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

(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.

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

0A

+

0A

·

x1

+

0A

·

x2

POL(isNat(x1)) =

-I

+

0A

·

x1

POL(ACTIVATE(x1)) =

-I

+

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__x(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(X(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(n__s(x1)) =

-I

+

0A

·

x1

POL(U311(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(s(x1)) =

-I

+

0A

·

x1

POL(U711(x1, x2, x3)) =

-I

+

0A

·

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)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

-I

+

0A

·

x1

POL(U51(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(U21(x1)) =

-I

+

0A

·

x1

POL(U11(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(U52(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

0A

+

-I

·

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

POL(U32(x1)) =

0A

+

0A

·

x1

POL(U12(x1)) =

0A

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(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(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)
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(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

0A

+

0A

·

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)) =

0A

+

0A

·

x1

POL(ACTIVATE(x1)) =

-I

+

0A

·

x1

POL(PLUS(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(0) =

0A

POL(U411(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(n__x(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

0A

+

0A

·

x1

POL(U311(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(s(x1)) =

0A

+

0A

·

x1

POL(U711(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U721(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(x(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(U511(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

-I

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

0A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(U21(x1)) =

-I

+

0A

·

x1

POL(U11(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(U52(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

-I

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U71(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U32(x1)) =

-I

+

0A

·

x1

POL(U12(x1)) =

-I

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(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(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))
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)
U71¹(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.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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(U111(x1, x2)) =

0A

+

-I

·

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

+

-I

·

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__x(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(X(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(n__s(x1)) =

-I

+

0A

·

x1

POL(U311(x1, x2)) =

0A

+

-I

·

x1

+

1A

·

x2

POL(s(x1)) =

-I

+

0A

·

x1

POL(U711(x1, x2, x3)) =

1A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U721(x1, x2, x3)) =

1A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(x(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(U511(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

0A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

0A

+

-I

·

x1

+

-I

·

x2

POL(U21(x1)) =

0A

+

-I

·

x1

POL(U11(x1, x2)) =

0A

+

-I

·

x1

+

-I

·

x2

POL(U52(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

1A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U71(x1, x2, x3)) =

-I

+

1A

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U32(x1)) =

0A

+

-I

·

x1

POL(U12(x1)) =

0A

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(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(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))
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))
U71¹(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.

(11) 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(U111(x1, x2)) =

2A

+

0A

·

x1

+

0A

·

x2

POL(tt) =

1A

POL(ISNAT(x1)) =

2A

+

0A

·

x1

POL(activate(x1)) =

-I

+

0A

·

x1

POL(n__plus(x1, x2)) =

-I

+

0A

·

x1

+

5A

·

x2

POL(isNat(x1)) =

0A

+

0A

·

x1

POL(ACTIVATE(x1)) =

2A

+

0A

·

x1

POL(PLUS(x1, x2)) =

2A

+

0A

·

x1

+

5A

·

x2

POL(0) =

2A

POL(U411(x1, x2)) =

2A

+

-I

·

x1

+

0A

·

x2

POL(n__x(x1, x2)) =

-I

+

5A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

2A

+

5A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

-I

+

0A

·

x1

POL(U311(x1, x2)) =

2A

+

-I

·

x1

+

0A

·

x2

POL(s(x1)) =

-I

+

0A

·

x1

POL(U711(x1, x2, x3)) =

2A

+

0A

·

x1

+

0A

·

x2

+

5A

·

x3

POL(U721(x1, x2, x3)) =

2A

+

-I

·

x1

+

0A

·

x2

+

5A

·

x3

POL(x(x1, x2)) =

-I

+

5A

·

x1

+

0A

·

x2

POL(U511(x1, x2, x3)) =

0A

+

2A

·

x1

+

5A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

2A

+

-I

·

x1

+

5A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

-I

+

0A

·

x1

+

5A

·

x2

POL(n__0) =

2A

POL(U61(x1)) =

2A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

-I

+

-I

·

x1

+

5A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(U21(x1)) =

0A

+

0A

·

x1

POL(U11(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(U52(x1, x2, x3)) =

-I

+

-I

·

x1

+

5A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

-I

+

-I

·

x1

+

0A

·

x2

+

5A

·

x3

POL(U71(x1, x2, x3)) =

-I

+

-I

·

x1

+

0A

·

x2

+

5A

·

x3

POL(U32(x1)) =

-I

+

0A

·

x1

POL(U12(x1)) =

1A

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(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(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))
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)
U72¹(tt, M, N) → X(activate(N), activate(M))
U71¹(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.

