YES(O(1), O(n^1)) 6.68/2.18 YES(O(1), O(n^1)) 7.10/2.28 7.10/2.28 7.10/2.28 7.10/2.28 7.10/2.28 7.10/2.28 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 7.10/2.28 7.10/2.28 7.10/2.28
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 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 (BOTH BOUNDS(ID, ID))
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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 (⇔)
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↳18 BOUNDS(O(1), O(1))
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7.10/2.28

(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2))) 7.10/2.28
U12(tt) → tt 7.10/2.28
U21(tt) → tt 7.10/2.28
U31(tt, N) → activate(N) 7.10/2.28
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N)) 7.10/2.28
U42(tt, M, N) → s(plus(activate(N), activate(M))) 7.10/2.28
isNat(n__0) → tt 7.10/2.28
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2)) 7.10/2.28
isNat(n__s(V1)) → U21(isNat(activate(V1))) 7.10/2.28
plus(N, 0) → U31(isNat(N), N) 7.10/2.28
plus(N, s(M)) → U41(isNat(M), M, N) 7.10/2.28
0 → n__0 7.10/2.28
plus(X1, X2) → n__plus(X1, X2) 7.10/2.28
s(X) → n__s(X) 7.10/2.28
activate(n__0) → 0 7.10/2.28
activate(n__plus(X1, X2)) → plus(X1, X2) 7.10/2.28
activate(n__s(X)) → s(X) 7.10/2.28
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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7.10/2.28

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.28
U12(tt) → tt 7.10/2.28
U21(tt) → tt 7.10/2.28
U31(tt, z0) → activate(z0) 7.10/2.28
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.28
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.28
isNat(n__0) → tt 7.10/2.28
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.28
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.28
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.28
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.28
plus(z0, z1) → n__plus(z0, z1) 7.10/2.28
0 → n__0 7.10/2.28
s(z0) → n__s(z0) 7.10/2.28
activate(n__0) → 0 7.10/2.28
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.28
activate(n__s(z0)) → s(z0) 7.10/2.28
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.28
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 7.10/2.28
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0)) 7.10/2.28
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1)) 7.10/2.28
ACTIVATE(n__0) → c14(0') 7.10/2.28
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1)) 7.10/2.28
ACTIVATE(n__s(z0)) → c16(S(z0))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.28
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 7.10/2.28
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0)) 7.10/2.28
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1)) 7.10/2.28
ACTIVATE(n__0) → c14(0') 7.10/2.28
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1)) 7.10/2.28
ACTIVATE(n__s(z0)) → c16(S(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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7.10/2.28

(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)) 7.10/2.28
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))) 7.10/2.28
U12(tt) → tt 7.10/2.28
U21(tt) → tt 7.10/2.28
U31(tt, z0) → activate(z0) 7.10/2.28
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.28
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.28
isNat(n__0) → tt 7.10/2.28
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.28
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.28
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.28
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.28
plus(z0, z1) → n__plus(z0, z1) 7.10/2.28
0 → n__0 7.10/2.28
s(z0) → n__s(z0) 7.10/2.28
activate(n__0) → 0 7.10/2.28
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.28
activate(n__s(z0)) → s(z0) 7.10/2.28
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.28
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 7.10/2.28
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
ACTIVATE(n__0) → c14(0') 7.10/2.28
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1)) 7.10/2.28
ACTIVATE(n__s(z0)) → c16(S(z0))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.28
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 7.10/2.28
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
ACTIVATE(n__0) → c14(0') 7.10/2.28
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1)) 7.10/2.28
ACTIVATE(n__s(z0)) → c16(S(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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7.10/2.28

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

Removed 7 trailing tuple parts
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7.10/2.28

