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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 QDP
        ↳10 QDPOrderProof (⇔)
        ↳11 QDP
        ↳12 PisEmptyProof (⇔)
        ↳13 TRUE
    ↳14 QDP
        ↳15 QDPOrderProof (⇔)
        ↳16 QDP
        ↳17 PisEmptyProof (⇔)
        ↳18 TRUE

(0) Obligation:

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

U111(tt, N) → ACTIVATE(N)
U211(tt, M, N) → S(plus(activate(N), activate(M)))
U211(tt, M, N) → PLUS(activate(N), activate(M))
U211(tt, M, N) → ACTIVATE(N)
U211(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(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)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U111(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U211(and(isNat(M), n__isNat(N)), M, N)
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 01
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__s(X)) → S(X)

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(4) Obligation:

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

ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U111(isNat(N), N)
U111(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U211(and(isNat(M), n__isNat(N)), M, N)
U211(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
PLUS(N, s(M)) → ISNAT(M)
U211(tt, M, N) → ACTIVATE(N)
U211(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → ISNAT(N)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVATE(x0, x1)  =  ACTIVATE(x0, x1)
PLUS(x0, x1, x2)  =  PLUS(x0, x1, x2)
U111(x0, x1, x2)  =  U111(x0, x1)
ISNAT(x0, x1)  =  ISNAT(x0, x1)
AND(x0, x1, x2)  =  AND(x0, x1, x2)
U211(x0, x1, x2, x3)  =  U211(x1, x2, x3)

Tags:
ACTIVATE has argument tags [0,4] and root tag 0
PLUS has argument tags [0,4,14] and root tag 0
U111 has argument tags [4,0,9] and root tag 0
ISNAT has argument tags [0,0] and root tag 0
AND has argument tags [0,0,4] and root tag 0
U211 has argument tags [11,0,15,4] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACTIVATE(x1)  =  ACTIVATE
n__plus(x1, x2)  =  n__plus(x1, x2)
PLUS(x1, x2)  =  PLUS
0  =  0
U111(x1, x2)  =  x2
isNat(x1)  =  x1
tt  =  tt
n__isNat(x1)  =  x1
ISNAT(x1)  =  ISNAT
AND(x1, x2)  =  AND
activate(x1)  =  x1
n__s(x1)  =  n__s(x1)
s(x1)  =  s(x1)
U211(x1, x2, x3)  =  U211(x3)
and(x1, x2)  =  x2
n__0  =  n__0
plus(x1, x2)  =  plus(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[0, n0] > [ACTIVATE, nplus2, PLUS, tt, ISNAT, AND, plus2, U213] > [ns1, s1] > U21^11
[0, n0] > [ACTIVATE, nplus2, PLUS, tt, ISNAT, AND, plus2, U213] > U112 > U21^11

Status:
ACTIVATE: multiset
nplus2: [1,2]
PLUS: multiset
0: multiset
tt: multiset
ISNAT: multiset
AND: multiset
ns1: [1]
s1: [1]
U21^11: multiset
n0: multiset
plus2: [1,2]
U112: multiset
U213: [3,2,1]


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

PLUS(N, 0) → U111(isNat(N), N)
U111(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U211(and(isNat(M), n__isNat(N)), M, N)
U211(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
PLUS(N, s(M)) → ISNAT(M)
U211(tt, M, N) → ACTIVATE(N)
U211(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNAT(x0, x1)  =  ISNAT(x0)
ACTIVATE(x0, x1)  =  ACTIVATE(x0)

Tags:
ISNAT has argument tags [2,0] and root tag 0
ACTIVATE has argument tags [2,2] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNAT(x1)  =  x1
n__s(x1)  =  n__s(x1)
activate(x1)  =  x1
ACTIVATE(x1)  =  x1
n__isNat(x1)  =  x1
n__0  =  n__0
0  =  0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
U11(x1, x2)  =  x2
isNat(x1)  =  x1
tt  =  tt
and(x1, x2)  =  and(x1, x2)
s(x1)  =  s(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[n0, 0] > tt
[nplus2, plus2, U213] > [ns1, s1] > and2

Status:
ns1: [1]
n0: multiset
0: multiset
nplus2: [2,1]
plus2: [2,1]
tt: multiset
and2: multiset
s1: [1]
U213: [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) → U11(isNat(N), N)
U11(tt, N) → activate(N)
activate(n__isNat(X)) → isNat(X)
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
and(tt, X) → activate(X)
isNat(n__s(V1)) → isNat(activate(V1))
activate(n__s(X)) → s(X)
activate(X) → X
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
plus(X1, X2) → n__plus(X1, X2)
isNat(n__0) → tt
isNat(X) → n__isNat(X)
s(X) → n__s(X)
0 → n__0

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

PLUS(N, s(M)) → U211(and(isNat(M), n__isNat(N)), M, N)
U211(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(N, s(M)) → U211(and(isNat(M), n__isNat(N)), M, N)
U211(tt, M, N) → PLUS(activate(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:
PLUS(x0, x1, x2)  =  PLUS(x0, x2)
U211(x0, x1, x2, x3)  =  U211(x0)

Tags:
PLUS has argument tags [2,3,1] and root tag 0
U211 has argument tags [2,4,0,1] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS
s(x1)  =  s(x1)
U211(x1, x2, x3)  =  x2
and(x1, x2)  =  and(x2)
isNat(x1)  =  isNat(x1)
n__isNat(x1)  =  n__isNat(x1)
tt  =  tt
activate(x1)  =  x1
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
U11(x1, x2)  =  U11(x1, x2)
n__s(x1)  =  n__s(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[isNat1, nisNat1, nplus2, plus2, U213] > [s1, ns1] > and1 > [PLUS, tt, n0, 0]
[isNat1, nisNat1, nplus2, plus2, U213] > U112 > [PLUS, tt, n0, 0]

Status:
PLUS: multiset
s1: [1]
and1: multiset
isNat1: [1]
nisNat1: [1]
tt: multiset
n0: multiset
nplus2: [1,2]
plus2: [1,2]
0: multiset
U112: multiset
ns1: [1]
U213: [3,2,1]


The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE