YES(O(1), O(n^2)) 22.68/6.38 YES(O(1), O(n^2)) 23.42/6.42 23.42/6.42 23.42/6.42 23.42/6.42 23.42/6.42 23.42/6.42 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 23.42/6.42 23.42/6.42 23.42/6.42
23.42/6.42 23.42/6.42 23.42/6.42
23.42/6.42
23.42/6.42

(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2))) 23.42/6.42
U12(tt) → tt 23.42/6.42
U21(tt) → tt 23.42/6.42
U31(tt, V2) → U32(isNat(activate(V2))) 23.42/6.42
U32(tt) → tt 23.42/6.42
U41(tt, N) → activate(N) 23.42/6.42
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N)) 23.42/6.42
U52(tt, M, N) → s(plus(activate(N), activate(M))) 23.42/6.42
U61(tt) → 0 23.42/6.42
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N)) 23.42/6.42
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N)) 23.42/6.42
isNat(n__0) → tt 23.42/6.42
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2)) 23.42/6.42
isNat(n__s(V1)) → U21(isNat(activate(V1))) 23.42/6.42
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2)) 23.42/6.42
plus(N, 0) → U41(isNat(N), N) 23.42/6.42
plus(N, s(M)) → U51(isNat(M), M, N) 23.42/6.42
x(N, 0) → U61(isNat(N)) 23.42/6.42
x(N, s(M)) → U71(isNat(M), M, N) 23.42/6.42
0n__0 23.42/6.42
plus(X1, X2) → n__plus(X1, X2) 23.42/6.42
s(X) → n__s(X) 23.42/6.42
x(X1, X2) → n__x(X1, X2) 23.42/6.42
activate(n__0) → 0 23.42/6.42
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2)) 23.42/6.42
activate(n__s(X)) → s(activate(X)) 23.42/6.42
activate(n__x(X1, X2)) → x(activate(X1), activate(X2)) 23.42/6.42
activate(X) → X

Rewrite Strategy: INNERMOST
23.42/6.42
23.42/6.42

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

Converted CpxTRS to CDT
23.42/6.42
23.42/6.42

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.42
U12(tt) → tt 23.42/6.42
U21(tt) → tt 23.42/6.42
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.42
U32(tt) → tt 23.42/6.42
U41(tt, z0) → activate(z0) 23.42/6.42
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.42
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.42
U61(tt) → 0 23.42/6.42
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.42
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.42
isNat(n__0) → tt 23.42/6.42
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.42
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.42
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.42
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.42
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.42
plus(z0, z1) → n__plus(z0, z1) 23.42/6.42
x(z0, 0) → U61(isNat(z0)) 23.42/6.42
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.42
x(z0, z1) → n__x(z0, z1) 23.42/6.42
0n__0 23.42/6.42
s(z0) → n__s(z0) 23.42/6.42
activate(n__0) → 0 23.42/6.42
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.42
activate(n__s(z0)) → s(activate(z0)) 23.42/6.42
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.42
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.42
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.42
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.42
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.42
U61'(tt) → c8(0') 23.42/6.42
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.42
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 23.42/6.42
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 23.42/6.44
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
ACTIVATE(n__0) → c23(0') 23.42/6.44
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.44
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.44
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.44
U61'(tt) → c8(0') 23.42/6.44
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 23.42/6.44
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 23.42/6.44
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
ACTIVATE(n__0) → c23(0') 23.42/6.44
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.44
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, PLUS, X, ACTIVATE

Compound Symbols:

c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c15, c16, c18, c19, c23, c24, c25, c26

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23.42/6.44

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 23.42/6.44
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 23.42/6.44
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
23.42/6.44
23.42/6.44

