YES(O(1), O(n^2)) 12.38/3.64 YES(O(1), O(n^2)) 12.38/3.69 12.38/3.69 12.38/3.69 12.38/3.69 12.38/3.69 12.38/3.69 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 12.38/3.69 12.38/3.69 12.38/3.69
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
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↳8 CdtProblem
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↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication)
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↳10 CdtProblem
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↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
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↳12 CdtProblem
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↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳14 CdtProblem
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↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳16 CdtProblem
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↳17 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
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↳18 CdtProblem
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↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
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↳20 CdtProblem
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↳21 CdtKnowledgeProof (⇔)
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↳22 BOUNDS(O(1), O(1))
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(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2))) 12.38/3.69
U12(tt) → tt 12.38/3.69
U21(tt) → tt 12.38/3.69
U31(tt, N) → activate(N) 12.38/3.69
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N)) 12.38/3.69
U42(tt, M, N) → s(plus(activate(N), activate(M))) 12.38/3.69
isNat(n__0) → tt 12.38/3.69
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2)) 12.38/3.69
isNat(n__s(V1)) → U21(isNat(activate(V1))) 12.38/3.69
plus(N, 0) → U31(isNat(N), N) 12.38/3.69
plus(N, s(M)) → U41(isNat(M), M, N) 12.38/3.69
0 → n__0 12.38/3.69
plus(X1, X2) → n__plus(X1, X2) 12.38/3.69
s(X) → n__s(X) 12.38/3.69
activate(n__0) → 0 12.38/3.69
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2)) 12.38/3.69
activate(n__s(X)) → s(activate(X)) 12.38/3.69
activate(X) → 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:

U11(tt, z0) → U12(isNat(activate(z0))) 12.38/3.69
U12(tt) → tt 12.38/3.69
U21(tt) → tt 12.38/3.69
U31(tt, z0) → activate(z0) 12.38/3.69
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.38/3.69
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.38/3.69
isNat(n__0) → tt 12.38/3.69
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.38/3.69
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.38/3.69
plus(z0, 0) → U31(isNat(z0), z0) 12.38/3.69
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.38/3.69
plus(z0, z1) → n__plus(z0, z1) 12.38/3.69
0 → n__0 12.38/3.69
s(z0) → n__s(z0) 12.38/3.69
activate(n__0) → 0 12.38/3.69
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.38/3.69
activate(n__s(z0)) → s(activate(z0)) 12.38/3.69
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.38/3.69
U31'(tt, z0) → c3(ACTIVATE(z0)) 12.38/3.69
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 12.38/3.69
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 12.38/3.69
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.38/3.69
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.38/3.69
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0)) 12.38/3.69
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1)) 12.38/3.69
ACTIVATE(n__0) → c14(0') 12.38/3.69
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 12.38/3.69
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.38/3.69
U31'(tt, z0) → c3(ACTIVATE(z0)) 12.38/3.69
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 12.38/3.69
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 12.38/3.69
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.38/3.69
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.38/3.69
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0)) 12.38/3.69
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1)) 12.38/3.69
ACTIVATE(n__0) → c14(0') 12.38/3.69
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 12.38/3.69
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U42', ISNAT, PLUS, ACTIVATE

Compound Symbols:

c, c3, c4, c5, c7, c8, c9, c10, c14, c15, c16

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0)) 12.38/3.69
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.73
U12(tt) → tt 12.97/3.73
U21(tt) → tt 12.97/3.73
U31(tt, z0) → activate(z0) 12.97/3.73
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.73
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.73
isNat(n__0) → tt 12.97/3.73
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.73
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.73
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.73
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.73
plus(z0, z1) → n__plus(z0, z1) 12.97/3.73
0 → n__0 12.97/3.73
s(z0) → n__s(z0) 12.97/3.73
activate(n__0) → 0 12.97/3.73
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.73
activate(n__s(z0)) → s(activate(z0)) 12.97/3.73
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14(0') 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14(0') 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U42', ISNAT, ACTIVATE

