Termination w.r.t. Q proof of /tmp/tmpd8jWU8/MYNAT_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 QDPOrderProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 DependencyGraphProof (⇔)

↳16 QDP

↳17 QDPOrderProof (⇔)

↳18 QDP

↳19 QDPOrderProof (⇔)

↳20 QDP

↳21 QDPOrderProof (⇔)

↳22 QDP

↳23 DependencyGraphProof (⇔)

↳24 QDP

↳25 QDPOrderProof (⇔)

↳26 QDP

↳27 QDPOrderProof (⇔)

↳28 QDP

↳29 QDPOrderProof (⇔)

↳30 QDP

↳31 DependencyGraphProof (⇔)

↳32 QDP

↳33 QDPOrderProof (⇔)

↳34 QDP

↳35 DependencyGraphProof (⇔)

↳36 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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
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, V2) → U32¹(isNat(activate(V2)))
U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
U41¹(tt, N) → ACTIVATE(N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U52¹(tt, M, N) → S(plus(activate(N), activate(M)))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U61¹(tt) → 0¹
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U71¹(tt, M, N) → ISNAT(activate(N))
U71¹(tt, M, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U72¹(tt, M, N) → X(activate(N), activate(M))
U72¹(tt, M, N) → ACTIVATE(N)
U72¹(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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U61¹(isNat(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U71¹(isNat(M), M, N)
X(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)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
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 8 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(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)
U72¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → ACTIVATE(M)
U71¹(tt, M, N) → ISNAT(activate(N))
U71¹(tt, M, N) → ACTIVATE(N)
U71¹(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
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.

U71¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U11¹(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(tt) = 0A

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = -I + 1A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(U41¹(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(n__s(x₁)) = -I + 0A · x₁

POL(n__x(x₁, x₂)) = 0A + 0A · x₁ + 3A · x₂

POL(X(x₁, x₂)) = 0A + 0A · x₁ + 3A · x₂

POL(U31¹(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = -I + 1A · x₁ + 3A · x₂ + 0A · x₃

POL(U72¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 3A · x₂ + 0A · x₃

POL(x(x₁, x₂)) = 0A + 0A · x₁ + 3A · x₂

POL(U51¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = -I + 0A · x₁

POL(U31(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(U51(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(n__0) = 0A

POL(U11(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(U52(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U61(x₁)) = 0A + -I · x₁

POL(U12(x₁)) = 0A + -I · x₁

POL(U32(x₁)) = 0A + -I · x₁

POL(U41(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 0A + -I · x₁ + 3A · x₂ + 0A · x₃

POL(U71(x₁, x₂, x₃)) = 0A + 0A · x₁ + 3A · x₂ + 0A · x₃

The following usable rules [FROCOS05] were oriented:

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

(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))
U11¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)
U72¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → ACTIVATE(M)
U71¹(tt, M, N) → ISNAT(activate(N))
U71¹(tt, M, N) → 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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U72¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U11¹(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(tt) = 0A

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(n__plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = -I + 0A · x₁

POL(PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(U41¹(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(n__s(x₁)) = -I + 0A · x₁

POL(n__x(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(X(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(U31¹(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 1A · x₂ + 0A · x₃

POL(U72¹(x₁, x₂, x₃)) = 0A + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(x(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(U51¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = 0A + -I · x₁

POL(U31(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(U51(x₁, x₂, x₃)) = 0A + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(n__0) = 0A

POL(U11(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(U52(x₁, x₂, x₃)) = -I + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(U61(x₁)) = 0A + -I · x₁

POL(U12(x₁)) = 0A + -I · x₁

POL(U32(x₁)) = -I + 0A · x₁

POL(U41(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U71(x₁, x₂, x₃)) = 0A + -I · x₁ + 1A · x₂ + 0A · x₃

The following usable rules [FROCOS05] were oriented:

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

(8) 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(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)
U72¹(tt, M, N) → ACTIVATE(N)
U71¹(tt, M, N) → ISNAT(activate(N))
U71¹(tt, M, N) → 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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(tt, M, N) → ISNAT(activate(N))
U71¹(tt, M, N) → ACTIVATE(N)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U11¹(x₁, x₂)) = 1A + -I · x₁ + 0A · x₂

