01
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.01
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.01
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.01
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.01
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.01
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.01
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.01
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.01
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.01
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.01
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.01
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.01
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.01
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.01
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.01
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.01
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.01
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.01
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.01
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.01
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.01
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.01
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.01
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.01
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.01
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.01
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.01
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.01
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.01
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.01
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.01
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.01
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.01
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.01
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.01
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.01
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.01
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25

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

Removed 4 trailing tuple parts
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258.20/82.01

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.01
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.01
active(and(tt, z0)) → mark(z0) 258.20/82.01
active(isNat(0)) → mark(tt) 258.20/82.01
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.01
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.01
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.01
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.01
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.01
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.01
active(s(z0)) → s(active(z0)) 258.20/82.01
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.01
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.01
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.01
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.01
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.01
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.01
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.01
s(mark(z0)) → mark(s(z0)) 258.20/82.01
s(ok(z0)) → ok(s(z0)) 258.20/82.01
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.01
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.01
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.01
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.01
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.01
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.01
proper(tt) → ok(tt) 258.20/82.01
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.01
proper(s(z0)) → s(proper(z0)) 258.20/82.01
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.01
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.01
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.01
proper(0) → ok(0) 258.20/82.01
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.01
top(mark(z0)) → top(proper(z0)) 258.20/82.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.01
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.01
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.01
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.01
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.01
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.01
S(mark(z0)) → c18(S(z0)) 258.20/82.01
S(ok(z0)) → c19(S(z0)) 258.20/82.01
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.01
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.01
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.01
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.01
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.01
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.01
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.01
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.01
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 258.20/82.02
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25

258.20/82.02
258.20/82.02

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

Use narrowing to replace PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt)) 258.20/82.02
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.02
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.02
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 258.20/82.02
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 258.20/82.02
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
258.20/82.02
258.20/82.02

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.02
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.02
active(and(tt, z0)) → mark(z0) 258.20/82.02
active(isNat(0)) → mark(tt) 258.20/82.02
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.02
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.02
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.02
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.02
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.02
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.02
active(s(z0)) → s(active(z0)) 258.20/82.02
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.02
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.02
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.02
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.02
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.02
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.02
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.02
s(mark(z0)) → mark(s(z0)) 258.20/82.02
s(ok(z0)) → ok(s(z0)) 258.20/82.02
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.02
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.02
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.02
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.02
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.02
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.02
proper(tt) → ok(tt) 258.20/82.02
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.02
proper(s(z0)) → s(proper(z0)) 258.20/82.02
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.02
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.02
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.02
proper(0) → ok(0) 258.20/82.02
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.02
top(mark(z0)) → top(proper(z0)) 258.20/82.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt)) 258.20/82.02
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.02
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.02
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 258.20/82.02
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 258.20/82.02
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt)) 258.20/82.02
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.02
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.02
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 258.20/82.02
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 258.20/82.02
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27

258.20/82.02
258.20/82.02

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

Removed 6 trailing tuple parts
258.20/82.02
258.20/82.02

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.02
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.02
active(and(tt, z0)) → mark(z0) 258.20/82.02
active(isNat(0)) → mark(tt) 258.20/82.02
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.02
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.02
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.02
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.02
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.02
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.02
active(s(z0)) → s(active(z0)) 258.20/82.02
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.02
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.02
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.02
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.02
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.02
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.02
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.02
s(mark(z0)) → mark(s(z0)) 258.20/82.02
s(ok(z0)) → ok(s(z0)) 258.20/82.02
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.02
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.02
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.02
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.02
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.02
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.02
proper(tt) → ok(tt) 258.20/82.02
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.02
proper(s(z0)) → s(proper(z0)) 258.20/82.02
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.02
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.02
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.02
proper(0) → ok(0) 258.20/82.02
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.02
top(mark(z0)) → top(proper(z0)) 258.20/82.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.02
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.02
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.02
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.02
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.02
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.02
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27

258.20/82.02
258.20/82.02

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

Use narrowing to replace PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) by

PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(s(tt)) → c28(S(ok(tt)), PROPER(tt)) 258.20/82.02
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.02
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.02
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.02
PROPER(s(0)) → c28(S(ok(0)), PROPER(0))
258.20/82.02
258.20/82.02

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.02
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.02
active(and(tt, z0)) → mark(z0) 258.20/82.02
active(isNat(0)) → mark(tt) 258.20/82.02
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.02
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.02
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.02
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.02
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.02
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.02
active(s(z0)) → s(active(z0)) 258.20/82.02
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.02
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.02
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.02
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.02
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.02
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.02
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.02
s(mark(z0)) → mark(s(z0)) 258.20/82.02
s(ok(z0)) → ok(s(z0)) 258.20/82.02
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.02
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.02
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.02
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.02
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.02
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.02
proper(tt) → ok(tt) 258.20/82.02
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.02
proper(s(z0)) → s(proper(z0)) 258.20/82.02
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.02
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.02
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.02
proper(0) → ok(0) 258.20/82.02
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.02
top(mark(z0)) → top(proper(z0)) 258.20/82.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.02
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.02
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.02
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.02
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.02
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.02
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.02
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.02
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.02
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.02
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.02
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.02
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.02
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.02
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.02
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.02
PROPER(s(tt)) → c28(S(ok(tt)), PROPER(tt)) 258.20/82.02
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.02
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.02
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.02
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.02
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.02
PROPER(s(0)) → c28(S(ok(0)), PROPER(0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.02
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.02
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.02
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.02
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.02
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.02
S(mark(z0)) → c18(S(z0)) 258.20/82.02
S(ok(z0)) → c19(S(z0)) 258.20/82.02
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.02
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.02
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.02
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.02
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.02
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.02
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.02
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.02
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.02
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.02
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.02
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.02
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.02
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.02
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.02
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.02
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.02
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.02
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.02
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.02
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.02
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.02
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.02
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.02
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.02
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.02
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.02
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.02
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.02
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.02
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.02
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.02
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.02
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt)), PROPER(tt)) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28