Compound Symbols:

c, c3, c4, c5, c7, c8, c14, c15, c16

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

Removed 7 trailing tuple parts
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.73
U12(tt) → tt 12.97/3.73
U21(tt) → tt 12.97/3.73
U31(tt, z0) → activate(z0) 12.97/3.73
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.73
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.73
isNat(n__0) → tt 12.97/3.73
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.73
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.73
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.73
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.73
plus(z0, z1) → n__plus(z0, z1) 12.97/3.73
0 → n__0 12.97/3.73
s(z0) → n__s(z0) 12.97/3.73
activate(n__0) → 0 12.97/3.73
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.73
activate(n__s(z0)) → s(activate(z0)) 12.97/3.73
activate(z0) → z0
Tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
U42'(tt, z0, z1) → c5(ACTIVATE(z1), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
U42'(tt, z0, z1) → c5(ACTIVATE(z1), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U31', U41', ISNAT, U11', U42', ACTIVATE

Compound Symbols:

c3, c4, c7, c, c5, c8, c14, c15, c16

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

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

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.73
U12(tt) → tt 12.97/3.73
U21(tt) → tt 12.97/3.73
U31(tt, z0) → activate(z0) 12.97/3.73
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.73
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.73
isNat(n__0) → tt 12.97/3.73
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.73
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.73
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.73
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.73
plus(z0, z1) → n__plus(z0, z1) 12.97/3.73
0 → n__0 12.97/3.73
s(z0) → n__s(z0) 12.97/3.73
activate(n__0) → 0 12.97/3.73
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.73
activate(n__s(z0)) → s(activate(z0)) 12.97/3.73
activate(z0) → z0
Tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U41'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U41'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U31', ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c3, c7, c, c8, c14, c15, c16, c1

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12.97/3.73

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 3 leading nodes:

U31'(tt, z0) → c3(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
Removed 1 trailing nodes:

ACTIVATE(n__0) → c14
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.73
U12(tt) → tt 12.97/3.73
U21(tt) → tt 12.97/3.73
U31(tt, z0) → activate(z0) 12.97/3.73
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.73
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.73
isNat(n__0) → tt 12.97/3.73
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.73
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.73
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.73
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.73
plus(z0, z1) → n__plus(z0, z1) 12.97/3.73
0 → n__0 12.97/3.73
s(z0) → n__s(z0) 12.97/3.73
activate(n__0) → 0 12.97/3.73
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.73
activate(n__s(z0)) → s(activate(z0)) 12.97/3.73
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c7, c, c8, c14, c15, c16, c1

12.97/3.73
12.97/3.73

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

The following tuples could be moved from S to K by knowledge propagation:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
12.97/3.73
12.97/3.73

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.73
U12(tt) → tt 12.97/3.73
U21(tt) → tt 12.97/3.73
U31(tt, z0) → activate(z0) 12.97/3.73
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.73
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.73
isNat(n__0) → tt 12.97/3.73
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.73
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.73
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.73
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.73
plus(z0, z1) → n__plus(z0, z1) 12.97/3.73
0 → n__0 12.97/3.73
s(z0) → n__s(z0) 12.97/3.73
activate(n__0) → 0 12.97/3.73
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.73
activate(n__s(z0)) → s(activate(z0)) 12.97/3.73
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.73
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.73
ACTIVATE(n__0) → c14 12.97/3.73
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.73
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.73
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.73
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c7, c, c8, c14, c15, c16, c1

12.97/3.73
12.97/3.73

(13) 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.

ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 12.97/3.74
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.74
activate(n__s(z0)) → s(activate(z0)) 12.97/3.74
activate(z0) → z0 12.97/3.74
s(z0) → n__s(z0) 12.97/3.74
plus(z0, z1) → n__plus(z0, z1) 12.97/3.74
0 → n__0 12.97/3.74
isNat(n__0) → tt 12.97/3.74
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.74
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.74
U21(tt) → tt 12.97/3.74
U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.74
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.74
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.74
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.74
ACTIVATE(n__0) → c14 12.97/3.74
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.74
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.74
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.74
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.74
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.74
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 12.97/3.74

