Runtime Complexity (innermost) proof of /tmp/tmptf5zDX/PEANO_complete-noand

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

↳11 CdtUsableRulesProof (⇔, 2 ms)

↳12 CdtProblem

↳13 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 2090 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 2099 ms)

↳16 CdtProblem

↳17 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CdtProblem

↳19 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 813 ms)

↳20 CdtProblem

↳21 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳22 CdtProblem

↳23 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 4187 ms)

↳24 CdtProblem

↳25 CdtKnowledgeProof (⇔, 0 ms)

↳26 BOUNDS(O(1), O(1))

(0) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U16(tt) → tt
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U31(tt, z0) → U32(isNatKind(activate(z0)))
U32(tt) → tt
U41(tt) → tt
U51(tt, z0) → U52(isNatKind(activate(z0)), activate(z0))
U52(tt, z0) → activate(z0)
U61(tt, z0, z1) → U62(isNatKind(activate(z0)), activate(z0), activate(z1))
U62(tt, z0, z1) → U63(isNat(activate(z1)), activate(z0), activate(z1))
U63(tt, z0, z1) → U64(isNatKind(activate(z1)), activate(z0), activate(z1))
U64(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
plus(z0, 0) → U51(isNat(z0), z0)
plus(z0, s(z1)) → U61(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U15'(tt, z0) → c4(U16'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U16'(tt) → c5
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U22'(tt, z0) → c7(U23'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U23'(tt) → c8
U31'(tt, z0) → c9(U32'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))
U32'(tt) → c10
U41'(tt) → c11
U51'(tt, z0) → c12(U52'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U52'(tt, z0) → c13(ACTIVATE(z0))
U61'(tt, z0, z1) → c14(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U62'(tt, z0, z1) → c15(U63'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U63'(tt, z0, z1) → c16(U64'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U64'(tt, z0, z1) → c17(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__0) → c18
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
ISNATKIND(n__0) → c21
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNATKIND(n__s(z0)) → c23(U41'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))
PLUS(z0, 0) → c24(U51'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c25(U61'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, z1) → c26
0' → c27
S(z0) → c28
ACTIVATE(n__0) → c29(0')
ACTIVATE(n__plus(z0, z1)) → c30(PLUS(z0, z1))
ACTIVATE(n__s(z0)) → c31(S(z0))
ACTIVATE(z0) → c32

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U15'(tt, z0) → c4(U16'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U16'(tt) → c5
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U22'(tt, z0) → c7(U23'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U23'(tt) → c8
U31'(tt, z0) → c9(U32'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))
U32'(tt) → c10
U41'(tt) → c11
U51'(tt, z0) → c12(U52'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U52'(tt, z0) → c13(ACTIVATE(z0))
U61'(tt, z0, z1) → c14(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U62'(tt, z0, z1) → c15(U63'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U63'(tt, z0, z1) → c16(U64'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U64'(tt, z0, z1) → c17(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__0) → c18
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
ISNATKIND(n__0) → c21
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNATKIND(n__s(z0)) → c23(U41'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))
PLUS(z0, 0) → c24(U51'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c25(U61'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, z1) → c26
0' → c27
S(z0) → c28
ACTIVATE(n__0) → c29(0')
ACTIVATE(n__plus(z0, z1)) → c30(PLUS(z0, z1))
ACTIVATE(n__s(z0)) → c31(S(z0))
ACTIVATE(z0) → c32

K tuples:none
Defined Rule Symbols:

U11, U12, U13, U14, U15, U16, U21, U22, U23, U31, U32, U41, U51, U52, U61, U62, U63, U64, isNat, isNatKind, plus, 0, s, activate

Defined Pair Symbols:

U11', U12', U13', U14', U15', U16', U21', U22', U23', U31', U32', U41', U51', U52', U61', U62', U63', U64', ISNAT, ISNATKIND, PLUS, 0', S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

PLUS(z0, 0) → c24(U51'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c25(U61'(isNat(z1), z1, z0), ISNAT(z1))

Removed 15 trailing nodes:

ACTIVATE(n__0) → c29(0')
U41'(tt) → c11
U64'(tt, z0, z1) → c17(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
S(z0) → c28
U16'(tt) → c5
ACTIVATE(n__plus(z0, z1)) → c30(PLUS(z0, z1))
U52'(tt, z0) → c13(ACTIVATE(z0))
U32'(tt) → c10
ACTIVATE(z0) → c32
ISNATKIND(n__0) → c21
ISNAT(n__0) → c18
ACTIVATE(n__s(z0)) → c31(S(z0))
PLUS(z0, z1) → c26
U23'(tt) → c8
0' → c27

