Termination w.r.t. Q proof of /tmp/tmpl218U6/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 DependencyGraphProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 PisEmptyProof (⇔)

↳12 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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, 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(X1, X2)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__x(X1, X2)) → X(X1, 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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, 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(X1, X2)
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(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)
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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, 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.

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) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(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)
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)) → ISNAT(M)
U52¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, M, N) → ACTIVATE(M)
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 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(x1, x2)
ISNAT(x0, x1) = ISNAT(x0, x1)
ACTIVATE(x0, x1) = ACTIVATE(x0)
PLUS(x0, x1, x2) = PLUS(x0, x1)
U41¹(x0, x1, x2) = U41¹(x2)
X(x0, x1, x2) = X(x0, x1)
U31¹(x0, x1, x2) = U31¹(x0, x1)
U71¹(x0, x1, x2, x3) = U71¹(x0)
U72¹(x0, x1, x2, x3) = U72¹(x0)
U51¹(x0, x1, x2, x3) = U51¹(x1, x2, x3)
U52¹(x0, x1, x2, x3) = U52¹(x0, x2, x3)

Tags:
U11¹ has argument tags [0,34,36] and root tag 0
ISNAT has argument tags [29,2] and root tag 4
ACTIVATE has argument tags [3,63] and root tag 4
PLUS has argument tags [60,32,62] and root tag 4
U41¹ has argument tags [2,60,32] and root tag 3
X has argument tags [2,60,34] and root tag 10
U31¹ has argument tags [34,60,0] and root tag 0
U71¹ has argument tags [32,0,32,60] and root tag 0
U72¹ has argument tags [0,0,0,58] and root tag 8
U51¹ has argument tags [0,29,62,32] and root tag 4
U52¹ has argument tags [62,33,30,32] and root tag 4

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹
tt = tt
ISNAT(x1) = ISNAT
activate(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
isNat(x1) = isNat
ACTIVATE(x1) = x1
PLUS(x1, x2) = x2
0 = 0
U41¹(x1, x2) = U41¹(x1)
n__x(x1, x2) = n__x(x1, x2)
X(x1, x2) = X(x1, x2)
n__s(x1) = n__s(x1)
U31¹(x1, x2) = 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)
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2, x3) = x2
n__0 = n__0
plus(x1, x2) = plus(x1, x2)
U41(x1, x2) = x2
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
U11(x1, x2) = U11
U21(x1) = x1
U31(x1, x2) = U31
U61(x1) = U61(x1)
U51(x1, x2, x3) = U51(x1, x2, x3)
U12(x1) = x1
U32(x1) = x1
U52(x1, x2, x3) = U52(x1, x2, x3)

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

U11^1 > [tt, ISNAT, isNat, n_{_s₁}, s₁, U11, U31]
[n_{_x₂}, X₂, U71^1₃, U72^1₃, x₂, U71₃, U72₃] > [n_{_plus₂}, plus₂, U51₃, U52₃] > [tt, ISNAT, isNat, n_{_s₁}, s₁, U11, U31]
[n_{_x₂}, X₂, U71^1₃, U72^1₃, x₂, U71₃, U72₃] > [0, n_₀, U61₁] > [tt, ISNAT, isNat, n_{_s₁}, s₁, U11, U31]
U51^1₂ > [tt, ISNAT, isNat, n_{_s₁}, s₁, U11, U31]

Status:

U11^1: multiset
tt: multiset
ISNAT: []
n_{_plus₂}: [1,2]
isNat: []
0: multiset
U41^1₁: multiset
n_{_x₂}: [1,2]
X₂: [1,2]
n_{_s₁}: [1]
s₁: [1]
U71^1₃: [3,2,1]
U72^1₃: [3,2,1]
x₂: [1,2]
U51^1₂: multiset
n_₀: multiset
plus₂: [1,2]
U71₃: [3,2,1]
U72₃: [3,2,1]
U11: []
U31: []
U61₁: [1]
U51₃: [3,2,1]
U52₃: [3,2,1]

The following usable rules [FROCOS05] were oriented:

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

(6) 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))
U51¹(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, 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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
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 1 SCC with 1 less node.

(8) Obligation:

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

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)) → U51¹(isNat(M), M, 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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, 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.

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)) → U51¹(isNat(M), M, N)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U51¹(x0, x1, x2, x3) = U51¹(x0, x1, x2)
U52¹(x0, x1, x2, x3) = U52¹(x0, x2)
PLUS(x0, x1, x2) = PLUS(x0, x2)

Tags:
U51¹ has argument tags [1,0,14,13] and root tag 2
U52¹ has argument tags [1,15,5,0] and root tag 0
PLUS has argument tags [1,10,2] and root tag 1

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U51¹(x1, x2, x3) = U51¹(x3)
tt = tt
U52¹(x1, x2, x3) = U52¹(x3)
isNat(x1) = isNat
activate(x1) = x1
PLUS(x1, x2) = PLUS(x1)
s(x1) = s(x1)
n__0 = n__0
0 = 0
n__plus(x1, x2) = n__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
U41(x1, x2) = x2
n__x(x1, x2) = n__x(x1, x2)
x(x1, x2) = x(x1, x2)
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2, x3) = U72(x1, x2, x3)
n__s(x1) = n__s(x1)
U11(x1, x2) = x1
U21(x1) = x1
U31(x1, x2) = U31
U51(x1, x2, x3) = U51(x1, x2, x3)
U61(x1) = U61
U12(x1) = U12
U32(x1) = x1
U52(x1, x2, x3) = U52(x1, x2, x3)

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

[n_₀, 0, U61] > [U51^1₁, tt, U52^1₁, isNat, PLUS₁, s₁, n_{_s₁}, U31, U12]
[n_{_x₂}, x₂, U71₃, U72₃] > [n_{_plus₂}, plus₂, U51₃, U52₃] > [U51^1₁, tt, U52^1₁, isNat, PLUS₁, s₁, n_{_s₁}, U31, U12]

Status:

U51^1₁: multiset
tt: multiset
U52^1₁: multiset
isNat: multiset
PLUS₁: multiset
s₁: multiset
n_₀: multiset
0: multiset
n_{_plus₂}: [2,1]
plus₂: [2,1]
n_{_x₂}: multiset
x₂: multiset
U71₃: multiset
U72₃: multiset
n_{_s₁}: multiset
U31: multiset
U51₃: [2,3,1]
U61: []
U12: multiset
U52₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

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

(10) Obligation:

Q DP problem:
P is empty.
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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

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

(11) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(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) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) PisEmptyProof (EQUIVALENT transformation)

(12) TRUE