POL(tt) = 0A

POL(ISNAT(x₁)) = 1A + 0A · x₁

POL(activate(x₁)) = 0A + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = 1A + 0A · x₁

POL(PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(0) = 1A

POL(U41¹(x₁, x₂)) = 1A + -I · x₁ + 0A · x₂

POL(n__s(x₁)) = 2A + 0A · x₁

POL(n__x(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(X(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(U31¹(x₁, x₂)) = 1A + -I · x₁ + 0A · x₂

POL(s(x₁)) = 2A + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 2A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U72¹(x₁, x₂, x₃)) = -I + 1A · x₁ + 0A · x₂ + 1A · x₃

POL(x(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(U51¹(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = 0A + -I · x₁

POL(U31(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(U51(x₁, x₂, x₃)) = 2A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(n__0) = 1A

POL(U11(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(U52(x₁, x₂, x₃)) = 2A + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(U61(x₁)) = -I + 1A · x₁

POL(U12(x₁)) = 0A + -I · x₁

POL(U32(x₁)) = 0A + 0A · x₁

POL(U41(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U71(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 1A · x₃

The following usable rules [FROCOS05] were oriented:

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

(10) 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(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)
U72¹(tt, M, N) → 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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(N, 0) → ISNAT(N)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U72¹(tt, M, N) → ACTIVATE(N)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U11¹(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(tt) = 0A

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(activate(x₁)) = 0A + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(U41¹(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(n__s(x₁)) = 0A + 0A · x₁

POL(n__x(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(X(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(U31¹(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 1A + 0A · x₁ + 0A · x₂ + 1A · x₃

POL(U72¹(x₁, x₂, x₃)) = 1A + 0A · x₁ + 0A · x₂ + 1A · x₃

POL(x(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(U51¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = 0A + 0A · x₁

POL(U31(x₁, x₂)) = 1A + -I · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(U51(x₁, x₂, x₃)) = -I + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(n__0) = 0A

POL(U11(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(U52(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U61(x₁)) = 0A + -I · x₁

POL(U12(x₁)) = 0A + 0A · x₁

POL(U32(x₁)) = 1A + -I · x₁

POL(U41(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U71(x₁, x₂, x₃)) = 1A + 0A · x₁ + 0A · x₂ + 1A · x₃

The following usable rules [FROCOS05] were oriented:

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

(12) 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(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)

The TRS R consists of the following rules:

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

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

(13) 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))
U11¹(tt, V2) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U11¹(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(tt) = 1A

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(isNat(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(PLUS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(0) = 5A

POL(U41¹(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(n__s(x₁)) = 2A + 0A · x₁

POL(n__x(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(X(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(U31¹(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(s(x₁)) = 2A + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U72¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(x(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(U51¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = 1A + -I · x₁

POL(U31(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(U51(x₁, x₂, x₃)) = 3A + -I · x₁ + 1A · x₂ + 0A · x₃

POL(n__0) = 5A

POL(U11(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(U52(x₁, x₂, x₃)) = 2A + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U61(x₁)) = 5A + 0A · x₁

POL(U12(x₁)) = 1A + -I · x₁

POL(U32(x₁)) = 0A + 0A · x₁

POL(U41(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U71(x₁, x₂, x₃)) = 0A + -I · x₁ + 0A · x₂ + 1A · x₃

The following usable rules [FROCOS05] were oriented:

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

(14) Obligation:

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

ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)

The TRS R consists of the following rules:

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

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)

The TRS R consists of the following rules:

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

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x₁)) = -I + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 2A · x₂

POL(ACTIVATE(x₁)) = -I + 0A · x₁

POL(PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(0) = 4A

POL(U41¹(x₁, x₂)) = 4A + -I · x₁ + 0A · x₂

POL(isNat(x₁)) = 0A + 0A · x₁

POL(tt) = 2A

POL(n__s(x₁)) = 2A + 0A · x₁

POL(n__x(x₁, x₂)) = 0A + 2A · x₁ + 0A · x₂

POL(X(x₁, x₂)) = 0A + 2A · x₁ + 0A · x₂

POL(s(x₁)) = 2A + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 2A + 0A · x₁ + 0A · x₂ + 2A · x₃