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U16(tt) → tt
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U31(tt, z0) → U32(isNatKind(activate(z0)))
U32(tt) → tt
U41(tt) → tt
U51(tt, z0) → U52(isNatKind(activate(z0)), activate(z0))
U52(tt, z0) → activate(z0)
U61(tt, z0, z1) → U62(isNatKind(activate(z0)), activate(z0), activate(z1))
U62(tt, z0, z1) → U63(isNat(activate(z1)), activate(z0), activate(z1))
U63(tt, z0, z1) → U64(isNatKind(activate(z1)), activate(z0), activate(z1))
U64(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
plus(z0, 0) → U51(isNat(z0), z0)
plus(z0, s(z1)) → U61(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U15'(tt, z0) → c4(U16'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U22'(tt, z0) → c7(U23'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c9(U32'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))
U51'(tt, z0) → c12(U52'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U61'(tt, z0, z1) → c14(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U62'(tt, z0, z1) → c15(U63'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U63'(tt, z0, z1) → c16(U64'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNATKIND(n__s(z0)) → c23(U41'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U15'(tt, z0) → c4(U16'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U22'(tt, z0) → c7(U23'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c9(U32'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))
U51'(tt, z0) → c12(U52'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
U61'(tt, z0, z1) → c14(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
U62'(tt, z0, z1) → c15(U63'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U63'(tt, z0, z1) → c16(U64'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z0))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNATKIND(n__s(z0)) → c23(U41'(isNatKind(activate(z0))), ISNATKIND(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U13, U14, U15, U16, U21, U22, U23, U31, U32, U41, U51, U52, U61, U62, U63, U64, isNat, isNatKind, plus, 0, s, activate

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U61', U62', U63', ISNAT, ISNATKIND

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c14, c15, c16, c19, c20, c22, c23

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

Removed 41 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U16(tt) → tt
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U31(tt, z0) → U32(isNatKind(activate(z0)))
U32(tt) → tt
U41(tt) → tt
U51(tt, z0) → U52(isNatKind(activate(z0)), activate(z0))
U52(tt, z0) → activate(z0)
U61(tt, z0, z1) → U62(isNatKind(activate(z0)), activate(z0), activate(z1))
U62(tt, z0, z1) → U63(isNat(activate(z1)), activate(z0), activate(z1))
U63(tt, z0, z1) → U64(isNatKind(activate(z1)), activate(z0), activate(z1))
U64(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
plus(z0, 0) → U51(isNat(z0), z0)
plus(z0, s(z1)) → U61(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c14(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c15(U63'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c14(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c15(U63'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:none
Defined Rule Symbols:

U11, U12, U13, U14, U15, U16, U21, U22, U23, U31, U32, U41, U51, U52, U61, U62, U63, U64, isNat, isNatKind, plus, 0, s, activate

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U61', U62', U63', ISNAT, ISNATKIND

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c14, c15, c16, c19, c20, c22, c23

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U16(tt) → tt
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U31(tt, z0) → U32(isNatKind(activate(z0)))
U32(tt) → tt
U41(tt) → tt
U51(tt, z0) → U52(isNatKind(activate(z0)), activate(z0))
U52(tt, z0) → activate(z0)
U61(tt, z0, z1) → U62(isNatKind(activate(z0)), activate(z0), activate(z1))
U62(tt, z0, z1) → U63(isNat(activate(z1)), activate(z0), activate(z1))
U63(tt, z0, z1) → U64(isNatKind(activate(z1)), activate(z0), activate(z1))
U64(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
plus(z0, 0) → U51(isNat(z0), z0)
plus(z0, s(z1)) → U61(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

K tuples:none
Defined Rule Symbols:

U11, U12, U13, U14, U15, U16, U21, U22, U23, U31, U32, U41, U51, U52, U61, U62, U63, U64, isNat, isNatKind, plus, 0, s, activate