258.20/82.03
258.20/82.03

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

Removed 2 trailing tuple parts
258.20/82.03
258.20/82.03

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.03
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.03
active(and(tt, z0)) → mark(z0) 258.20/82.03
active(isNat(0)) → mark(tt) 258.20/82.03
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.03
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.03
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.03
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.03
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.03
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.03
active(s(z0)) → s(active(z0)) 258.20/82.03
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.03
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.03
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.03
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.03
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.03
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.03
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.03
s(mark(z0)) → mark(s(z0)) 258.20/82.03
s(ok(z0)) → ok(s(z0)) 258.20/82.03
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.03
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.03
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.03
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.03
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.03
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.03
proper(tt) → ok(tt) 258.20/82.03
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.03
proper(s(z0)) → s(proper(z0)) 258.20/82.03
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.03
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.03
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.03
proper(0) → ok(0) 258.20/82.03
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.03
top(mark(z0)) → top(proper(z0)) 258.20/82.03
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.03
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.03
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28

258.20/82.03
258.20/82.03

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

Use narrowing to replace PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.03
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.03
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.03
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
258.20/82.03
258.20/82.03

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.03
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.03
active(and(tt, z0)) → mark(z0) 258.20/82.03
active(isNat(0)) → mark(tt) 258.20/82.03
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.03
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.03
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.03
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.03
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.03
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.03
active(s(z0)) → s(active(z0)) 258.20/82.03
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.03
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.03
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.03
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.03
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.03
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.03
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.03
s(mark(z0)) → mark(s(z0)) 258.20/82.03
s(ok(z0)) → ok(s(z0)) 258.20/82.03
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.03
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.03
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.03
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.03
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.03
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.03
proper(tt) → ok(tt) 258.20/82.03
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.03
proper(s(z0)) → s(proper(z0)) 258.20/82.03
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.03
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.03
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.03
proper(0) → ok(0) 258.20/82.03
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.03
top(mark(z0)) → top(proper(z0)) 258.20/82.03
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.03
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.03
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.03
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.03
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.03
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.03
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.03
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.03
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.03
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.03
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29

258.20/82.03
258.20/82.03

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

Removed 4 trailing tuple parts
258.20/82.03
258.20/82.03

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.03
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.03
active(and(tt, z0)) → mark(z0) 258.20/82.03
active(isNat(0)) → mark(tt) 258.20/82.03
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.03
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.03
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.03
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.03
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.03
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.03
active(s(z0)) → s(active(z0)) 258.20/82.03
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.03
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.03
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.03
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.03
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.03
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.03
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.03
s(mark(z0)) → mark(s(z0)) 258.20/82.03
s(ok(z0)) → ok(s(z0)) 258.20/82.03
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.03
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.03
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.03
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.03
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.03
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.03
proper(tt) → ok(tt) 258.20/82.03
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.03
proper(s(z0)) → s(proper(z0)) 258.20/82.03
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.03
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.03
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.03
proper(0) → ok(0) 258.20/82.03
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.03
top(mark(z0)) → top(proper(z0)) 258.20/82.03
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.03
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.03
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.03
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.03
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.03
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.03
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.03
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.03
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.03
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.03
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.03
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.03
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.03
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.03
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.03
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.03
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.03
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.03
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.03
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29

258.20/82.03
258.20/82.03

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

Use narrowing to replace PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.03
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.03
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.03
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.03
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.03
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.03
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.03
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.03
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.03
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.03
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.03
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.03
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.03
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
258.20/82.03
258.20/82.03

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.03
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.03
active(and(tt, z0)) → mark(z0) 258.20/82.03
active(isNat(0)) → mark(tt) 258.20/82.03
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.03
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.03
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.03
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.03
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.03
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.03
active(s(z0)) → s(active(z0)) 258.20/82.03
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.03
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.03
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.03
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.03
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.03
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.03
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.03
s(mark(z0)) → mark(s(z0)) 258.20/82.03
s(ok(z0)) → ok(s(z0)) 258.20/82.03
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.03
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.03
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.03
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.03
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.03
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.03
proper(tt) → ok(tt) 258.20/82.03
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.03
proper(s(z0)) → s(proper(z0)) 258.20/82.03
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.03
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.03
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.03
proper(0) → ok(0) 258.20/82.03
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.03
top(mark(z0)) → top(proper(z0)) 258.20/82.03
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.03
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.03
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.03
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.03
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.03
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.03
S(mark(z0)) → c18(S(z0)) 258.20/82.03
S(ok(z0)) → c19(S(z0)) 258.20/82.03
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.03
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.03
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.03
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.03
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.03
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.03
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.03
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.03
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.03
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.03
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.03
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.03
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.03
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.03
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.03
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.03
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.03
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.03
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.03
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.03
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.03
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.03
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.03
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.03
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.03
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.03
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.03
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.03
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.03
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30

258.20/82.04
258.20/82.04

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

Removed 4 trailing tuple parts
258.20/82.04
258.20/82.04

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.04
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.04
active(and(tt, z0)) → mark(z0) 258.20/82.04
active(isNat(0)) → mark(tt) 258.20/82.04
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.04
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.04
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.04
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.04
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.04
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.04
active(s(z0)) → s(active(z0)) 258.20/82.04
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.04
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.04
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.04
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.04
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.04
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.04
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.04
s(mark(z0)) → mark(s(z0)) 258.20/82.04
s(ok(z0)) → ok(s(z0)) 258.20/82.04
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.04
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.04
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.04
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.04
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.04
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.04
proper(tt) → ok(tt) 258.20/82.04
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.04
proper(s(z0)) → s(proper(z0)) 258.20/82.04
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.04
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.04
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.04
proper(0) → ok(0) 258.20/82.04
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.04
top(mark(z0)) → top(proper(z0)) 258.20/82.04
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30