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.44
U12(tt) → tt 23.42/6.44
U21(tt) → tt 23.42/6.44
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.44
U32(tt) → tt 23.42/6.44
U41(tt, z0) → activate(z0) 23.42/6.44
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.44
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.44
U61(tt) → 0 23.42/6.44
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.44
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.44
isNat(n__0) → tt 23.42/6.44
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.44
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.44
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.44
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.44
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.44
plus(z0, z1) → n__plus(z0, z1) 23.42/6.44
x(z0, 0) → U61(isNat(z0)) 23.42/6.44
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.44
x(z0, z1) → n__x(z0, z1) 23.42/6.44
0n__0 23.42/6.44
s(z0) → n__s(z0) 23.42/6.44
activate(n__0) → 0 23.42/6.44
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.44
activate(n__s(z0)) → s(activate(z0)) 23.42/6.44
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.44
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.44
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.44
U61'(tt) → c8(0') 23.42/6.44
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__0) → c23(0') 23.42/6.45
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.45
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.45
U61'(tt) → c8(0') 23.42/6.45
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__0) → c23(0') 23.42/6.45
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, ACTIVATE

Compound Symbols:

c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c23, c24, c25, c26

23.42/6.45
23.42/6.45

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

Removed 12 trailing tuple parts
23.42/6.45
23.42/6.45

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.45
U12(tt) → tt 23.42/6.45
U21(tt) → tt 23.42/6.45
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.45
U32(tt) → tt 23.42/6.45
U41(tt, z0) → activate(z0) 23.42/6.45
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.45
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.45
U61(tt) → 0 23.42/6.45
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.45
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.45
isNat(n__0) → tt 23.42/6.45
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.45
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.45
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.45
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.45
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.45
plus(z0, z1) → n__plus(z0, z1) 23.42/6.45
x(z0, 0) → U61(isNat(z0)) 23.42/6.45
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.45
x(z0, z1) → n__x(z0, z1) 23.42/6.45
0n__0 23.42/6.45
s(z0) → n__s(z0) 23.42/6.45
activate(n__0) → 0 23.42/6.45
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.45
activate(n__s(z0)) → s(activate(z0)) 23.42/6.45
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.45
activate(z0) → z0
Tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.45
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U41', U51', U71', ISNAT, U11', U31', U52', U61', U72', ACTIVATE

Compound Symbols:

c5, c6, c9, c12, c14, c, c3, c7, c8, c10, c13, c23, c24, c25, c26

23.42/6.46
23.42/6.46

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

Split RHS of tuples not part of any SCC
23.42/6.46
23.42/6.46

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.46
U12(tt) → tt 23.42/6.46
U21(tt) → tt 23.42/6.46
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.46
U32(tt) → tt 23.42/6.46
U41(tt, z0) → activate(z0) 23.42/6.46
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.46
U61(tt) → 0 23.42/6.46
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.46
isNat(n__0) → tt 23.42/6.46
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.46
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.46
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.46
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.46
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.46
plus(z0, z1) → n__plus(z0, z1) 23.42/6.46
x(z0, 0) → U61(isNat(z0)) 23.42/6.46
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.46
x(z0, z1) → n__x(z0, z1) 23.42/6.46
0n__0 23.42/6.46
s(z0) → n__s(z0) 23.42/6.46
activate(n__0) → 0 23.42/6.46
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.46
activate(n__s(z0)) → s(activate(z0)) 23.42/6.46
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.46
activate(z0) → z0
Tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U41', ISNAT, U11', U31', U61', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c5, c12, c14, c, c3, c8, c13, c23, c24, c25, c26, c1

23.42/6.46
23.42/6.46

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 5 leading nodes:

U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
Removed 2 trailing nodes:

U61'(tt) → c8 23.42/6.46
ACTIVATE(n__0) → c23
23.42/6.46
23.42/6.46

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.46
U12(tt) → tt 23.42/6.46
U21(tt) → tt 23.42/6.46
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.46
U32(tt) → tt 23.42/6.46
U41(tt, z0) → activate(z0) 23.42/6.46
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.46
U61(tt) → 0 23.42/6.46
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.46
isNat(n__0) → tt 23.42/6.46
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.46
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.46
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.46
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.46
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.46
plus(z0, z1) → n__plus(z0, z1) 23.42/6.46
x(z0, 0) → U61(isNat(z0)) 23.42/6.46
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.46
x(z0, z1) → n__x(z0, z1) 23.42/6.46
0n__0 23.42/6.46
s(z0) → n__s(z0) 23.42/6.46
activate(n__0) → 0 23.42/6.46
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.46
activate(n__s(z0)) → s(activate(z0)) 23.42/6.46
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.46
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