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

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

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

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U16(tt) → tt
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U31(tt, z0) → U32(isNatKind(activate(z0)))
U32(tt) → tt
U41(tt) → tt
U51(tt, z0) → U52(isNatKind(activate(z0)), activate(z0))
U52(tt, z0) → activate(z0)
U61(tt, z0, z1) → U62(isNatKind(activate(z0)), activate(z0), activate(z1))
U62(tt, z0, z1) → U63(isNat(activate(z1)), activate(z0), activate(z1))
U63(tt, z0, z1) → U64(isNatKind(activate(z1)), activate(z0), activate(z1))
U64(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
plus(z0, 0) → U51(isNat(z0), z0)
plus(z0, s(z1)) → U61(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))

Defined Rule Symbols:

U11, U12, U13, U14, U15, U16, U21, U22, U23, U31, U32, U41, U51, U52, U61, U62, U63, U64, isNat, isNatKind, plus, 0, s, activate

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U51(tt, z0) → U52(isNatKind(activate(z0)), activate(z0))
U52(tt, z0) → activate(z0)
U61(tt, z0, z1) → U62(isNatKind(activate(z0)), activate(z0), activate(z1))
U62(tt, z0, z1) → U63(isNat(activate(z1)), activate(z0), activate(z1))
U63(tt, z0, z1) → U64(isNatKind(activate(z1)), activate(z0), activate(z1))
U64(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
plus(z0, 0) → U51(isNat(z0), z0)
plus(z0, s(z1)) → U61(isNat(z1), z1, z0)

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

(13) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

U22'(tt, z0) → c7(ISNAT(activate(z0)))

We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(z0)
plus(z0, z1) → n__plus(z0, z1)
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__0) → 0
s(z0) → n__s(z0)
0 → n__0

And the Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

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

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = [2] + [3]x₂ + [5]x₃
POL(U11'(x₁, x₂, x₃)) = x₂ + x₃
POL(U12(x₁, x₂, x₃)) = [2]x₃
POL(U12'(x₁, x₂, x₃)) = x₂ + x₃
POL(U13(x₁, x₂, x₃)) = [2] + [3]x₃
POL(U13'(x₁, x₂, x₃)) = x₂ + x₃
POL(U14(x₁, x₂, x₃)) = [2] + x₂
POL(U14'(x₁, x₂, x₃)) = x₂ + x₃
POL(U15(x₁, x₂)) = [3] + [2]x₁
POL(U15'(x₁, x₂)) = x₂
POL(U16(x₁)) = [5] + [2]x₁
POL(U21(x₁, x₂)) = [3] + [2]x₁
POL(U21'(x₁, x₂)) = [1] + x₂
POL(U22(x₁, x₂)) = [4]
POL(U22'(x₁, x₂)) = [1] + x₂
POL(U23(x₁)) = [2]
POL(U31(x₁, x₂)) = [3]x₂
POL(U31'(x₁, x₂)) = 0
POL(U32(x₁)) = [3]
POL(U41(x₁)) = [3] + [5]x₁
POL(U51'(x₁, x₂)) = [4] + [4]x₁
POL(U61'(x₁, x₂, x₃)) = [5] + [2]x₁ + [3]x₂ + [3]x₃
POL(U62'(x₁, x₂, x₃)) = [3]x₂ + x₃
POL(U63'(x₁, x₂, x₃)) = [3]x₂
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = [1] + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = [1] + x₁
POL(tt) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

(15) CdtRuleRemovalProof (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)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))

We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(z0)
plus(z0, z1) → n__plus(z0, z1)
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__0) → 0
s(z0) → n__s(z0)
0 → n__0

And the Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

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

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = [2] + x₂ + [2]x₃
POL(U11'(x₁, x₂, x₃)) = x₂ + x₃
POL(U12(x₁, x₂, x₃)) = [2] + [2]x₂ + [2]x₃
POL(U12'(x₁, x₂, x₃)) = x₂ + x₃
POL(U13(x₁, x₂, x₃)) = [2] + [2]x₁ + x₂ + [4]x₃
POL(U13'(x₁, x₂, x₃)) = x₂ + x₃
POL(U14(x₁, x₂, x₃)) = [1] + [4]x₁ + [4]x₂
POL(U14'(x₁, x₂, x₃)) = x₂ + x₃
POL(U15(x₁, x₂)) = [2] + [2]x₁ + [2]x₂
POL(U15'(x₁, x₂)) = x₂
POL(U16(x₁)) = [3]
POL(U21(x₁, x₂)) = [3] + [3]x₁ + [5]x₂
POL(U21'(x₁, x₂)) = [1] + x₂
POL(U22(x₁, x₂)) = [3] + [2]x₁ + [2]x₂
POL(U22'(x₁, x₂)) = [1] + x₂
POL(U23(x₁)) = [3] + [5]x₁
POL(U31(x₁, x₂)) = [3]x₁
POL(U31'(x₁, x₂)) = 0
POL(U32(x₁)) = [5]
POL(U41(x₁)) = [2]
POL(U51'(x₁, x₂)) = [4] + [2]x₁ + [3]x₂
POL(U61'(x₁, x₂, x₃)) = [5] + [3]x₁ + [3]x₂ + [5]x₃
POL(U62'(x₁, x₂, x₃)) = [5] + [3]x₂ + [3]x₃
POL(U63'(x₁, x₂, x₃)) = [2]x₃
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = [4] + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = [4] + x₁
POL(tt) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