POL(U72¹(x₁, x₂, x₃)) = 0A + 0A · x₁ + 0A · x₂ + 2A · x₃

POL(x(x₁, x₂)) = 0A + 2A · x₁ + 0A · x₂

POL(U31¹(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(U51¹(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = 0A + 0A · x₁

POL(U31(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 2A · x₂

POL(U51(x₁, x₂, x₃)) = -I + 0A · x₁ + 2A · x₂ + 0A · x₃

POL(n__0) = 4A

POL(U11(x₁, x₂)) = 0A + 0A · x₁ + 2A · x₂

POL(U52(x₁, x₂, x₃)) = 2A + 0A · x₁ + 2A · x₂ + 0A · x₃

POL(U61(x₁)) = -I + 2A · x₁

POL(U12(x₁)) = 0A + 2A · x₁

POL(U32(x₁)) = 2A + -I · x₁

POL(U41(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = -I + 0A · x₁ + 0A · x₂ + 2A · x₃

POL(U71(x₁, x₂, x₃)) = 0A + 0A · x₁ + 0A · x₂ + 2A · x₃

The following usable rules [FROCOS05] were oriented:

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

(18) Obligation:

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

ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)

The TRS R consists of the following rules:

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

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x₁)) = 1A + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(PLUS(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(0) = 1A

POL(U41¹(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = -I + 0A · x₁

POL(tt) = 0A

POL(n__s(x₁)) = 0A + 0A · x₁

POL(n__x(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(X(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U72¹(x₁, x₂, x₃)) = 1A + 0A · x₁ + 0A · x₂ + 1A · x₃

POL(x(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(U31¹(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(U51¹(x₁, x₂, x₃)) = 1A + -I · x₁ + 1A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = 0A + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U21(x₁)) = -I + 0A · x₁

POL(U31(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(U51(x₁, x₂, x₃)) = 0A + -I · x₁ + 1A · x₂ + 0A · x₃

POL(n__0) = 1A

POL(U11(x₁, x₂)) = -I + -I · x₁ + 1A · x₂

POL(U52(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U61(x₁)) = -I + 1A · x₁

POL(U12(x₁)) = -I + 1A · x₁

POL(U32(x₁)) = 1A + 0A · x₁

POL(U41(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 1A · x₃

POL(U71(x₁, x₂, x₃)) = 1A + -I · x₁ + 0A · x₂ + 1A · x₃

The following usable rules [FROCOS05] were oriented:

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

(20) Obligation:

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

ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → ISNAT(M)

The TRS R consists of the following rules:

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

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(tt, V2) → ISNAT(activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
X(N, s(M)) → ISNAT(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x₁)) = -I + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(ACTIVATE(x₁)) = -I + 0A · x₁

POL(PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(0) = 0A

POL(U41¹(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = 0A + -I · x₁

POL(tt) = 0A

POL(n__s(x₁)) = -I + 0A · x₁

POL(n__x(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(X(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = -I + -I · x₁ + 1A · x₂ + 0A · x₃

POL(U72¹(x₁, x₂, x₃)) = -I + -I · x₁ + 1A · x₂ + 0A · x₃

POL(x(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(U31¹(x₁, x₂)) = -I + -I · x₁ + 1A · x₂

POL(U51¹(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U21(x₁)) = 0A + -I · x₁

POL(U31(x₁, x₂)) = 0A + 0A · x₁ + -I · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(U51(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(n__0) = 0A

POL(U11(x₁, x₂)) = 0A + 0A · x₁ + -I · x₂

POL(U52(x₁, x₂, x₃)) = -I + -I · x₁ + 0A · x₂ + 0A · x₃

POL(U61(x₁)) = 1A + -I · x₁

POL(U12(x₁)) = 0A + -I · x₁

POL(U32(x₁)) = 0A + -I · x₁

POL(U41(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = -I + -I · x₁ + 1A · x₂ + 0A · x₃

POL(U71(x₁, x₂, x₃)) = -I + -I · x₁ + 1A · x₂ + 0A · x₃

The following usable rules [FROCOS05] were oriented:

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

(22) Obligation:

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

ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → X(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) Obligation:

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

ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → X(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__plus(x₁, x₂)) = 4A + 0A · x₁ + 1A · x₂