23.42/6.46
23.42/6.46

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

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

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
23.42/6.46
23.42/6.46

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.46
U12(tt) → tt 23.42/6.46
U21(tt) → tt 23.42/6.46
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.46
U32(tt) → tt 23.42/6.46
U41(tt, z0) → activate(z0) 23.42/6.46
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.46
U61(tt) → 0 23.42/6.46
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.46
isNat(n__0) → tt 23.42/6.46
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.46
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.46
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.46
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.46
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.46
plus(z0, z1) → n__plus(z0, z1) 23.42/6.46
x(z0, 0) → U61(isNat(z0)) 23.42/6.46
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.46
x(z0, z1) → n__x(z0, z1) 23.42/6.46
0n__0 23.42/6.46
s(z0) → n__s(z0) 23.42/6.46
activate(n__0) → 0 23.42/6.46
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.46
activate(n__s(z0)) → s(activate(z0)) 23.42/6.46
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.46
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__0) → c23 23.42/6.49
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__0) → c23 23.42/6.49
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

23.42/6.49
23.42/6.49

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

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

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 23.42/6.49
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.49
activate(n__s(z0)) → s(activate(z0)) 23.42/6.49
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.49
activate(z0) → z0 23.42/6.49
x(z0, z1) → n__x(z0, z1) 23.42/6.49
s(z0) → n__s(z0) 23.42/6.49
plus(z0, z1) → n__plus(z0, z1) 23.42/6.49
0n__0 23.42/6.49
isNat(n__0) → tt 23.42/6.49
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.49
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.49
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.49
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.49
U32(tt) → tt 23.42/6.49
U21(tt) → tt 23.42/6.49
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.49
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__0) → c23 23.42/6.49
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.42/6.49

POL(0) = 0    23.42/6.49
POL(ACTIVATE(x1)) = 0    23.42/6.49
POL(ISNAT(x1)) = [4]x1    23.42/6.49
POL(U11(x1, x2)) = [3]    23.42/6.49
POL(U11'(x1, x2)) = [3] + [4]x2    23.42/6.49
POL(U12(x1)) = [3]    23.42/6.49
POL(U21(x1)) = [3]    23.42/6.49
POL(U31(x1, x2)) = [3]    23.42/6.49
POL(U31'(x1, x2)) = [4]x2    23.42/6.49
POL(U32(x1)) = [3] + [3]x1    23.42/6.49
POL(U51'(x1, x2, x3)) = x1 + [5]x2 + [4]x3    23.42/6.49
POL(U52'(x1, x2, x3)) = 0    23.42/6.49
POL(U71'(x1, x2, x3)) = x1 + [3]x2 + [4]x3    23.42/6.49
POL(U72'(x1, x2, x3)) = 0    23.42/6.49
POL(activate(x1)) = x1    23.42/6.49
POL(c(x1, x2)) = x1 + x2    23.42/6.49
POL(c1(x1)) = x1    23.42/6.49
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.42/6.49
POL(c13(x1, x2)) = x1 + x2    23.42/6.49
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.42/6.49
POL(c23) = 0    23.42/6.49
POL(c24(x1, x2)) = x1 + x2    23.42/6.49
POL(c25(x1)) = x1    23.42/6.49
POL(c26(x1, x2)) = x1 + x2    23.42/6.49
POL(c3(x1, x2)) = x1 + x2    23.42/6.49
POL(isNat(x1)) = 0    23.42/6.49
POL(n__0) = 0    23.42/6.49
POL(n__plus(x1, x2)) = [1] + x1 + x2    23.42/6.49
POL(n__s(x1)) = [4] + x1    23.42/6.49
POL(n__x(x1, x2)) = x1 + x2    23.42/6.49
POL(plus(x1, x2)) = [1] + x1 + x2    23.42/6.49
POL(s(x1)) = [4] + x1    23.42/6.49
POL(tt) = 0    23.42/6.49
POL(x(x1, x2)) = x1 + x2   
23.42/6.49
23.42/6.49

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.49
U12(tt) → tt 23.42/6.49
U21(tt) → tt 23.42/6.49
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.49
U32(tt) → tt 23.42/6.49
U41(tt, z0) → activate(z0) 23.42/6.49
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.49
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.49
U61(tt) → 0 23.42/6.49
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.49
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.49
isNat(n__0) → tt 23.42/6.49
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.49
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.49
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.49
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.49
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

23.80/6.52
23.80/6.52

(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.

