Termination w.r.t. Q proof of /tmp/tmpoTFcWM/PEANO_nokinds-noand

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(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

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, V2) → U12¹(isNat(activate(V2)))
U11¹(tt, V2) → ISNAT(activate(V2))
U11¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, N) → ACTIVATE(N)
U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N))
U41¹(tt, M, N) → ISNAT(activate(N))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
U42¹(tt, M, N) → S(plus(activate(N), activate(M)))
U42¹(tt, M, N) → PLUS(activate(N), activate(M))
U42¹(tt, M, N) → ACTIVATE(N)
U42¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U31¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U41¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__s(X)) → S(X)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(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

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 5 less nodes.

(4) Obligation:

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

U11¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U11¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U31¹(isNat(N), N)
U31¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(isNat(M), M, N)
U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N))
U42¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U42¹(tt, M, N) → ACTIVATE(N)
U42¹(tt, M, N) → ACTIVATE(M)
U41¹(tt, M, N) → ISNAT(activate(N))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
U31¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(isNat(M), M, N)
U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N))
PLUS(N, s(M)) → ISNAT(M)
U42¹(tt, M, N) → ACTIVATE(N)
U42¹(tt, M, N) → ACTIVATE(M)
U41¹(tt, M, N) → ISNAT(activate(N))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = x2
tt = tt
ISNAT(x1) = x1
activate(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
isNat(x1) = isNat
ACTIVATE(x1) = x1
PLUS(x1, x2) = PLUS(x1, x2)
0 = 0
U31¹(x1, x2) = U31¹(x1, x2)
n__s(x1) = n__s(x1)
s(x1) = s(x1)
U41¹(x1, x2, x3) = U41¹(x1, x2, x3)
U42¹(x1, x2, x3) = U42¹(x2, x3)
U11(x1, x2) = x1
U12(x1) = x1
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U21(x1) = x1
n__0 = n__0
U31(x1, x2) = x2

Recursive Path Order [RPO].
Precedence:

[n_{_plus₂}, U41₃, U42₃, plus₂] > [PLUS₂, U31^1₂, U41^1₃, U42^1₂] > [tt, isNat, 0, n_{_s₁}, s₁, n_₀]

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

U11¹(tt, V2) → ISNAT(activate(V2))
U11¹(tt, V2) → ACTIVATE(V2)
PLUS(N, 0) → U31¹(isNat(N), N)
U42¹(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(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

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) TRUE