Termination w.r.t. Q proof of /tmp/tmpPxVNQB/PEANO_complete-noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

(0) Obligation:

Q restricted rewrite system:
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(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U15¹(tt, V2) → U16¹(isNat(activate(V2)))
U15¹(tt, V2) → ISNAT(activate(V2))
U15¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U22¹(tt, V1) → U23¹(isNat(activate(V1)))
U22¹(tt, V1) → ISNAT(activate(V1))
U22¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, V2) → U32¹(isNatKind(activate(V2)))
U31¹(tt, V2) → ISNATKIND(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U51¹(tt, N) → ISNATKIND(activate(N))
U51¹(tt, N) → ACTIVATE(N)
U52¹(tt, N) → ACTIVATE(N)
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U61¹(tt, M, N) → ISNATKIND(activate(M))
U61¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U62¹(tt, M, N) → ISNAT(activate(N))
U62¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ACTIVATE(N)
U63¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → S(plus(activate(N), activate(M)))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U64¹(tt, M, N) → ACTIVATE(N)
U64¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → U41¹(isNatKind(activate(V1)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U51¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)

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(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U51¹(isNat(N), N)
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U52¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U51¹(tt, N) → ISNATKIND(activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U51¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U64¹(tt, M, N) → ACTIVATE(N)
U64¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ACTIVATE(N)
U63¹(tt, M, N) → ACTIVATE(M)
U62¹(tt, M, N) → ISNAT(activate(N))
U62¹(tt, M, N) → ACTIVATE(N)
U62¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ISNATKIND(activate(M))
U61¹(tt, M, N) → ACTIVATE(M)
U61¹(tt, M, N) → ACTIVATE(N)
U31¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U15¹(tt, V2) → ACTIVATE(V2)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)

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(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.