258.20/82.04
258.20/82.04

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

Use narrowing to replace PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) by

PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)), PROPER(tt)) 258.20/82.04
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(isNat(0)) → c31(ISNAT(ok(0)), PROPER(0))
258.20/82.04
258.20/82.04

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.04
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.04
active(and(tt, z0)) → mark(z0) 258.20/82.04
active(isNat(0)) → mark(tt) 258.20/82.04
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.04
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.04
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.04
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.04
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.04
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.04
active(s(z0)) → s(active(z0)) 258.20/82.04
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.04
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.04
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.04
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.04
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.04
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.04
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.04
s(mark(z0)) → mark(s(z0)) 258.20/82.04
s(ok(z0)) → ok(s(z0)) 258.20/82.04
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.04
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.04
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.04
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.04
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.04
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.04
proper(tt) → ok(tt) 258.20/82.04
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.04
proper(s(z0)) → s(proper(z0)) 258.20/82.04
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.04
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.04
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.04
proper(0) → ok(0) 258.20/82.04
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.04
top(mark(z0)) → top(proper(z0)) 258.20/82.04
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)), PROPER(tt)) 258.20/82.04
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(isNat(0)) → c31(ISNAT(ok(0)), PROPER(0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)), PROPER(tt)) 258.20/82.04
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(isNat(0)) → c31(ISNAT(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31

258.20/82.04
258.20/82.04

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

Removed 2 trailing tuple parts
258.20/82.04
258.20/82.04

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.04
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.04
active(and(tt, z0)) → mark(z0) 258.20/82.04
active(isNat(0)) → mark(tt) 258.20/82.04
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.04
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.04
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.04
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.04
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.04
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.04
active(s(z0)) → s(active(z0)) 258.20/82.04
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.04
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.04
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.04
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.04
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.04
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.04
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.04
s(mark(z0)) → mark(s(z0)) 258.20/82.04
s(ok(z0)) → ok(s(z0)) 258.20/82.04
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.04
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.04
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.04
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.04
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.04
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.04
proper(tt) → ok(tt) 258.20/82.04
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.04
proper(s(z0)) → s(proper(z0)) 258.20/82.04
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.04
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.04
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.04
proper(0) → ok(0) 258.20/82.04
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.04
top(mark(z0)) → top(proper(z0)) 258.20/82.04
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.04
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.04
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.04
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.04
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.04
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.04
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.04
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.04
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.04
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.04
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.04
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.04
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.04
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.04
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.04
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.04
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.04
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.04
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.04
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.04
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.04
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.04
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.04
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.04
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.04
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.04
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.04
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.04
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.04
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.04
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.04
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.04
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.04
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.04
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.04
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.04
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.04
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.04
S(mark(z0)) → c18(S(z0)) 258.20/82.04
S(ok(z0)) → c19(S(z0)) 258.20/82.04
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.04
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.04
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.04
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.04
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.04
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.04
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) 258.20/82.04
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.04
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.04
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.04
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.04
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.04
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.04
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.04
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.04
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.04
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.04
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.04
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.04
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.04
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.04
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.04
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.04
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.05
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.05
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.05
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.05
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.05
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.05
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31

258.20/82.05
258.20/82.05

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

Use narrowing to replace TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
TOP(mark(tt)) → c34(TOP(ok(tt)), PROPER(tt)) 258.20/82.05
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
TOP(mark(0)) → c34(TOP(ok(0)), PROPER(0))
258.20/82.05
258.20/82.05

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.05
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.05
active(and(tt, z0)) → mark(z0) 258.20/82.05
active(isNat(0)) → mark(tt) 258.20/82.05
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.05
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.05
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.05
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.05
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.05
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.05
active(s(z0)) → s(active(z0)) 258.20/82.05
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.05
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.05
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.05
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.05
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.05
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.05
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.05
s(mark(z0)) → mark(s(z0)) 258.20/82.05
s(ok(z0)) → ok(s(z0)) 258.20/82.05
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.05
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.05
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.05
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.05
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.05
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.05
proper(tt) → ok(tt) 258.20/82.05
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.05
proper(s(z0)) → s(proper(z0)) 258.20/82.05
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.05
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.05
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.05
proper(0) → ok(0) 258.20/82.05
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.05
top(mark(z0)) → top(proper(z0)) 258.20/82.05
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.05
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.05
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.05
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.05
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.05
S(mark(z0)) → c18(S(z0)) 258.20/82.05
S(ok(z0)) → c19(S(z0)) 258.20/82.05
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.05
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.05
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.05
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.05
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.05
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.05
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.05
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.05
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.05
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.05
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.05
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.05
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.05
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
TOP(mark(tt)) → c34(TOP(ok(tt)), PROPER(tt)) 258.20/82.05
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
TOP(mark(0)) → c34(TOP(ok(0)), PROPER(0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.05
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.05
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.05
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.05
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.05
S(mark(z0)) → c18(S(z0)) 258.20/82.05
S(ok(z0)) → c19(S(z0)) 258.20/82.05
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.05
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.05
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.05
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.05
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.05
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.05
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.05
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.05
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.05
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.05
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.05
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.05
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.05
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
TOP(mark(tt)) → c34(TOP(ok(tt)), PROPER(tt)) 258.20/82.05
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
TOP(mark(0)) → c34(TOP(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34

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258.20/82.05

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

Removed 2 trailing tuple parts
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258.20/82.05