POL(0) = 0    12.97/3.74
POL(ACTIVATE(x1)) = 0    12.97/3.74
POL(ISNAT(x1)) = [2]x1    12.97/3.74
POL(U11(x1, x2)) = [3]    12.97/3.74
POL(U11'(x1, x2)) = [2]x2    12.97/3.74
POL(U12(x1)) = [3]    12.97/3.74
POL(U21(x1)) = [3]    12.97/3.74
POL(U41'(x1, x2, x3)) = [3] + x1 + [5]x2 + [5]x3    12.97/3.74
POL(U42'(x1, x2, x3)) = [2] + [2]x2    12.97/3.74
POL(activate(x1)) = x1    12.97/3.74
POL(c(x1, x2)) = x1 + x2    12.97/3.74
POL(c1(x1)) = x1    12.97/3.74
POL(c14) = 0    12.97/3.74
POL(c15(x1, x2)) = x1 + x2    12.97/3.74
POL(c16(x1)) = x1    12.97/3.74
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    12.97/3.74
POL(c8(x1, x2)) = x1 + x2    12.97/3.74
POL(isNat(x1)) = 0    12.97/3.76
POL(n__0) = 0    12.97/3.76
POL(n__plus(x1, x2)) = x1 + x2    12.97/3.76
POL(n__s(x1)) = [1] + x1    12.97/3.76
POL(plus(x1, x2)) = x1 + x2    12.97/3.76
POL(s(x1)) = [1] + x1    12.97/3.76
POL(tt) = 0   
12.97/3.76
12.97/3.76

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.76
U12(tt) → tt 12.97/3.76
U21(tt) → tt 12.97/3.76
U31(tt, z0) → activate(z0) 12.97/3.76
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.76
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.76
isNat(n__0) → tt 12.97/3.76
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.76
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.76
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.76
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.76
plus(z0, z1) → n__plus(z0, z1) 12.97/3.76
0 → n__0 12.97/3.76
s(z0) → n__s(z0) 12.97/3.76
activate(n__0) → 0 12.97/3.76
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.76
activate(n__s(z0)) → s(activate(z0)) 12.97/3.76
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.76
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c7, c, c8, c14, c15, c16, c1

12.97/3.76
12.97/3.76

(15) 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.

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 12.97/3.76
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.76
activate(n__s(z0)) → s(activate(z0)) 12.97/3.76
activate(z0) → z0 12.97/3.76
s(z0) → n__s(z0) 12.97/3.76
plus(z0, z1) → n__plus(z0, z1) 12.97/3.76
0 → n__0 12.97/3.76
isNat(n__0) → tt 12.97/3.76
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.76
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.76
U21(tt) → tt 12.97/3.76
U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.76
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.76
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 12.97/3.76

POL(0) = 0    12.97/3.76
POL(ACTIVATE(x1)) = 0    12.97/3.76
POL(ISNAT(x1)) = [4]x1    12.97/3.76
POL(U11(x1, x2)) = [3]    12.97/3.76
POL(U11'(x1, x2)) = [4]x2    12.97/3.76
POL(U12(x1)) = [3]    12.97/3.76
POL(U21(x1)) = [3]    12.97/3.76
POL(U41'(x1, x2, x3)) = [4] + x1 + [3]x2 + [4]x3    12.97/3.76
POL(U42'(x1, x2, x3)) = [4] + x2    12.97/3.76
POL(activate(x1)) = x1    12.97/3.76
POL(c(x1, x2)) = x1 + x2    12.97/3.76
POL(c1(x1)) = x1    12.97/3.76
POL(c14) = 0    12.97/3.76
POL(c15(x1, x2)) = x1 + x2    12.97/3.76
POL(c16(x1)) = x1    12.97/3.76
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    12.97/3.76
POL(c8(x1, x2)) = x1 + x2    12.97/3.76
POL(isNat(x1)) = 0    12.97/3.76
POL(n__0) = 0    12.97/3.76
POL(n__plus(x1, x2)) = [1] + x1 + x2    12.97/3.76
POL(n__s(x1)) = x1    12.97/3.76
POL(plus(x1, x2)) = [1] + x1 + x2    12.97/3.76
POL(s(x1)) = x1    12.97/3.76
POL(tt) = 0   
12.97/3.76
12.97/3.76

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.76
U12(tt) → tt 12.97/3.76
U21(tt) → tt 12.97/3.76
U31(tt, z0) → activate(z0) 12.97/3.76
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.76
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.76
isNat(n__0) → tt 12.97/3.76
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.76
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.76
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.76
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.76
plus(z0, z1) → n__plus(z0, z1) 12.97/3.76
0 → n__0 12.97/3.76
s(z0) → n__s(z0) 12.97/3.76
activate(n__0) → 0 12.97/3.76
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.76
activate(n__s(z0)) → s(activate(z0)) 12.97/3.76
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.76
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c7, c, c8, c14, c15, c16, c1