(13) 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)
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(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

+

1A

·

x2

POL(isNat(x1)) =

0A

+

-I

·

x1

POL(ACTIVATE(x1)) =

0A

+

0A

·

x1

POL(PLUS(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(0) =

5A

POL(U411(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(n__x(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

0A

+

0A

·

x1

POL(U311(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(s(x1)) =

0A

+

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)) =

0A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

0A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(n__0) =

5A

POL(U61(x1)) =

5A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

1A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

0A

+

-I

·

x1

+

-I

·

x2

POL(U21(x1)) =

0A

+

-I

·

x1

POL(U11(x1, x2)) =

0A

+

0A

·

x1

+

-I

·

x2

POL(U52(x1, x2, x3)) =

0A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

-I

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U71(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U32(x1)) =

0A

+

-I

·

x1

POL(U12(x1)) =

0A

+

-I

·

x1

The following usable rules [FROCOS05] were oriented:

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

(14) 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)
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))
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))
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))
U71¹(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.

(15) 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))

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)) =

-I

+

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)) =

0A

+

0A

·

x1

POL(PLUS(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(0) =

5A

POL(U411(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(n__x(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(X(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(n__s(x1)) =

0A

+

0A

·

x1

POL(U311(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(s(x1)) =

0A

+

0A

·

x1

POL(U711(x1, x2, x3)) =

1A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U721(x1, x2, x3)) =

0A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(x(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(U511(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(n__0) =

5A

POL(U61(x1)) =

5A

+

0A

·

x1

POL(U51(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

0A

+

-I

·

x1

+

-I

·

x2

POL(U21(x1)) =

0A

+

-I

·

x1

POL(U11(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(U52(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

5A

+

-I

·

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

POL(U32(x1)) =

0A

+

-I

·

x1

POL(U12(x1)) =

0A

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(16) 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)
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))
ISNAT(n__x(V1, V2)) → ISNAT(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))
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))
U71¹(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.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U52¹(tt, M, N) → ACTIVATE(M)
U71¹(tt, M, N) → ISNAT(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)) =

0A

+

0A

·

x1

POL(activate(x1)) =

1A

+

0A

·

x1

POL(n__plus(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(isNat(x1)) =

0A

+

0A

·

x1

POL(ACTIVATE(x1)) =

0A

+

0A

·

x1

POL(PLUS(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(0) =

1A

POL(U411(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(n__x(x1, x2)) =

1A

+

1A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

2A

+

0A

·

x1

POL(U311(x1, x2)) =

0A

+

-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)) =

2A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(x(x1, x2)) =

1A

+

1A

·

x1

+

0A

·

x2

POL(U511(x1, x2, x3)) =

2A

+

0A

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

2A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

1A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

2A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(U21(x1)) =

0A

+

0A

·

x1

POL(U11(x1, x2)) =

1A

+

0A

·

x1

+

0A

·

x2

POL(U52(x1, x2, x3)) =

2A

+

0A

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

2A

+

0A

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

2A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U71(x1, x2, x3)) =

2A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U32(x1)) =

0A

+

-I

·

x1

POL(U12(x1)) =

0A

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(18) 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)
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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
ISNAT(n__x(V1, V2)) → ISNAT(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))
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(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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))

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

POL(U111(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(tt) =

0A

POL(ISNAT(x1)) =

-I

+

0A

·

x1

POL(activate(x1)) =

-I

+

0A

·

x1

POL(n__plus(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(isNat(x1)) =

0A

+

-I

·

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__x(x1, x2)) =

3A

+

1A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

0A

+

1A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

3A

+

0A

·

x1

POL(U311(x1, x2)) =

3A

+

-I

·

x1

+

0A

·

x2

POL(s(x1)) =

3A

+

0A

·

x1

POL(U711(x1, x2, x3)) =

3A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U721(x1, x2, x3)) =

-I

+

3A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(x(x1, x2)) =

3A

+

1A

·

x1

+

0A

·

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(plus(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

3A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

-I

+

3A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

0A

+

0A

·

x1

+

-I

·

x2

POL(U21(x1)) =

0A

+

0A

·

x1

POL(U11(x1, x2)) =

0A

+

0A

·

x1

+

-I

·

x2

POL(U52(x1, x2, x3)) =

-I

+

3A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

-I

+

3A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U71(x1, x2, x3)) =

3A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U32(x1)) =

0A

+

0A

·

x1

POL(U12(x1)) =

0A

+

-I

·

x1

The following usable rules [FROCOS05] were oriented:

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

(20) 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)
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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), 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))
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(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.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, V2) → ISNAT(activate(V2))

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

POL(U111(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(tt) =

0A

POL(ISNAT(x1)) =

-I

+

0A

·

x1

POL(activate(x1)) =

-I

+

0A

·

x1

POL(n__plus(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(isNat(x1)) =

-I

+

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__x(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

0A

+

1A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

0A

+

0A

·

x1

POL(U311(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(s(x1)) =

0A

+

0A

·

x1

POL(U711(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U721(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(x(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(U511(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

-I

+

0A

·

x1

+

0A

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

0A

+

1A

·

x1

POL(U51(x1, x2, x3)) =

0A

+

0A

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(U21(x1)) =

0A

+

0A

·

x1

POL(U11(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(U52(x1, x2, x3)) =

0A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(U72(x1, x2, x3)) =

0A

+

0A

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U71(x1, x2, x3)) =

0A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(U32(x1)) =

0A

+

-I

·

x1

POL(U12(x1)) =

0A

+

0A

·

x1

The following usable rules [FROCOS05] were oriented:

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

(22) Obligation:

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

ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U11¹(tt, V2) → 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)
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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), 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))
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(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.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, V2) → ACTIVATE(V2)

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)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(U111(x1, x2)) =

1A

+

-I

·

x1

+

1A

·

x2

POL(isNat(x1)) =

0A

+

0A

·

x1

POL(activate(x1)) =

0A

+

0A

·

x1

POL(tt) =

1A

POL(ACTIVATE(x1)) =

0A

+

0A

·

x1

POL(PLUS(x1, x2)) =

1A

+

0A

·

x1

+

-I

·

x2

POL(0) =

0A

POL(U411(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(n__x(x1, x2)) =

1A

+

1A

·

x1

+

0A

·

x2

POL(X(x1, x2)) =

-I

+

1A

·

x1

+

0A

·

x2

POL(n__s(x1)) =

2A

+

0A

·

x1

POL(U311(x1, x2)) =

0A

+

-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)) =

2A

+

-I

·

x1

+

0A

·

x2

+

1A

·

x3

POL(x(x1, x2)) =

1A

+

1A

·

x1

+

0A

·

x2

POL(U511(x1, x2, x3)) =

1A

+

-I

·

x1

+

-I

·

x2

+

0A

·

x3

POL(U521(x1, x2, x3)) =

1A

+

-I

·

x1

+

-I

·

x2

+

0A

·

x3

POL(plus(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(n__0) =

0A

POL(U61(x1)) =

0A

+

-I

·

x1

POL(U51(x1, x2, x3)) =

3A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U31(x1, x2)) =

1A

+

-I

·

x1

+

0A

·

x2

POL(U21(x1)) =

1A

+

-I

·

x1

POL(U11(x1, x2)) =

1A

+

-I

·

x1

+

1A

·

x2

POL(U52(x1, x2, x3)) =

3A

+

-I

·

x1

+

1A

·

x2

+

0A

·

x3

POL(U41(x1, x2)) =

0A

+

-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

POL(U32(x1)) =

1A

+

-I

·

x1

POL(U12(x1)) =

1A

+

-I

·

x1

The following usable rules [FROCOS05] were oriented:

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

(24) 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(X1, X2)
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
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)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), 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))
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(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.

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) 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(X1, X2)
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, 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) → 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))
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(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.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(tt, V2) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(ACTIVATE(x₁)) = x₁   
POL(ISNAT(x₁)) = x₁   
POL(PLUS(x₁, x₂)) = x₁   
POL(U11(x₁, x₂)) = x₁   
POL(U12(x₁)) = 1   
POL(U21(x₁)) = x₁   
POL(U31(x₁, x₂)) = x₁   
POL(U31¹(x₁, x₂)) = x₁ + x₂   
POL(U32(x₁)) = 1   
POL(U41(x₁, x₂)) = x₂   
POL(U41¹(x₁, x₂)) = x₂   
POL(U51(x₁, x₂, x₃)) = x₃   
POL(U51¹(x₁, x₂, x₃)) = x₃   
POL(U52(x₁, x₂, x₃)) = x₃   
POL(U52¹(x₁, x₂, x₃)) = x₃   
POL(U61(x₁)) = 1   
POL(U71(x₁, x₂, x₃)) = x₂ + x₃   
POL(U71¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U72(x₁, x₂, x₃)) = x₂ + x₃   
POL(U72¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(X(x₁, x₂)) = x₁ + x₂   
POL(activate(x₁)) = x₁   
POL(isNat(x₁)) = x₁   
POL(n__0) = 1   
POL(n__plus(x₁, x₂)) = x₁   
POL(n__s(x₁)) = x₁   
POL(n__x(x₁, x₂)) = x₁ + x₂   
POL(plus(x₁, x₂)) = x₁   
POL(s(x₁)) = x₁   
POL(tt) = 1   
POL(x(x₁, x₂)) = x₁ + x₂   