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.05
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.05
active(and(tt, z0)) → mark(z0) 258.20/82.05
active(isNat(0)) → mark(tt) 258.20/82.05
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.05
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.05
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.05
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.05
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.05
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.05
active(s(z0)) → s(active(z0)) 258.20/82.05
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.05
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.05
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.05
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.05
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.05
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.05
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.05
s(mark(z0)) → mark(s(z0)) 258.20/82.05
s(ok(z0)) → ok(s(z0)) 258.20/82.05
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.05
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.05
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.05
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.05
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.05
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.05
proper(tt) → ok(tt) 258.20/82.05
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.05
proper(s(z0)) → s(proper(z0)) 258.20/82.05
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.05
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.05
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.05
proper(0) → ok(0) 258.20/82.05
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.05
top(mark(z0)) → top(proper(z0)) 258.20/82.05
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.05
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.05
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.05
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.05
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.05
S(mark(z0)) → c18(S(z0)) 258.20/82.05
S(ok(z0)) → c19(S(z0)) 258.20/82.05
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.05
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.05
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.05
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.05
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.05
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.05
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.05
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.05
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.05
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.05
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.05
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.05
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.05
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.05
TOP(mark(0)) → c34(TOP(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.05
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.05
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.05
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.05
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.05
S(mark(z0)) → c18(S(z0)) 258.20/82.05
S(ok(z0)) → c19(S(z0)) 258.20/82.05
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.05
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.05
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.05
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.05
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.05
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.05
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.05
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.05
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.05
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.05
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.05
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.05
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.05
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.05
TOP(mark(0)) → c34(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c34

258.20/82.05
258.20/82.05

(59) 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(0)) → c34(TOP(ok(0)))
We considered the (Usable) Rules:

proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.05
proper(tt) → ok(tt) 258.20/82.05
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.05
proper(s(z0)) → s(proper(z0)) 258.20/82.05
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.05
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.05
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.05
proper(0) → ok(0) 258.20/82.05
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.05
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.05
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.05
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.05
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.05
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.05
s(mark(z0)) → mark(s(z0)) 258.20/82.05
s(ok(z0)) → ok(s(z0)) 258.20/82.05
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.05
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.05
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.05
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.05
active(U11(tt, z0)) → mark(z0) 258.20/82.05
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.05
active(and(tt, z0)) → mark(z0) 258.20/82.05
active(isNat(0)) → mark(tt) 258.20/82.05
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.05
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.05
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.05
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.05
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.05
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.05
active(s(z0)) → s(active(z0)) 258.20/82.05
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.05
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.05
active(and(z0, z1)) → and(active(z0), z1)
And the Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.05
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.05
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.05
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.05
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.05
S(mark(z0)) → c18(S(z0)) 258.20/82.05
S(ok(z0)) → c19(S(z0)) 258.20/82.05
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.05
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.05
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.05
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.05
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.05
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.05
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.05
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.05
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.05
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.05
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.05
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.05
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.05
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.05
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.05
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.05
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.05
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.05
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.05
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.05
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.05
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.05
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.05
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.05
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.05
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.05
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.05
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.05
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.05
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.05
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.05
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.05
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.05
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.05
TOP(mark(0)) → c34(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.05

POL(0) = 0    258.20/82.05
POL(ACTIVE(x1)) = 0    258.20/82.05
POL(AND(x1, x2)) = 0    258.20/82.05
POL(ISNAT(x1)) = 0    258.20/82.05
POL(PLUS(x1, x2)) = 0    258.20/82.05
POL(PROPER(x1)) = 0    258.20/82.05
POL(S(x1)) = 0    258.20/82.05
POL(TOP(x1)) = [4]x1    258.20/82.05
POL(U11(x1, x2)) = [1]    258.20/82.05
POL(U11'(x1, x2)) = 0    258.20/82.05
POL(U21(x1, x2, x3)) = [1]    258.20/82.05
POL(U21'(x1, x2, x3)) = 0    258.20/82.05
POL(active(x1)) = x1    258.20/82.05
POL(and(x1, x2)) = [1]    258.20/82.05
POL(c(x1)) = x1    258.20/82.05
POL(c1(x1)) = x1    258.20/82.05
POL(c10(x1)) = x1    258.20/82.05
POL(c10(x1, x2)) = x1 + x2    258.20/82.05
POL(c11(x1, x2)) = x1 + x2    258.20/82.05
POL(c12(x1, x2)) = x1 + x2    258.20/82.05
POL(c13(x1)) = x1    258.20/82.05
POL(c13(x1, x2)) = x1 + x2    258.20/82.05
POL(c14(x1)) = x1    258.20/82.05
POL(c15(x1)) = x1    258.20/82.05
POL(c16(x1)) = x1    258.20/82.05
POL(c17(x1)) = x1    258.20/82.05
POL(c18(x1)) = x1    258.20/82.05
POL(c19(x1)) = x1    258.20/82.05
POL(c2(x1)) = x1    258.20/82.05
POL(c20(x1)) = x1    258.20/82.05
POL(c21(x1)) = x1    258.20/82.05
POL(c22(x1)) = x1    258.20/82.05
POL(c23(x1)) = x1    258.20/82.05
POL(c24(x1)) = x1    258.20/82.05
POL(c25(x1, x2)) = x1 + x2    258.20/82.05
POL(c25(x1, x2, x3)) = x1 + x2 + x3    258.20/82.05
POL(c27(x1, x2, x3)) = x1 + x2 + x3    258.20/82.05
POL(c27(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    258.20/82.05
POL(c28(x1)) = x1    258.20/82.05
POL(c28(x1, x2)) = x1 + x2    258.20/82.05
POL(c29(x1, x2)) = x1 + x2    258.20/82.05
POL(c29(x1, x2, x3)) = x1 + x2 + x3    258.20/82.05
POL(c3(x1)) = x1    258.20/82.05
POL(c30(x1, x2)) = x1 + x2    258.20/82.05
POL(c30(x1, x2, x3)) = x1 + x2 + x3    258.20/82.05
POL(c31(x1)) = x1    258.20/82.05
POL(c31(x1, x2)) = x1 + x2    258.20/82.05
POL(c33(x1)) = x1    258.20/82.05
POL(c34(x1)) = x1    258.20/82.05
POL(c34(x1, x2)) = x1 + x2    258.20/82.05
POL(c35(x1, x2)) = x1 + x2    258.20/82.05
POL(c4(x1)) = x1    258.20/82.05
POL(c5(x1)) = x1    258.20/82.05
POL(c8(x1)) = x1    258.20/82.05
POL(c8(x1, x2)) = x1 + x2    258.20/82.05
POL(c9(x1)) = x1    258.20/82.05
POL(c9(x1, x2)) = x1 + x2    258.20/82.05
POL(isNat(x1)) = [1]    258.20/82.05
POL(mark(x1)) = [1]    258.20/82.05
POL(ok(x1)) = x1    258.20/82.05
POL(plus(x1, x2)) = [1]    258.20/82.05
POL(proper(x1)) = 0    258.20/82.05
POL(s(x1)) = [1]    258.20/82.05
POL(tt) = [1]   
258.20/82.05
258.20/82.05