12.97/3.76
12.97/3.76

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

The following tuples could be moved from S to K by knowledge propagation:

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
12.97/3.76
12.97/3.76

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.76
U12(tt) → tt 12.97/3.76
U21(tt) → tt 12.97/3.76
U31(tt, z0) → activate(z0) 12.97/3.76
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.76
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.76
isNat(n__0) → tt 12.97/3.76
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.76
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.76
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.76
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.76
plus(z0, z1) → n__plus(z0, z1) 12.97/3.76
0 → n__0 12.97/3.76
s(z0) → n__s(z0) 12.97/3.76
activate(n__0) → 0 12.97/3.76
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.76
activate(n__s(z0)) → s(activate(z0)) 12.97/3.76
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.76
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c7, c, c8, c14, c15, c16, c1

12.97/3.76
12.97/3.76

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 12.97/3.76
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.76
activate(n__s(z0)) → s(activate(z0)) 12.97/3.76
activate(z0) → z0 12.97/3.76
s(z0) → n__s(z0) 12.97/3.76
plus(z0, z1) → n__plus(z0, z1) 12.97/3.76
0 → n__0 12.97/3.76
isNat(n__0) → tt 12.97/3.76
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.76
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.76
U21(tt) → tt 12.97/3.76
U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.76
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.76
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 12.97/3.76

POL(0) = 0    12.97/3.76
POL(ACTIVATE(x1)) = [2]x1    12.97/3.76
POL(ISNAT(x1)) = [2]x12    12.97/3.76
POL(U11(x1, x2)) = [2]    12.97/3.76
POL(U11'(x1, x2)) = [3]x2 + [2]x22 + x1·x2    12.97/3.76
POL(U12(x1)) = [2]    12.97/3.76
POL(U21(x1)) = [2]    12.97/3.76
POL(U41'(x1, x2, x3)) = [2] + [2]x1 + x2 + [2]x3 + [3]x32 + x2·x3 + [2]x1·x3 + [3]x12 + [2]x1·x2 + [2]x22    12.97/3.76
POL(U42'(x1, x2, x3)) = [2]x3 + x1·x2    12.97/3.76
POL(activate(x1)) = x1    12.97/3.76
POL(c(x1, x2)) = x1 + x2    12.97/3.76
POL(c1(x1)) = x1    12.97/3.76
POL(c14) = 0    12.97/3.76
POL(c15(x1, x2)) = x1 + x2    12.97/3.76
POL(c16(x1)) = x1    12.97/3.76
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    12.97/3.76
POL(c8(x1, x2)) = x1 + x2    12.97/3.76
POL(isNat(x1)) = [3]    12.97/3.76
POL(n__0) = 0    12.97/3.76
POL(n__plus(x1, x2)) = [2] + x1 + x2    12.97/3.76
POL(n__s(x1)) = [2] + x1    12.97/3.76
POL(plus(x1, x2)) = [2] + x1 + x2    12.97/3.76
POL(s(x1)) = [2] + x1    12.97/3.76
POL(tt) = [2]   
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(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 12.97/3.76
U12(tt) → tt 12.97/3.76
U21(tt) → tt 12.97/3.76
U31(tt, z0) → activate(z0) 12.97/3.76
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 12.97/3.76
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 12.97/3.76
isNat(n__0) → tt 12.97/3.76
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 12.97/3.76
isNat(n__s(z0)) → U21(isNat(activate(z0))) 12.97/3.76
plus(z0, 0) → U31(isNat(z0), z0) 12.97/3.76
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 12.97/3.76
plus(z0, z1) → n__plus(z0, z1) 12.97/3.76
0 → n__0 12.97/3.76
s(z0) → n__s(z0) 12.97/3.76
activate(n__0) → 0 12.97/3.76
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 12.97/3.76
activate(n__s(z0)) → s(activate(z0)) 12.97/3.76
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__0) → c14 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0)) 12.97/3.76
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ACTIVATE(n__0) → c14
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 12.97/3.76
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 12.97/3.76
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 12.97/3.76
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 12.97/3.76
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 12.97/3.76
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', ACTIVATE, U41', U42'

Compound Symbols:

c7, c, c8, c14, c15, c16, c1

12.97/3.76
12.97/3.76

(21) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

ACTIVATE(n__0) → c14
Now S is empty
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(22) BOUNDS(O(1), O(1))

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12.97/3.79 EOF