The following usable rules [FROCOS05] were oriented:

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

(28) 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(X1, X2)
PLUS(N, 0) → U41¹(isNat(N), N)
U41¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__x(X1, X2)) → X(X1, 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))
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(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.

(29) DependencyGraphProof (EQUIVALENT transformation)

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

(30) Obligation:

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

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, 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(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.

(31) 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(X1, X2)
ACTIVATE(n__x(X1, X2)) → X(X1, 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)
0  =  0
U41¹(x1, x2)  =  U41¹(x2)
isNat(x1)  =  isNat
tt  =  tt
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)
activate(x1)  =  x1
x(x1, x2)  =  x(x1, x2)
ISNAT(x1)  =  ISNAT(x1)
n__s(x1)  =  n__s(x1)
U51¹(x1, x2, x3)  =  U51¹(x3)
U52¹(x1, x2, x3)  =  U52¹(x3)
plus(x1, x2)  =  plus(x1, x2)
n__0  =  n__0
U61(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U31(x1, x2)  =  U31
U21(x1)  =  x1
U11(x1, x2)  =  U11
U52(x1, x2, x3)  =  U52(x1, x2, x3)
U32(x1)  =  U32
U12(x1)  =  U12
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, U41₂] > [ACTIVATE₁, PLUS₁, U41^1₁, ISNAT₁, U51^1₁, U52^1₁] > [0, isNat, tt, n₀, U31, U11, U32, U12] > [s₁, n_s1]

Status:

n_plus2: [1,2]
U41^1₁: multiset
U31: multiset
U11: multiset
x₂: [2,1]
n_s1: [1]
U51^1₁: multiset
PLUS₁: multiset
U71^1₃: [2,3,1]
isNat: multiset
tt: multiset
s₁: [1]
U51₃: [3,2,1]
ACTIVATE₁: multiset
plus₂: [1,2]
X₂: [2,1]
U32: multiset
U12: multiset
U72^1₃: [2,3,1]
0: multiset
ISNAT₁: multiset
U52₃: [3,2,1]
U52^1₁: multiset
U41₂: [2,1]
n₀: multiset
n_x2: [2,1]
U72₃: [2,3,1]
U71₃: [2,3,1]

The following usable rules [FROCOS05] were oriented:

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

(32) 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(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.

(33) DependencyGraphProof (EQUIVALENT transformation)

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

(34) 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(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.

(35) 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)
U52¹(tt, M, N) → PLUS(activate(N), activate(M))

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

[PLUS₂, U51^1₃, U52^1₃] > [isNat, tt, U21, U32] > [s₁, n_s1]
[PLUS₂, U51^1₃, U52^1₃] > [isNat, tt, U21, U32] > [0, n₀, U61, U41₁]
[x₂, n_x2, U72₃, U71₃] > [plus₂, n_plus2, U51₃, U52₃] > [isNat, tt, U21, U32] > [s₁, n_s1]
[x₂, n_x2, U72₃, U71₃] > [plus₂, n_plus2, U51₃, U52₃] > [isNat, tt, U21, U32] > [0, n₀, U61, U41₁]

Status:

n_plus2: [1,2]
plus₂: [1,2]
U21: []
U32: []
U41₁: [1]
x₂: [1,2]
n_s1: multiset
0: multiset
U51^1₃: [2,3,1]
isNat: []
U52₃: [3,2,1]
PLUS₂: [2,1]
U52^1₃: [2,3,1]
U61: multiset
tt: multiset
n₀: multiset
n_x2: [1,2]
s₁: multiset
U72₃: [3,2,1]
U51₃: [3,2,1]
U71₃: [3,2,1]

The following usable rules [FROCOS05] were oriented:

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

(36) Obligation:

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

U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), 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.

(37) DependencyGraphProof (EQUIVALENT transformation)

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

(38) TRUE