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.05
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.05
active(and(tt, z0)) → mark(z0) 258.20/82.05
active(isNat(0)) → mark(tt) 258.20/82.05
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.05
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.05
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.05
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.05
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.05
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.05
active(s(z0)) → s(active(z0)) 258.20/82.05
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.05
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.05
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.05
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.05
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.05
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.05
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.05
s(mark(z0)) → mark(s(z0)) 258.20/82.05
s(ok(z0)) → ok(s(z0)) 258.20/82.05
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.05
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.05
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.05
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.05
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.05
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.05
proper(tt) → ok(tt) 258.20/82.05
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.05
proper(s(z0)) → s(proper(z0)) 258.20/82.05
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.05
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.05
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.05
proper(0) → ok(0) 258.20/82.05
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.05
top(mark(z0)) → top(proper(z0)) 258.20/82.05
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.05
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.05
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.05
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.05
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.05
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.05
S(mark(z0)) → c18(S(z0)) 258.20/82.05
S(ok(z0)) → c19(S(z0)) 258.20/82.05
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.05
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.05
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.05
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.05
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.05
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.05
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.05
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.05
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.05
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.05
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.05
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.05
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.05
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.05
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.05
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.05
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.05
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.05
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.05
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.05
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.05
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.05
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.05
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.05
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.05
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.05
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.05
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.05
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.05
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.05
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.06
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.06
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.06
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.06
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.06
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.06
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.06
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.06
TOP(mark(0)) → c34(TOP(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.06
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.06
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.06
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.06
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.06
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.06
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.06
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.06
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
TOP(mark(tt)) → c34(TOP(ok(tt)))
K tuples:

TOP(mark(0)) → c34(TOP(ok(0)))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c34

258.20/82.06
258.20/82.06

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

Use narrowing to replace TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0))) 258.20/82.06
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 258.20/82.06
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.06
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0))) 258.20/82.06
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.06
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
258.20/82.06
258.20/82.06

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.06
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.06
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.06
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.06
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.06
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.06
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.06
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.06
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.06
TOP(mark(0)) → c34(TOP(ok(0))) 258.20/82.06
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0))) 258.20/82.06
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 258.20/82.06
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.06
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0))) 258.20/82.06
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.06
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.06
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.06
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.06
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.06
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.06
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.06
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.06
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.06
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.06
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0))) 258.20/82.06
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 258.20/82.06
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.06
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0))) 258.20/82.06
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.06
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:

TOP(mark(0)) → c34(TOP(ok(0)))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, PROPER, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c34, c35

258.20/82.06
258.20/82.06

(63) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0)) 258.20/82.06
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1)) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0))) 258.20/82.06
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1))) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0)) 258.20/82.06
ACTIVE(plus(z0, 0)) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1)) 258.20/82.06
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0)) 258.20/82.06
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2)) 258.20/82.06
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2)) 258.20/82.06
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(s(isNat(0))) → c10(S(mark(tt))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0))))) 258.20/82.06
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1))))) 258.20/82.06
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0)))) 258.20/82.06
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0)))) 258.20/82.06
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0))) 258.20/82.06
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1))) 258.20/82.06
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 258.20/82.06
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 258.20/82.06
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1)) 258.20/82.06
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2)) 258.20/82.06
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0))) 258.20/82.06
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1)) 258.20/82.06
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1)) 258.20/82.06
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1))) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1)) 258.20/82.06
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1)) 258.20/82.06
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0)))) 258.20/82.06
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 258.20/82.06
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1))) 258.20/82.06
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0))) 258.20/82.06
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2)) 258.20/82.06
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 258.20/82.06
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 258.20/82.06
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 258.20/82.06
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(s(tt)) → c28(S(ok(tt))) 258.20/82.06
PROPER(s(0)) → c28(S(ok(0))) 258.20/82.06
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 258.20/82.06
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 258.20/82.06
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0))) 258.20/82.06
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1)) 258.20/82.06
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 258.20/82.06
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1)) 258.20/82.06
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0)) 258.20/82.06
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0)) 258.20/82.06
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1)) 258.20/82.06
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
PROPER(isNat(tt)) → c31(ISNAT(ok(tt))) 258.20/82.06
PROPER(isNat(0)) → c31(ISNAT(ok(0))) 258.20/82.06
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1))) 258.20/82.06
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2))) 258.20/82.06
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0))) 258.20/82.06
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 258.20/82.06
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 258.20/82.06
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0))) 258.20/82.06
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0))) 258.20/82.06
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1))) 258.20/82.06
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0))) 258.20/82.06
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0))) 258.20/82.06
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1)))) 258.20/82.06
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0)))) 258.20/82.06
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0))) 258.20/82.06
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1)))) 258.20/82.06
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1))) 258.20/82.06
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2))) 258.20/82.06
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 258.20/82.06
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
258.20/82.06
258.20/82.06