U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
0n__0 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.80/6.52

POL(0) = 0    23.80/6.52
POL(ACTIVATE(x1)) = 0    23.80/6.52
POL(ISNAT(x1)) = [3] + [4]x1    23.80/6.52
POL(U11(x1, x2)) = [2]    23.80/6.52
POL(U11'(x1, x2)) = [3] + x1 + [4]x2    23.80/6.52
POL(U12(x1)) = [1]    23.80/6.52
POL(U21(x1)) = [4]    23.80/6.52
POL(U31(x1, x2)) = [4]    23.80/6.52
POL(U31'(x1, x2)) = [4] + [4]x2    23.80/6.52
POL(U32(x1)) = 0    23.80/6.52
POL(U51'(x1, x2, x3)) = [3] + x1 + [5]x2 + [4]x3    23.80/6.52
POL(U52'(x1, x2, x3)) = [2]x2    23.80/6.52
POL(U71'(x1, x2, x3)) = [3] + x1 + [4]x3    23.80/6.52
POL(U72'(x1, x2, x3)) = 0    23.80/6.52
POL(activate(x1)) = x1    23.80/6.52
POL(c(x1, x2)) = x1 + x2    23.80/6.52
POL(c1(x1)) = x1    23.80/6.52
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.80/6.52
POL(c13(x1, x2)) = x1 + x2    23.80/6.52
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.80/6.52
POL(c23) = 0    23.80/6.52
POL(c24(x1, x2)) = x1 + x2    23.80/6.52
POL(c25(x1)) = x1    23.80/6.52
POL(c26(x1, x2)) = x1 + x2    23.80/6.52
POL(c3(x1, x2)) = x1 + x2    23.80/6.52
POL(isNat(x1)) = [4]    23.80/6.52
POL(n__0) = 0    23.80/6.52
POL(n__plus(x1, x2)) = [4] + x1 + x2    23.80/6.52
POL(n__s(x1)) = x1    23.80/6.52
POL(n__x(x1, x2)) = [1] + x1 + x2    23.80/6.52
POL(plus(x1, x2)) = [4] + x1 + x2    23.80/6.52
POL(s(x1)) = x1    23.80/6.52
POL(tt) = 0    23.80/6.52
POL(x(x1, x2)) = [1] + x1 + x2   
23.80/6.52
23.80/6.52

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U41(tt, z0) → activate(z0) 23.80/6.52
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.80/6.52
U61(tt) → 0 23.80/6.52
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
plus(z0, 0) → U41(isNat(z0), z0) 23.80/6.52
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

23.80/6.52
23.80/6.52

(17) 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__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
0n__0 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.80/6.52

POL(0) = 0    23.80/6.52
POL(ACTIVATE(x1)) = 0    23.80/6.52
POL(ISNAT(x1)) = x1    23.80/6.52
POL(U11(x1, x2)) = [3]    23.80/6.52
POL(U11'(x1, x2)) = x2    23.80/6.52
POL(U12(x1)) = [3]    23.80/6.52
POL(U21(x1)) = [3]    23.80/6.52
POL(U31(x1, x2)) = [3]    23.80/6.52
POL(U31'(x1, x2)) = [3] + x2    23.80/6.52
POL(U32(x1)) = [3] + [3]x1    23.80/6.52
POL(U51'(x1, x2, x3)) = [5]x2 + [3]x3    23.80/6.52
POL(U52'(x1, x2, x3)) = 0    23.80/6.52
POL(U71'(x1, x2, x3)) = [2]x1 + [5]x2 + [2]x3    23.80/6.52
POL(U72'(x1, x2, x3)) = 0    23.80/6.52
POL(activate(x1)) = x1    23.80/6.52
POL(c(x1, x2)) = x1 + x2    23.80/6.52
POL(c1(x1)) = x1    23.80/6.52
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.80/6.52
POL(c13(x1, x2)) = x1 + x2    23.80/6.52
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.80/6.52
POL(c23) = 0    23.80/6.52
POL(c24(x1, x2)) = x1 + x2    23.80/6.52
POL(c25(x1)) = x1    23.80/6.52
POL(c26(x1, x2)) = x1 + x2    23.80/6.52
POL(c3(x1, x2)) = x1 + x2    23.80/6.52
POL(isNat(x1)) = 0    23.80/6.52
POL(n__0) = 0    23.80/6.52
POL(n__plus(x1, x2)) = x1 + x2    23.80/6.52
POL(n__s(x1)) = x1    23.80/6.52
POL(n__x(x1, x2)) = [4] + x1 + x2    23.80/6.52
POL(plus(x1, x2)) = x1 + x2    23.80/6.52
POL(s(x1)) = x1    23.80/6.52
POL(tt) = [5]    23.80/6.52
POL(x(x1, x2)) = [4] + x1 + x2   
23.80/6.52
23.80/6.52