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.28
U12(tt) → tt 7.10/2.28
U21(tt) → tt 7.10/2.28
U31(tt, z0) → activate(z0) 7.10/2.28
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.28
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.28
isNat(n__0) → tt 7.10/2.28
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.28
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.28
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.28
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.28
plus(z0, z1) → n__plus(z0, z1) 7.10/2.28
0 → n__0 7.10/2.28
s(z0) → n__s(z0) 7.10/2.28
activate(n__0) → 0 7.10/2.28
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.28
activate(n__s(z0)) → s(z0) 7.10/2.28
activate(z0) → z0
Tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.28
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
U42'(tt, z0, z1) → c5(ACTIVATE(z1), ACTIVATE(z0)) 7.10/2.28
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
ACTIVATE(n__0) → c14 7.10/2.28
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.28
ACTIVATE(n__s(z0)) → c16
S tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.28
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.28
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
U42'(tt, z0, z1) → c5(ACTIVATE(z1), ACTIVATE(z0)) 7.10/2.28
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.28
ACTIVATE(n__0) → c14 7.10/2.28
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.28
ACTIVATE(n__s(z0)) → c16
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))) 7.10/2.29
U12(tt) → tt 7.10/2.29
U21(tt) → tt 7.10/2.29
U31(tt, z0) → activate(z0) 7.10/2.29
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.29
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.29
isNat(n__0) → tt 7.10/2.29
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.29
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.29
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.29
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.29
plus(z0, z1) → n__plus(z0, z1) 7.10/2.29
0 → n__0 7.10/2.29
s(z0) → n__s(z0) 7.10/2.29
activate(n__0) → 0 7.10/2.29
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.29
activate(n__s(z0)) → s(z0) 7.10/2.29
activate(z0) → z0
Tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.29
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.29
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.29
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.29
ACTIVATE(n__0) → c14 7.10/2.29
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.29
ACTIVATE(n__s(z0)) → c16 7.10/2.29
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.29
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.29
U41'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.29
U41'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.29
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.29
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.29
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.29
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.29
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.29
ACTIVATE(n__0) → c14 7.10/2.29
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.29
ACTIVATE(n__s(z0)) → c16 7.10/2.29
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.29
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.29
U41'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.29
U41'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.29
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.29
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

7.10/2.29
7.10/2.29

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

Removed 9 trailing nodes:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.29
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.29
U31'(tt, z0) → c3(ACTIVATE(z0)) 7.10/2.29
ACTIVATE(n__s(z0)) → c16 7.10/2.29
U41'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.29
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.29
ACTIVATE(n__0) → c14 7.10/2.29
U41'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.29
ACTIVATE(n__plus(z0, z1)) → c15
7.10/2.29
7.10/2.29

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.29
U12(tt) → tt 7.10/2.29
U21(tt) → tt 7.10/2.29
U31(tt, z0) → activate(z0) 7.10/2.29
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.29
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.29
isNat(n__0) → tt 7.10/2.29
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.29
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.29
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.29
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.29
plus(z0, z1) → n__plus(z0, z1) 7.10/2.30
0 → n__0 7.10/2.30
s(z0) → n__s(z0) 7.10/2.30
activate(n__0) → 0 7.10/2.30
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.30
activate(n__s(z0)) → s(z0) 7.10/2.30
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
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)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
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

7.10/2.30
7.10/2.30

(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))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
7.10/2.30
7.10/2.30

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.30
U12(tt) → tt 7.10/2.30
U21(tt) → tt 7.10/2.30
U31(tt, z0) → activate(z0) 7.10/2.30
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.30
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.30
isNat(n__0) → tt 7.10/2.30
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.30
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.30
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.30
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.30
plus(z0, z1) → n__plus(z0, z1) 7.10/2.30
0 → n__0 7.10/2.30
s(z0) → n__s(z0) 7.10/2.30
activate(n__0) → 0 7.10/2.30
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.30
activate(n__s(z0)) → s(z0) 7.10/2.30
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
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)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
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

7.10/2.30
7.10/2.30

(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.

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 7.10/2.30
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.30
activate(n__s(z0)) → s(z0) 7.10/2.30
activate(z0) → z0 7.10/2.30
s(z0) → n__s(z0) 7.10/2.30
plus(z0, z1) → n__plus(z0, z1) 7.10/2.30
0 → n__0 7.10/2.30
isNat(n__0) → tt 7.10/2.30
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.30
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.30
U21(tt) → tt 7.10/2.30
U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.30
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)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.10/2.30

POL(0) = 0    7.10/2.30
POL(ACTIVATE(x1)) = 0    7.10/2.30
POL(ISNAT(x1)) = x1    7.10/2.30
POL(U11(x1, x2)) = [3]    7.10/2.30
POL(U11'(x1, x2)) = [1] + x2    7.10/2.30
POL(U12(x1)) = [3]    7.10/2.30
POL(U21(x1)) = [3]    7.10/2.30
POL(U41'(x1, x2, x3)) = [1] + x1 + [3]x3    7.10/2.30
POL(U42'(x1, x2, x3)) = [1] + x3    7.10/2.30
POL(activate(x1)) = x1    7.10/2.30
POL(c(x1, x2)) = x1 + x2    7.10/2.30
POL(c1(x1)) = x1    7.10/2.30
POL(c14) = 0    7.10/2.30
POL(c15) = 0    7.10/2.30
POL(c16) = 0    7.10/2.30
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    7.10/2.30
POL(c8(x1, x2)) = x1 + x2    7.10/2.30
POL(isNat(x1)) = 0    7.10/2.30
POL(n__0) = 0    7.10/2.30
POL(n__plus(x1, x2)) = [1] + x1 + x2    7.10/2.30
POL(n__s(x1)) = x1    7.10/2.30
POL(plus(x1, x2)) = [1] + x1 + x2    7.10/2.30
POL(s(x1)) = x1    7.10/2.30
POL(tt) = 0   
7.10/2.30
7.10/2.30