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
TOP(mark(tt)) → c34(TOP(ok(tt))) 258.20/82.06
TOP(mark(0)) → c34(TOP(ok(0)))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
TOP(mark(tt)) → c34(TOP(ok(tt)))
K tuples:

TOP(mark(0)) → c34(TOP(ok(0)))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c34

258.20/82.06
258.20/82.06

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

Removed 2 trailing tuple parts
258.20/82.06
258.20/82.06

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
TOP(mark(tt)) → c34 258.20/82.06
TOP(mark(0)) → c34
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
TOP(mark(tt)) → c34
K tuples:

TOP(mark(0)) → c34
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c34

258.20/82.06
258.20/82.06

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

Removed 2 trailing nodes:

TOP(mark(tt)) → c34 258.20/82.06
TOP(mark(0)) → c34
258.20/82.06
258.20/82.06

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.06
258.20/82.06

(69) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.06

POL(AND(x1, x2)) = x1 + x1·x2 + x12·x2    258.20/82.06
POL(ISNAT(x1)) = x1 + x12 + x13    258.20/82.06
POL(PLUS(x1, x2)) = x1·x22    258.20/82.06
POL(S(x1)) = x1 + x12 + x13    258.20/82.06
POL(U11'(x1, x2)) = 0    258.20/82.06
POL(U21'(x1, x2, x3)) = 0    258.20/82.06
POL(c14(x1)) = x1    258.20/82.06
POL(c15(x1)) = x1    258.20/82.06
POL(c16(x1)) = x1    258.20/82.06
POL(c17(x1)) = x1    258.20/82.06
POL(c18(x1)) = x1    258.20/82.06
POL(c19(x1)) = x1    258.20/82.06
POL(c20(x1)) = x1    258.20/82.06
POL(c21(x1)) = x1    258.20/82.06
POL(c22(x1)) = x1    258.20/82.06
POL(c23(x1)) = x1    258.20/82.06
POL(c24(x1)) = x1    258.20/82.06
POL(c33(x1)) = x1    258.20/82.06
POL(mark(x1)) = [1] + x1    258.20/82.06
POL(ok(x1)) = [1] + x1   
258.20/82.06
258.20/82.06

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.06
258.20/82.06

(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.

U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.06

POL(AND(x1, x2)) = [5]x1 + [5]x2    258.20/82.06
POL(ISNAT(x1)) = [5]x1    258.20/82.06
POL(PLUS(x1, x2)) = 0    258.20/82.06
POL(S(x1)) = [3]x1    258.20/82.06
POL(U11'(x1, x2)) = 0    258.20/82.06
POL(U21'(x1, x2, x3)) = x3    258.20/82.06
POL(c14(x1)) = x1    258.20/82.06
POL(c15(x1)) = x1    258.20/82.06
POL(c16(x1)) = x1    258.20/82.06
POL(c17(x1)) = x1    258.20/82.06
POL(c18(x1)) = x1    258.20/82.06
POL(c19(x1)) = x1    258.20/82.06
POL(c20(x1)) = x1    258.20/82.06
POL(c21(x1)) = x1    258.20/82.06
POL(c22(x1)) = x1    258.20/82.06
POL(c23(x1)) = x1    258.20/82.06
POL(c24(x1)) = x1    258.20/82.06
POL(c33(x1)) = x1    258.20/82.06
POL(mark(x1)) = x1    258.20/82.06
POL(ok(x1)) = [1] + x1   
258.20/82.06
258.20/82.06

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.06
258.20/82.06

(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.

U11'(mark(z0), z1) → c14(U11'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.06

POL(AND(x1, x2)) = [5]x1 + [5]x2    258.20/82.06
POL(ISNAT(x1)) = [5]x1    258.20/82.06
POL(PLUS(x1, x2)) = 0    258.20/82.06
POL(S(x1)) = [3]x1    258.20/82.06
POL(U11'(x1, x2)) = [4]x1    258.20/82.06
POL(U21'(x1, x2, x3)) = [5]x2 + [5]x3    258.20/82.06
POL(c14(x1)) = x1    258.20/82.06
POL(c15(x1)) = x1    258.20/82.06
POL(c16(x1)) = x1    258.20/82.06
POL(c17(x1)) = x1    258.20/82.06
POL(c18(x1)) = x1    258.20/82.06
POL(c19(x1)) = x1    258.20/82.06
POL(c20(x1)) = x1    258.20/82.06
POL(c21(x1)) = x1    258.20/82.06
POL(c22(x1)) = x1    258.20/82.06
POL(c23(x1)) = x1    258.20/82.06
POL(c24(x1)) = x1    258.20/82.06
POL(c33(x1)) = x1    258.20/82.06
POL(mark(x1)) = [1] + x1    258.20/82.06
POL(ok(x1)) = x1   
258.20/82.06
258.20/82.06

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.06
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.06
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.06
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.06
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.06
active(s(z0)) → s(active(z0)) 258.20/82.06
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.06
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.06
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.06
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.06
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.06
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.06
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.06
s(mark(z0)) → mark(s(z0)) 258.20/82.06
s(ok(z0)) → ok(s(z0)) 258.20/82.06
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.06
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.06
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.06
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.06
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.06
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.06
proper(tt) → ok(tt) 258.20/82.06
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.06
proper(s(z0)) → s(proper(z0)) 258.20/82.06
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.06
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.06
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.06
proper(0) → ok(0) 258.20/82.06
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.06
top(mark(z0)) → top(proper(z0)) 258.20/82.06
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
U11'(mark(z0), z1) → c14(U11'(z0, z1))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.06
258.20/82.06

(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.