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U41(tt, z0) → activate(z0) 23.80/6.52
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.80/6.52
U61(tt) → 0 23.80/6.52
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
plus(z0, 0) → U41(isNat(z0), z0) 23.80/6.52
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

23.80/6.52
23.80/6.52

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

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

ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
0n__0 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.80/6.52

POL(0) = 0    23.80/6.52
POL(ACTIVATE(x1)) = [2]x1    23.80/6.52
POL(ISNAT(x1)) = x12    23.80/6.52
POL(U11(x1, x2)) = [3] + [3]x1    23.80/6.52
POL(U11'(x1, x2)) = [2]x2 + x22    23.80/6.52
POL(U12(x1)) = [3]    23.80/6.52
POL(U21(x1)) = [3] + [3]x1    23.80/6.52
POL(U31(x1, x2)) = [3] + [3]x1    23.80/6.52
POL(U31'(x1, x2)) = [2] + [2]x2 + x22    23.80/6.52
POL(U32(x1)) = [3] + [3]x1    23.80/6.52
POL(U51'(x1, x2, x3)) = [2] + [3]x1 + [3]x2 + [3]x3 + [3]x32 + [2]x2·x3 + [3]x1·x3 + [3]x12 + [3]x1·x2    23.80/6.52
POL(U52'(x1, x2, x3)) = [2]x2 + [2]x3 + [2]x2·x3    23.80/6.52
POL(U71'(x1, x2, x3)) = [2] + [3]x1 + [3]x2 + [3]x3 + [3]x32 + [3]x1·x3 + [3]x12 + [3]x1·x2 + x22    23.80/6.52
POL(U72'(x1, x2, x3)) = [2]x2 + [2]x3 + [2]x32    23.80/6.52
POL(activate(x1)) = x1    23.80/6.52
POL(c(x1, x2)) = x1 + x2    23.80/6.52
POL(c1(x1)) = x1    23.80/6.52
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.80/6.52
POL(c13(x1, x2)) = x1 + x2    23.80/6.52
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    23.80/6.52
POL(c23) = 0    23.80/6.52
POL(c24(x1, x2)) = x1 + x2    23.80/6.52
POL(c25(x1)) = x1    23.80/6.52
POL(c26(x1, x2)) = x1 + x2    23.80/6.52
POL(c3(x1, x2)) = x1 + x2    23.80/6.52
POL(isNat(x1)) = 0    23.80/6.52
POL(n__0) = 0    23.80/6.52
POL(n__plus(x1, x2)) = [3] + x1 + x2    23.80/6.52
POL(n__s(x1)) = [1] + x1    23.80/6.52
POL(n__x(x1, x2)) = [2] + x1 + x2    23.80/6.52
POL(plus(x1, x2)) = [3] + x1 + x2    23.80/6.52
POL(s(x1)) = [1] + x1    23.80/6.52
POL(tt) = 0    23.80/6.52
POL(x(x1, x2)) = [2] + x1 + x2   
23.80/6.52
23.80/6.52

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U41(tt, z0) → activate(z0) 23.80/6.52
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.80/6.52
U61(tt) → 0 23.80/6.52
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
plus(z0, 0) → U41(isNat(z0), z0) 23.80/6.52
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ACTIVATE(n__0) → c23
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

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(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

ACTIVATE(n__0) → c23
Now S is empty
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(22) BOUNDS(O(1), O(1))

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23.99/6.68 EOF