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

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

U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

(19) CdtRuleRemovalProof (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)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))

We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(z0)
plus(z0, z1) → n__plus(z0, z1)
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__0) → 0
s(z0) → n__s(z0)
0 → n__0

And the Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

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

POL(0) = 0
POL(ISNAT(x₁)) = [4]x₁
POL(ISNATKIND(x₁)) = 0
POL(U11(x₁, x₂, x₃)) = [2] + x₁ + [4]x₃
POL(U11'(x₁, x₂, x₃)) = [4]x₂ + [4]x₃
POL(U12(x₁, x₂, x₃)) = [3] + [3]x₂ + [3]x₃
POL(U12'(x₁, x₂, x₃)) = [4]x₂ + [4]x₃
POL(U13(x₁, x₂, x₃)) = [2] + [2]x₁ + [3]x₂ + [2]x₃
POL(U13'(x₁, x₂, x₃)) = [4]x₂ + [4]x₃
POL(U14(x₁, x₂, x₃)) = [2] + [2]x₂ + [3]x₃
POL(U14'(x₁, x₂, x₃)) = [4]x₂ + [4]x₃
POL(U15(x₁, x₂)) = [3] + [3]x₁ + [5]x₂
POL(U15'(x₁, x₂)) = [4]x₂
POL(U16(x₁)) = [5] + [2]x₁
POL(U21(x₁, x₂)) = [2]
POL(U21'(x₁, x₂)) = [3] + [4]x₂
POL(U22(x₁, x₂)) = [2] + [2]x₁ + [3]x₂
POL(U22'(x₁, x₂)) = [4]x₂
POL(U23(x₁)) = [1] + [5]x₁
POL(U31(x₁, x₂)) = [5] + [5]x₁ + [3]x₂
POL(U31'(x₁, x₂)) = 0
POL(U32(x₁)) = [5] + [3]x₁
POL(U41(x₁)) = [3] + [3]x₁
POL(U51'(x₁, x₂)) = [1] + [5]x₁ + [5]x₂
POL(U61'(x₁, x₂, x₃)) = [5] + [3]x₁ + [5]x₂ + [5]x₃
POL(U62'(x₁, x₂, x₃)) = [2] + [2]x₂ + [4]x₃
POL(U63'(x₁, x₂, x₃)) = [2]x₂
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(isNat(x₁)) = [2]
POL(isNatKind(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = [1] + x₁ + x₂
POL(n__s(x₁)) = [4] + x₁
POL(plus(x₁, x₂)) = [1] + x₁ + x₂
POL(s(x₁)) = [4] + x₁
POL(tt) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

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

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

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(z0)
plus(z0, z1) → n__plus(z0, z1)
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__0) → 0
s(z0) → n__s(z0)
0 → n__0