U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.06
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.06
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.06
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.06
S(mark(z0)) → c18(S(z0)) 258.20/82.06
S(ok(z0)) → c19(S(z0)) 258.20/82.06
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.06
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.06
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.06
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.06
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.06
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.06

POL(AND(x1, x2)) = [3]x1 + [5]x2    258.20/82.06
POL(ISNAT(x1)) = [5]x1    258.20/82.06
POL(PLUS(x1, x2)) = 0    258.20/82.06
POL(S(x1)) = [5]x1    258.20/82.06
POL(U11'(x1, x2)) = [2]x1    258.20/82.06
POL(U21'(x1, x2, x3)) = [5]x2 + [3]x3    258.20/82.06
POL(c14(x1)) = x1    258.20/82.06
POL(c15(x1)) = x1    258.20/82.06
POL(c16(x1)) = x1    258.20/82.06
POL(c17(x1)) = x1    258.20/82.06
POL(c18(x1)) = x1    258.20/82.06
POL(c19(x1)) = x1    258.20/82.06
POL(c20(x1)) = x1    258.20/82.06
POL(c21(x1)) = x1    258.20/82.06
POL(c22(x1)) = x1    258.20/82.06
POL(c23(x1)) = x1    258.20/82.06
POL(c24(x1)) = x1    258.20/82.06
POL(c33(x1)) = x1    258.20/82.06
POL(mark(x1)) = x1    258.20/82.06
POL(ok(x1)) = [1] + x1   
258.20/82.06
258.20/82.06

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.06
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.06
active(and(tt, z0)) → mark(z0) 258.20/82.06
active(isNat(0)) → mark(tt) 258.20/82.06
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.06
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.07
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.07
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.07
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.07
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.07
active(s(z0)) → s(active(z0)) 258.20/82.07
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.07
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.07
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.07
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.07
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.07
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.07
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.07
s(mark(z0)) → mark(s(z0)) 258.20/82.07
s(ok(z0)) → ok(s(z0)) 258.20/82.07
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.07
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.07
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.07
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.07
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.07
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.07
proper(tt) → ok(tt) 258.20/82.07
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.07
proper(s(z0)) → s(proper(z0)) 258.20/82.07
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.07
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.07
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.07
proper(0) → ok(0) 258.20/82.07
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.07
top(mark(z0)) → top(proper(z0)) 258.20/82.07
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.07
258.20/82.07

(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.

U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.07

POL(AND(x1, x2)) = [3]x1 + [3]x2    258.20/82.07
POL(ISNAT(x1)) = [5]x1    258.20/82.07
POL(PLUS(x1, x2)) = 0    258.20/82.07
POL(S(x1)) = [3]x1    258.20/82.07
POL(U11'(x1, x2)) = [3]x1 + [5]x2    258.20/82.07
POL(U21'(x1, x2, x3)) = [4]x1 + [5]x2 + [5]x3    258.20/82.07
POL(c14(x1)) = x1    258.20/82.07
POL(c15(x1)) = x1    258.20/82.07
POL(c16(x1)) = x1    258.20/82.07
POL(c17(x1)) = x1    258.20/82.07
POL(c18(x1)) = x1    258.20/82.07
POL(c19(x1)) = x1    258.20/82.07
POL(c20(x1)) = x1    258.20/82.07
POL(c21(x1)) = x1    258.20/82.07
POL(c22(x1)) = x1    258.20/82.07
POL(c23(x1)) = x1    258.20/82.07
POL(c24(x1)) = x1    258.20/82.07
POL(c33(x1)) = x1    258.20/82.07
POL(mark(x1)) = [4] + x1    258.20/82.07
POL(ok(x1)) = x1   
258.20/82.07
258.20/82.07

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.07
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.07
active(and(tt, z0)) → mark(z0) 258.20/82.07
active(isNat(0)) → mark(tt) 258.20/82.07
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.07
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.07
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.07
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.07
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.07
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.07
active(s(z0)) → s(active(z0)) 258.20/82.07
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.07
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.07
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.07
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.07
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.07
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.07
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.07
s(mark(z0)) → mark(s(z0)) 258.20/82.07
s(ok(z0)) → ok(s(z0)) 258.20/82.07
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.07
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.07
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.07
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.07
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.07
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.07
proper(tt) → ok(tt) 258.20/82.07
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.07
proper(s(z0)) → s(proper(z0)) 258.20/82.07
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.07
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.07
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.07
proper(0) → ok(0) 258.20/82.07
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.07
top(mark(z0)) → top(proper(z0)) 258.20/82.07
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.07
258.20/82.07

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

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

PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.07

POL(AND(x1, x2)) = [3]x1 + [5]x2    258.20/82.07
POL(ISNAT(x1)) = [5]x1    258.20/82.07
POL(PLUS(x1, x2)) = x2    258.20/82.07
POL(S(x1)) = [5]x1    258.20/82.07
POL(U11'(x1, x2)) = [3]x1 + [5]x2    258.20/82.07
POL(U21'(x1, x2, x3)) = [5]x1 + [5]x2 + [5]x3    258.20/82.07
POL(c14(x1)) = x1    258.20/82.07
POL(c15(x1)) = x1    258.20/82.07
POL(c16(x1)) = x1    258.20/82.07
POL(c17(x1)) = x1    258.20/82.07
POL(c18(x1)) = x1    258.20/82.07
POL(c19(x1)) = x1    258.20/82.07
POL(c20(x1)) = x1    258.20/82.07
POL(c21(x1)) = x1    258.20/82.07
POL(c22(x1)) = x1    258.20/82.07
POL(c23(x1)) = x1    258.20/82.07
POL(c24(x1)) = x1    258.20/82.07
POL(c33(x1)) = x1    258.20/82.07
POL(mark(x1)) = [1] + x1    258.20/82.07
POL(ok(x1)) = x1   
258.20/82.07
258.20/82.07