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.30
U12(tt) → tt 7.10/2.30
U21(tt) → tt 7.10/2.30
U31(tt, z0) → activate(z0) 7.10/2.30
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.30
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.30
isNat(n__0) → tt 7.10/2.30
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.30
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.30
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.30
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.30
plus(z0, z1) → n__plus(z0, z1) 7.10/2.30
0 → n__0 7.10/2.30
s(z0) → n__s(z0) 7.10/2.30
activate(n__0) → 0 7.10/2.30
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.30
activate(n__s(z0)) → s(z0) 7.10/2.30
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
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)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.30
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

7.10/2.30
7.10/2.30

(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)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 7.10/2.30
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.30
activate(n__s(z0)) → s(z0) 7.10/2.30
activate(z0) → z0 7.10/2.30
s(z0) → n__s(z0) 7.10/2.30
plus(z0, z1) → n__plus(z0, z1) 7.10/2.30
0 → n__0 7.10/2.30
isNat(n__0) → tt 7.10/2.30
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.30
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.30
U21(tt) → tt 7.10/2.30
U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.30
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)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 7.10/2.30

POL(0) = [4]    7.10/2.30
POL(ACTIVATE(x1)) = 0    7.10/2.30
POL(ISNAT(x1)) = [2]x1    7.10/2.30
POL(U11(x1, x2)) = [4] + [2]x2    7.10/2.30
POL(U11'(x1, x2)) = [2]x2    7.10/2.30
POL(U12(x1)) = [4]    7.10/2.30
POL(U21(x1)) = [5] + x1    7.10/2.30
POL(U41'(x1, x2, x3)) = [1] + [3]x1 + [4]x2 + [3]x3    7.10/2.30
POL(U42'(x1, x2, x3)) = [4] + [4]x2    7.10/2.30
POL(activate(x1)) = x1    7.10/2.30
POL(c(x1, x2)) = x1 + x2    7.10/2.30
POL(c1(x1)) = x1    7.10/2.30
POL(c14) = 0    7.10/2.30
POL(c15) = 0    7.10/2.30
POL(c16) = 0    7.10/2.30
POL(c7(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    7.10/2.30
POL(c8(x1, x2)) = x1 + x2    7.10/2.30
POL(isNat(x1)) = [2]x1    7.10/2.30
POL(n__0) = [4]    7.10/2.30
POL(n__plus(x1, x2)) = [2] + x1 + x2    7.10/2.30
POL(n__s(x1)) = [4] + x1    7.10/2.30
POL(plus(x1, x2)) = [2] + x1 + x2    7.10/2.30
POL(s(x1)) = [4] + x1    7.10/2.30
POL(tt) = [3]   
7.10/2.30
7.10/2.30

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 7.10/2.30
U12(tt) → tt 7.10/2.30
U21(tt) → tt 7.10/2.30
U31(tt, z0) → activate(z0) 7.10/2.30
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1)) 7.10/2.30
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 7.10/2.30
isNat(n__0) → tt 7.10/2.30
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 7.10/2.30
isNat(n__s(z0)) → U21(isNat(activate(z0))) 7.10/2.30
plus(z0, 0) → U31(isNat(z0), z0) 7.10/2.30
plus(z0, s(z1)) → U41(isNat(z1), z1, z0) 7.10/2.30
plus(z0, z1) → n__plus(z0, z1) 7.10/2.30
0 → n__0 7.10/2.30
s(z0) → n__s(z0) 7.10/2.30
activate(n__0) → 0 7.10/2.30
activate(n__plus(z0, z1)) → plus(z0, z1) 7.10/2.30
activate(n__s(z0)) → s(z0) 7.10/2.30
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16 7.10/2.30
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1))) 7.10/2.30
U41'(tt, z0, z1) → c1(ISNAT(activate(z1))) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z1)) 7.10/2.30
U42'(tt, z0, z1) → c1(ACTIVATE(z0)) 7.10/2.30
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 7.10/2.30
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 7.10/2.30
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

7.10/2.30
7.10/2.30

(17) CdtKnowledgeProof (EQUIVALENT transformation)

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

ACTIVATE(n__0) → c14 7.10/2.30
ACTIVATE(n__plus(z0, z1)) → c15 7.10/2.30
ACTIVATE(n__s(z0)) → c16
Now S is empty
7.10/2.30
7.10/2.30

(18) BOUNDS(O(1), O(1))

7.10/2.30
7.10/2.30
7.44/2.37 EOF