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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPSizeChangeProof (⇔)
↳6 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) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Combined order from the following AFS and order.
activate(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)  =  U11(x1, x2)
isNat(x1)  =  x1
tt  =  tt
n__isNat(x1)  =  x1
and(x1, x2)  =  x2
n__s(x1)  =  n__s(x1)
s(x1)  =  s(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)

Lexicographic path order with status [LPO].
Quasi-Precedence:

[n0, 0, tt]
[nplus2, plus2, U213] > U112
[nplus2, plus2, U213] > [ns1, s1]

Status:
n0: []
0: []
nplus2: [1,2]
plus2: [1,2]
U112: [2,1]
tt: []
ns1: [1]
s1: [1]
U213: [3,2,1]

AFS:
activate(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)  =  U11(x1, x2)
isNat(x1)  =  x1
tt  =  tt
n__isNat(x1)  =  x1
and(x1, x2)  =  x2
n__s(x1)  =  n__s(x1)
s(x1)  =  s(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)

From the DPs we obtained the following set of size-change graphs:

  • U111(tt, N) → ACTIVATE(N) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • PLUS(N, 0) → U111(isNat(N), N) (allowed arguments on rhs = {2})
    The graph contains the following edges 1 >= 2

  • ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 1 > 1, 1 > 2

  • U211(tt, M, N) → PLUS(activate(N), activate(M)) (allowed arguments on rhs = {1, 2})
    The graph contains the following edges 3 >= 1, 2 >= 2

  • ACTIVATE(n__isNat(X)) → ISNAT(X) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 >= 1

  • AND(tt, X) → ACTIVATE(X) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2))) (allowed arguments on rhs = {2})
    The graph contains the following edges 1 > 2

  • PLUS(N, s(M)) → AND(isNat(M), n__isNat(N)) (allowed arguments on rhs = {2})
    The graph contains the following edges 1 >= 2

  • PLUS(N, s(M)) → U211(and(isNat(M), n__isNat(N)), M, N) (allowed arguments on rhs = {2, 3})
    The graph contains the following edges 2 > 2, 1 >= 3

  • PLUS(N, 0) → ISNAT(N) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 >= 1

  • PLUS(N, s(M)) → ISNAT(M) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 > 1

  • ISNAT(n__plus(V1, V2)) → ACTIVATE(V1) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • ISNAT(n__plus(V1, V2)) → ACTIVATE(V2) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • ISNAT(n__s(V1)) → ACTIVATE(V1) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • U211(tt, M, N) → ACTIVATE(N) (allowed arguments on rhs = {1})
    The graph contains the following edges 3 >= 1

  • U211(tt, M, N) → ACTIVATE(M) (allowed arguments on rhs = {1})
    The graph contains the following edges 2 >= 1

  • ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1)) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • ISNAT(n__s(V1)) → ISNAT(activate(V1)) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05].


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

(6) TRUE