Termination w.r.t. Q proof of /tmp/tmpc-uUmx/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 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔)

        ↳11 QDP

        ↳12 DependencyGraphProof (⇔)

        ↳13 TRUE

    ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 DependencyGraphProof (⇔)

        ↳18 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.

U11¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
U31¹(tt, N) → ACTIVATE(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)) → ISNAT(M)
U42¹(tt, M, N) → ACTIVATE(N)
U42¹(tt, M, N) → ACTIVATE(M)
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x0, x1, x2) = U11¹(x0, x1)
ISNAT(x0, x1) = ISNAT(x1)
ACTIVATE(x0, x1) = ACTIVATE(x0, x1)
PLUS(x0, x1, x2) = PLUS(x0, x1)
U31¹(x0, x1, x2) = U31¹(x0, x1)
U41¹(x0, x1, x2, x3) = U41¹(x0, x1, x2, x3)
U42¹(x0, x1, x2, x3) = U42¹(x1, x2, x3)

Tags:
U11¹ has argument tags [0,0,0] and root tag 1
ISNAT has argument tags [2,0] and root tag 1
ACTIVATE has argument tags [0,0] and root tag 0
PLUS has argument tags [16,0,18] and root tag 1
U31¹ has argument tags [0,0,16] and root tag 1
U41¹ has argument tags [4,0,20,0] and root tag 1
U42¹ has argument tags [1,0,16,0] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(ACTIVATE(x₁)) = 1
POL(ISNAT(x₁)) = 1 + x₁
POL(PLUS(x₁, x₂)) = x₂
POL(U11(x₁, x₂)) = x₁
POL(U11¹(x₁, x₂)) = x₂
POL(U12(x₁)) = x₁
POL(U21(x₁)) = 1
POL(U31(x₁, x₂)) = x₂
POL(U31¹(x₁, x₂)) = x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U41¹(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42¹(x₁, x₂, x₃)) = x₃
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = 1
POL(n__0) = 1
POL(n__plus(x₁, x₂)) = 1 + x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 1

The following usable rules [FROCOS05] were oriented:

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

(6) 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))
PLUS(N, 0) → U31¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
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))
U41¹(tt, M, N) → ISNAT(activate(N))

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 2 SCCs with 3 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U11¹(tt, V2) → ISNAT(activate(V2))

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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, V2) → ISNAT(activate(V2))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNAT(x0, x1) = ISNAT(x1)
U11¹(x0, x1, x2) = U11¹(x0, x2)

Tags:
ISNAT has argument tags [1,0] and root tag 0
U11¹ has argument tags [0,6,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(ISNAT(x₁)) = 1 + x₁
POL(U11(x₁, x₂)) = x₂
POL(U11¹(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁)) = 1
POL(U31(x₁, x₂)) = x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₃
POL(U42(x₁, x₂, x₃)) = 1
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = x₁
POL(n__0) = 1
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = 1
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1
POL(tt) = 1

The following usable rules [FROCOS05] were oriented:

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

(11) Obligation:

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

ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))

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.

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(13) TRUE

(14) Obligation:

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

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))

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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → U41¹(isNat(M), M, N)
U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PLUS(x0, x1, x2) = PLUS(x0, x2)
U41¹(x0, x1, x2, x3) = U41¹(x0)
U42¹(x0, x1, x2, x3) = U42¹(x0, x1, x3)

Tags:
PLUS has argument tags [0,4,0] and root tag 1
U41¹ has argument tags [0,8,12,0] and root tag 2
U42¹ has argument tags [0,0,11,0] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(PLUS(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂)) = 1 + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U41¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U42(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U42¹(x₁, x₂, x₃)) = x₂ + x₃
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(n__0) = 1
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

The following usable rules [FROCOS05] were oriented:

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

(16) Obligation:

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

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.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(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) Complex Obligation (AND)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) DependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) TRUE