And the Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

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

POL(0) = 0
POL(ISNAT(x₁)) = x₁²
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = [1] + x₂ + [2]x₃ + x₃² + [2]x₂·x₃ + x₂²
POL(U12(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = [1] + [2]x₃ + x₃² + x₂·x₃ + x₂²
POL(U13(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = [1] + x₃ + x₃² + x₂²
POL(U14(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = x₃² + x₂²
POL(U15(x₁, x₂)) = 0
POL(U15'(x₁, x₂)) = x₂²
POL(U16(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21'(x₁, x₂)) = [2]x₂ + x₂²
POL(U22(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = x₂ + x₂²
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U31'(x₁, x₂)) = x₂
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(U51'(x₁, x₂)) = x₂ + x₂²
POL(U61'(x₁, x₂, x₃)) = [1] + [2]x₂ + [2]x₃ + [3]x₃² + [3]x₂²
POL(U62'(x₁, x₂, x₃)) = [1] + [2]x₃ + [2]x₃²
POL(U63'(x₁, x₂, x₃)) = [1] + [2]x₃ + x₃²
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = [1] + x₁ + x₂
POL(n__s(x₁)) = [2] + x₁
POL(plus(x₁, x₂)) = [1] + x₁ + x₂
POL(s(x₁)) = [2] + x₁
POL(tt) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNatKind(n__0) → tt
isNatKind(n__plus(z0, z1)) → U31(isNatKind(activate(z0)), activate(z1))
isNatKind(n__s(z0)) → U41(isNatKind(activate(z0)))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
U31(tt, z0) → U32(isNatKind(activate(z0)))
U41(tt) → tt
U32(tt) → tt
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNatKind(activate(z0)), activate(z0), activate(z1))
isNat(n__s(z0)) → U21(isNatKind(activate(z0)), activate(z0))
U11(tt, z0, z1) → U12(isNatKind(activate(z0)), activate(z0), activate(z1))
U12(tt, z0, z1) → U13(isNatKind(activate(z1)), activate(z0), activate(z1))
U13(tt, z0, z1) → U14(isNatKind(activate(z1)), activate(z0), activate(z1))
U14(tt, z0, z1) → U15(isNat(activate(z0)), activate(z1))
U15(tt, z0) → U16(isNat(activate(z0)))
U21(tt, z0) → U22(isNatKind(activate(z0)), activate(z0))
U22(tt, z0) → U23(isNat(activate(z0)))
U23(tt) → tt
U16(tt) → tt

Tuples:

U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))

S tuples:

U31'(tt, z0) → c9(ISNATKIND(activate(z0)))

K tuples:

U51'(tt, z0) → c12(ISNATKIND(activate(z0)))
U61'(tt, z0, z1) → c5(U62'(isNatKind(activate(z0)), activate(z0), activate(z1)))
U61'(tt, z0, z1) → c5(ISNATKIND(activate(z0)))
U62'(tt, z0, z1) → c5(U63'(isNat(activate(z1)), activate(z0), activate(z1)))
U62'(tt, z0, z1) → c5(ISNAT(activate(z1)))
U63'(tt, z0, z1) → c16(ISNATKIND(activate(z1)))
U22'(tt, z0) → c7(ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c20(U21'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
U21'(tt, z0) → c6(U22'(isNatKind(activate(z0)), activate(z0)), ISNATKIND(activate(z0)))
ISNAT(n__plus(z0, z1)) → c19(U11'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U11'(tt, z0, z1) → c(U12'(isNatKind(activate(z0)), activate(z0), activate(z1)), ISNATKIND(activate(z0)))
U12'(tt, z0, z1) → c1(U13'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U13'(tt, z0, z1) → c2(U14'(isNatKind(activate(z1)), activate(z0), activate(z1)), ISNATKIND(activate(z1)))
U14'(tt, z0, z1) → c3(U15'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
U15'(tt, z0) → c4(ISNAT(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

Defined Rule Symbols:

isNatKind, activate, 0, plus, s, U31, U41, U32, isNat, U11, U12, U13, U14, U15, U21, U22, U23, U16

Defined Pair Symbols:

U11', U12', U13', U14', U15', U21', U22', U31', U51', U63', ISNAT, ISNATKIND, U61', U62'

Compound Symbols:

c, c1, c2, c3, c4, c6, c7, c9, c12, c16, c19, c20, c22, c23, c5

(25) CdtKnowledgeProof (EQUIVALENT transformation)

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

U31'(tt, z0) → c9(ISNATKIND(activate(z0)))
ISNATKIND(n__plus(z0, z1)) → c22(U31'(isNatKind(activate(z0)), activate(z1)), ISNATKIND(activate(z0)))
ISNATKIND(n__s(z0)) → c23(ISNATKIND(activate(z0)))

Now S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtUsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

(13) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(14) Obligation:

(15) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(16) Obligation:

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

(18) Obligation:

(19) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(20) Obligation:

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

(22) Obligation:

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

(24) Obligation:

(25) CdtKnowledgeProof (EQUIVALENT transformation)

(26) BOUNDS(O(1), O(1))