Termination w.r.t. Q proof of /tmp/tmpwqgC8v/PEANO_complete

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

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 QTRSRRRProof (⇔)

↳6 QTRS

↳7 QTRSRRRProof (⇔)

↳8 QTRS

↳9 QTRSRRRProof (⇔)

↳10 QTRS

  ↳11 RisEmptyProof (⇔)

    ↳12 TRUE

  ↳13 RisEmptyProof (⇔)

    ↳14 TRUE

(0) Obligation:

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

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
U11(x1, x2, x3) = U11(x1, x2, x3)
tt = tt
U12(x1, x2) = U12(x1, x2)
isNat(x1) = isNat(x1)
activate(x1) = x1
U13(x1) = x1
U21(x1, x2) = U21(x1, x2)
U22(x1) = x1
U31(x1, x2) = U31(x1, x2)
U41(x1, x2, x3) = U41(x1, x2, x3)
s(x1) = s(x1)
plus(x1, x2) = plus(x1, x2)
and(x1, x2) = and(x1, x2)
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
isNatKind(x1) = x1
n__isNatKind(x1) = x1
n__s(x1) = n__s(x1)
0 = 0
n__and(x1, x2) = n__and(x1, x2)
n__isNat(x1) = n__isNat(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[n_₀, 0] > tt > [U41₃, plus₂, n_{_plus₂}] > U11₃ > [U12₂, isNat₁, U21₂, n_{_isNat₁}] > [s₁, n_{_s₁}]
[n_₀, 0] > tt > [U41₃, plus₂, n_{_plus₂}] > U31₂ > [s₁, n_{_s₁}]
[n_₀, 0] > tt > [U41₃, plus₂, n_{_plus₂}] > [and₂, n_{_and₂}] > [s₁, n_{_s₁}]

Status:

n_{_plus₂}: [2,1]
plus₂: [2,1]
U31₂: multiset
n_{_isNat₁}: multiset
U41₃: [2,3,1]
U11₃: multiset
U12₂: multiset
n_{_s₁}: multiset
and₂: multiset
0: multiset
U21₂: multiset
n_{_and₂}: multiset
tt: multiset
n_₀: multiset
s₁: multiset
isNat₁: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U21(tt, V1) → U22(isNat(activate(V1)))
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)

(2) Obligation:

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

U13(tt) → tt
U22(tt) → tt
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U13(x₁)) = 1 + x₁
POL(U22(x₁)) = 1 + x₁
POL(activate(x₁)) = 1 + 2·x₁
POL(and(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2 + 2·x₁
POL(isNatKind(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(n__isNat(x₁)) = 2 + x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__plus(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(s(x₁)) = 2 + x₁
POL(tt) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U13(tt) → tt
U22(tt) → tt
0 → n__0
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

(4) Obligation:

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

plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = x₁
POL(n__0) = 1
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatKind(x₁)) = 1 + x₁
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))

(6) Obligation:

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

plus(X1, X2) → n__plus(X1, X2)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(activate(x₁)) = 2·x₁
POL(n__plus(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(plus(x₁, x₂)) = 1 + 2·x₁ + 2·x₂

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))

(8) Obligation:

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

plus(X1, X2) → n__plus(X1, X2)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(plus(x₁, x₂)) = 1 + x₁ + x₂

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

plus(X1, X2) → n__plus(X1, X2)

(10) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(11) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(12) TRUE

(13) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) QTRSRRRProof (EQUIVALENT transformation)

(10) Obligation:

(11) RisEmptyProof (EQUIVALENT transformation)

(12) TRUE

(13) RisEmptyProof (EQUIVALENT transformation)

(14) TRUE