POL(PLUS(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(0) = 1A

POL(U41¹(x₁, x₂)) = 2A + -I · x₁ + 0A · x₂

POL(isNat(x₁)) = -I + 0A · x₁

POL(tt) = 1A

POL(n__s(x₁)) = -I + 0A · x₁

POL(n__x(x₁, x₂)) = 0A + 1A · x₁ + 5A · x₂

POL(X(x₁, x₂)) = 0A + 1A · x₁ + 5A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 5A · x₂ + 1A · x₃

POL(U72¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 5A · x₂ + 1A · x₃

POL(x(x₁, x₂)) = 0A + 1A · x₁ + 5A · x₂

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(U51¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U21(x₁)) = -I + 0A · x₁

POL(U31(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(plus(x₁, x₂)) = 4A + 0A · x₁ + 1A · x₂

POL(U51(x₁, x₂, x₃)) = 4A + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(n__0) = 1A

POL(U11(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(U52(x₁, x₂, x₃)) = 4A + -I · x₁ + 1A · x₂ + 0A · x₃

POL(U61(x₁)) = 2A + -I · x₁

POL(U12(x₁)) = 1A + -I · x₁

POL(U32(x₁)) = 1A + -I · x₁

POL(U41(x₁, x₂)) = 4A + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 4A + 0A · x₁ + 5A · x₂ + 1A · x₃

POL(U71(x₁, x₂, x₃)) = -I + 3A · x₁ + 5A · x₂ + 1A · x₃

The following usable rules [FROCOS05] were oriented:

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

(26) Obligation:

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

ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → X(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → ISNAT(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__plus(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(PLUS(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(0) = 0A

POL(U41¹(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(isNat(x₁)) = 0A + 0A · x₁

POL(tt) = 0A

POL(n__s(x₁)) = 0A + 0A · x₁

POL(n__x(x₁, x₂)) = 0A + 1A · x₁ + 1A · x₂

POL(X(x₁, x₂)) = 0A + 1A · x₁ + 1A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(U71¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 1A · x₃

POL(U72¹(x₁, x₂, x₃)) = 0A + -I · x₁ + 1A · x₂ + 1A · x₃

POL(x(x₁, x₂)) = 0A + 1A · x₁ + 1A · x₂

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(U51¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U52¹(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U21(x₁)) = 0A + -I · x₁

POL(U31(x₁, x₂)) = 0A + -I · x₁ + 1A · x₂

POL(plus(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(U51(x₁, x₂, x₃)) = 1A + -I · x₁ + 1A · x₂ + 0A · x₃

POL(n__0) = 0A

POL(U11(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(U52(x₁, x₂, x₃)) = 0A + 0A · x₁ + 1A · x₂ + 0A · x₃

POL(U61(x₁)) = 1A + -I · x₁

POL(U12(x₁)) = 0A + -I · x₁

POL(U32(x₁)) = -I + 0A · x₁

POL(U41(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(U72(x₁, x₂, x₃)) = 0A + -I · x₁ + 1A · x₂ + 1A · x₃

POL(U71(x₁, x₂, x₃)) = -I + 0A · x₁ + 1A · x₂ + 1A · x₃

The following usable rules [FROCOS05] were oriented:

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

(28) Obligation:

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

ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
U52¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → ACTIVATE(N)
U72¹(tt, M, N) → X(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, s(M)) → U71¹(isNat(M), M, N)
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U72¹(tt, M, N) → X(activate(N), activate(M))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = ACTIVATE(x1)
n__plus(x1, x2) = n__plus(x1, x2)
PLUS(x1, x2) = PLUS(x1)
activate(x1) = x1
0 = 0
U41¹(x1, x2) = U41¹(x2)
isNat(x1) = isNat
tt = tt
n__s(x1) = n__s(x1)
n__x(x1, x2) = n__x(x1, x2)
X(x1, x2) = X(x1, x2)
s(x1) = s(x1)
U71¹(x1, x2, x3) = U71¹(x1, x2, x3)
U72¹(x1, x2, x3) = U72¹(x1, x2, x3)
x(x1, x2) = x(x1, x2)
ISNAT(x1) = ISNAT(x1)
U51¹(x1, x2, x3) = U51¹(x3)
U52¹(x1, x2, x3) = U52¹(x3)
U21(x1) = U21
U31(x1, x2) = U31
plus(x1, x2) = plus(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
n__0 = n__0
U11(x1, x2) = U11
U52(x1, x2, x3) = U52(x1, x2, x3)
U61(x1) = U61
U12(x1) = U12
U32(x1) = U32
U41(x1, x2) = U41(x1, x2)
U72(x1, x2, x3) = U72(x1, x2, x3)
U71(x1, x2, x3) = U71(x1, x2, x3)