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.07
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.07
active(and(tt, z0)) → mark(z0) 258.20/82.07
active(isNat(0)) → mark(tt) 258.20/82.07
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.07
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.07
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.07
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.07
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.07
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.07
active(s(z0)) → s(active(z0)) 258.20/82.07
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.07
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.07
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.07
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.07
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.07
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.07
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.07
s(mark(z0)) → mark(s(z0)) 258.20/82.07
s(ok(z0)) → ok(s(z0)) 258.20/82.07
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.07
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.07
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.07
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.07
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.07
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.07
proper(tt) → ok(tt) 258.20/82.07
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.07
proper(s(z0)) → s(proper(z0)) 258.20/82.07
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.07
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.07
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.07
proper(0) → ok(0) 258.20/82.07
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.07
top(mark(z0)) → top(proper(z0)) 258.20/82.07
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

258.20/82.07
258.20/82.07

(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.

PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 258.20/82.07

POL(AND(x1, x2)) = [3]x1 + [3]x2    258.20/82.07
POL(ISNAT(x1)) = [3]x1    258.20/82.07
POL(PLUS(x1, x2)) = [3]x1 + [3]x2    258.20/82.07
POL(S(x1)) = [3]x1    258.20/82.07
POL(U11'(x1, x2)) = [3]x1 + [5]x2    258.20/82.07
POL(U21'(x1, x2, x3)) = [5]x1 + [3]x2 + [5]x3    258.20/82.07
POL(c14(x1)) = x1    258.20/82.07
POL(c15(x1)) = x1    258.20/82.07
POL(c16(x1)) = x1    258.20/82.07
POL(c17(x1)) = x1    258.20/82.07
POL(c18(x1)) = x1    258.20/82.07
POL(c19(x1)) = x1    258.20/82.07
POL(c20(x1)) = x1    258.20/82.07
POL(c21(x1)) = x1    258.20/82.07
POL(c22(x1)) = x1    258.20/82.07
POL(c23(x1)) = x1    258.20/82.07
POL(c24(x1)) = x1    258.20/82.07
POL(c33(x1)) = x1    258.20/82.07
POL(mark(x1)) = [1] + x1    258.20/82.07
POL(ok(x1)) = x1   
258.20/82.07
258.20/82.07

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0) 258.20/82.07
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0))) 258.20/82.07
active(and(tt, z0)) → mark(z0) 258.20/82.07
active(isNat(0)) → mark(tt) 258.20/82.07
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1))) 258.20/82.07
active(isNat(s(z0))) → mark(isNat(z0)) 258.20/82.07
active(plus(z0, 0)) → mark(U11(isNat(z0), z0)) 258.20/82.07
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0)) 258.20/82.07
active(U11(z0, z1)) → U11(active(z0), z1) 258.20/82.07
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2) 258.20/82.07
active(s(z0)) → s(active(z0)) 258.20/82.07
active(plus(z0, z1)) → plus(active(z0), z1) 258.20/82.07
active(plus(z0, z1)) → plus(z0, active(z1)) 258.20/82.07
active(and(z0, z1)) → and(active(z0), z1) 258.20/82.07
U11(mark(z0), z1) → mark(U11(z0, z1)) 258.20/82.07
U11(ok(z0), ok(z1)) → ok(U11(z0, z1)) 258.20/82.07
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2)) 258.20/82.07
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2)) 258.20/82.07
s(mark(z0)) → mark(s(z0)) 258.20/82.07
s(ok(z0)) → ok(s(z0)) 258.20/82.07
plus(mark(z0), z1) → mark(plus(z0, z1)) 258.20/82.07
plus(z0, mark(z1)) → mark(plus(z0, z1)) 258.20/82.07
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 258.20/82.07
and(mark(z0), z1) → mark(and(z0, z1)) 258.20/82.07
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 258.20/82.07
proper(U11(z0, z1)) → U11(proper(z0), proper(z1)) 258.20/82.07
proper(tt) → ok(tt) 258.20/82.07
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2)) 258.20/82.07
proper(s(z0)) → s(proper(z0)) 258.20/82.07
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 258.20/82.07
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 258.20/82.07
proper(isNat(z0)) → isNat(proper(z0)) 258.20/82.07
proper(0) → ok(0) 258.20/82.07
isNat(ok(z0)) → ok(isNat(z0)) 258.20/82.07
top(mark(z0)) → top(proper(z0)) 258.20/82.07
top(ok(z0)) → top(active(z0))
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:none
K tuples:

S(mark(z0)) → c18(S(z0)) 258.20/82.07
S(ok(z0)) → c19(S(z0)) 258.20/82.07
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1)) 258.20/82.07
AND(mark(z0), z1) → c23(AND(z0, z1)) 258.20/82.07
AND(ok(z0), ok(z1)) → c24(AND(z0, z1)) 258.20/82.07
ISNAT(ok(z0)) → c33(ISNAT(z0)) 258.20/82.07
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2)) 258.20/82.07
U11'(mark(z0), z1) → c14(U11'(z0, z1)) 258.20/82.07
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1)) 258.20/82.07
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2)) 258.20/82.07
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1)) 258.20/82.07
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

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

The set S is empty
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(84) BOUNDS(O(1), O(1))

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258.59/82.24 EOF