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

[n_{_x₂}, X₂, U71^1₃, U72^1₃, x₂, U72₃, U71₃] > [n_{_plus₂}, plus₂, U51₃, U52₃] > [ACTIVATE₁, PLUS₁, U41^1₁, ISNAT₁, U51^1₁, U52^1₁] > [isNat, tt, U21, U31, U11, U12, U32] > [n_{_s₁}, s₁, U41₂]
[n_{_x₂}, X₂, U71^1₃, U72^1₃, x₂, U72₃, U71₃] > U61 > [0, n_₀] > [isNat, tt, U21, U31, U11, U12, U32] > [n_{_s₁}, s₁, U41₂]

Status:

n_{_plus₂}: [1,2]
U41^1₁: multiset
U31: multiset
U11: multiset
x₂: [2,1]
n_{_s₁}: multiset
U51^1₁: multiset
PLUS₁: multiset
U71^1₃: [2,3,1]
isNat: multiset
U61: multiset
tt: multiset
s₁: multiset
U51₃: [3,2,1]
ACTIVATE₁: multiset
plus₂: [1,2]
U21: multiset
X₂: [2,1]
U32: multiset
U72^1₃: [2,3,1]
U12: multiset
0: multiset
ISNAT₁: multiset
U52₃: [3,2,1]
U52^1₁: multiset
U41₂: multiset
n_₀: multiset
n_{_x₂}: [2,1]
U72₃: [2,3,1]
U71₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

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

(30) Obligation:

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

PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
U52¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ISNAT(activate(N))
U51¹(tt, M, N) → 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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(31) DependencyGraphProof (EQUIVALENT transformation)

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

(32) Obligation:

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

PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → U51¹(isNat(M), M, N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2) = PLUS(x2)
s(x1) = s(x1)
U51¹(x1, x2, x3) = U51¹(x1, x2)
isNat(x1) = isNat
tt = tt
U52¹(x1, x2, x3) = U52¹(x2)
activate(x1) = x1
n__s(x1) = n__s(x1)
U21(x1) = U21
n__x(x1, x2) = n__x(x1, x2)
U31(x1, x2) = U31
plus(x1, x2) = plus(x1, x2)
U51(x1, x2, x3) = U51(x1, x2, x3)
n__0 = n__0
n__plus(x1, x2) = n__plus(x1, x2)
U11(x1, x2) = x1
U52(x1, x2, x3) = U52(x1, x2, x3)
U61(x1) = U61
0 = 0
U12(x1) = x1
U32(x1) = U32
x(x1, x2) = x(x1, x2)
U41(x1, x2) = x2
U72(x1, x2, x3) = U72(x1, x2, x3)
U71(x1, x2, x3) = U71(x1, x2, x3)

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

[n_{_x₂}, x₂, U72₃, U71₃] > [plus₂, U51₃, n_{_plus₂}, U52₃] > [s₁, n_{_s₁}] > [PLUS₁, U51^1₂, isNat, tt, U52^1₁, U21, U31, U32]
[n_{_x₂}, x₂, U72₃, U71₃] > U61 > [n_₀, 0] > [PLUS₁, U51^1₂, isNat, tt, U52^1₁, U21, U31, U32]

Status:

n_{_plus₂}: multiset
U51^1₂: multiset
U21: multiset
plus₂: multiset
U32: multiset
U31: multiset
x₂: multiset
n_{_s₁}: multiset
0: multiset
PLUS₁: multiset
isNat: multiset
U52^1₁: multiset
U52₃: multiset
U61: multiset
tt: multiset
n_₀: multiset
n_{_x₂}: multiset
s₁: multiset
U72₃: multiset
U51₃: multiset
U71₃: multiset

The following usable rules [FROCOS05] were oriented:

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

(34) Obligation:

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

U52¹(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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(35) 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) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) QDPOrderProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) DependencyGraphProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) DependencyGraphProof (EQUIVALENT transformation)

(36) TRUE