(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(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, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → U131(isNat(activate(V2)))
U121(tt, V2) → ISNAT(activate(V2))
U121(tt, V2) → ACTIVATE(V2)
U211(tt, V1) → U221(isNat(activate(V1)))
U211(tt, V1) → ISNAT(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U311(tt, N) → ACTIVATE(N)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
U411(tt, M, N) → PLUS(activate(N), activate(M))
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 01
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(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 5 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U311(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(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.
U121(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(U121(x1, x2)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 |
POL(ISNAT(x1)) = | -I | + | 0A | · | x1 |
POL(activate(x1)) = | -I | + | 0A | · | x1 |
POL(n__plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 5A | · | x2 |
POL(U111(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 1A | · | x3 |
POL(and(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(isNatKind(x1)) = | -I | + | 0A | · | x1 |
POL(n__isNatKind(x1)) = | -I | + | 0A | · | x1 |
POL(isNat(x1)) = | 0A | + | 0A | · | x1 |
POL(ACTIVATE(x1)) = | 0A | + | 0A | · | x1 |
POL(PLUS(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U311(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(ISNATKIND(x1)) = | -I | + | 0A | · | x1 |
POL(AND(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(n__s(x1)) = | 0A | + | 0A | · | x1 |
POL(n__and(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(n__isNat(x1)) = | 0A | + | 0A | · | x1 |
POL(U211(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U411(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U11(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | -I | · | x3 |
POL(U12(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
POL(U13(x1)) = | 0A | + | -I | · | x1 |
POL(U21(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
POL(U22(x1)) = | 0A | + | -I | · | x1 |
POL(U41(x1, x2, x3)) = | 5A | + | -I | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 5A | · | x2 |
POL(U31(x1, x2)) = | 5A | + | 0A | · | x1 | + | 0A | · | x2 |
The following usable rules [FROCOS05] were oriented:
activate(n__isNat(X)) → isNat(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U41(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
0 → n__0
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
isNat(X) → n__isNat(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U311(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(ISNAT(x1)) = | 0A | + | 0A | · | x1 |
POL(n__plus(x1, x2)) = | 0A | + | 5A | · | x1 | + | 1A | · | x2 |
POL(U111(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(and(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(isNatKind(x1)) = | -I | + | 0A | · | x1 |
POL(activate(x1)) = | -I | + | 0A | · | x1 |
POL(n__isNatKind(x1)) = | -I | + | 0A | · | x1 |
POL(U121(x1, x2)) = | 0A | + | 2A | · | x1 | + | 0A | · | x2 |
POL(isNat(x1)) = | -I | + | 0A | · | x1 |
POL(ACTIVATE(x1)) = | 0A | + | 0A | · | x1 |
POL(PLUS(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U311(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(ISNATKIND(x1)) = | 0A | + | 0A | · | x1 |
POL(AND(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(n__s(x1)) = | 0A | + | 0A | · | x1 |
POL(n__and(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(n__isNat(x1)) = | -I | + | 0A | · | x1 |
POL(U211(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U411(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(U11(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 5A | · | x2 | + | 1A | · | x3 |
POL(U12(x1, x2)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 |
POL(U13(x1)) = | -I | + | 1A | · | x1 |
POL(U21(x1, x2)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 |
POL(U22(x1)) = | -I | + | 0A | · | x1 |
POL(U41(x1, x2, x3)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 | + | 5A | · | x3 |
POL(plus(x1, x2)) = | 0A | + | 5A | · | x1 | + | 1A | · | x2 |
POL(U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | 2A | · | x2 |
The following usable rules [FROCOS05] were oriented:
activate(n__isNat(X)) → isNat(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U41(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
0 → n__0
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
isNat(X) → n__isNat(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U311(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes.
(10) Complex Obligation (AND)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(AND(x1, x2)) = x2
POL(ISNAT(x1)) = x1
POL(ISNATKIND(x1)) = x1
POL(U11(x1, x2, x3)) = 1 + x2
POL(U111(x1, x2, x3)) = x2 + x3
POL(U12(x1, x2)) = 1
POL(U121(x1, x2)) = x2
POL(U13(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U211(x1, x2)) = x2
POL(U22(x1)) = 0
POL(U31(x1, x2)) = x2
POL(U41(x1, x2, x3)) = 1 + x2 + x3
POL(activate(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatKind(x1)) = x1
POL(n__0) = 0
POL(n__and(x1, x2)) = x1 + x2
POL(n__isNat(x1)) = x1
POL(n__isNatKind(x1)) = x1
POL(n__plus(x1, x2)) = 1 + x1 + x2
POL(n__s(x1)) = x1
POL(plus(x1, x2)) = 1 + x1 + x2
POL(s(x1)) = x1
POL(tt) = 0
The following usable rules [FROCOS05] were oriented:
activate(n__isNat(X)) → isNat(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U41(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
0 → n__0
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
isNat(X) → n__isNat(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
(13) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
U111(tt, V1, V2) → ISNAT(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
(15) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(16) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(AND(x1, x2)) = x2
POL(ISNAT(x1)) = x1
POL(ISNATKIND(x1)) = x1
POL(U11(x1, x2, x3)) = 1 + x1
POL(U12(x1, x2)) = 0
POL(U13(x1)) = 0
POL(U21(x1, x2)) = 1 + x2
POL(U211(x1, x2)) = 1 + x2
POL(U22(x1)) = 0
POL(U31(x1, x2)) = x2
POL(U41(x1, x2, x3)) = 1 + x2 + x3
POL(activate(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(isNat(x1)) = 1 + x1
POL(isNatKind(x1)) = x1
POL(n__0) = 0
POL(n__and(x1, x2)) = x1 + x2
POL(n__isNat(x1)) = 1 + x1
POL(n__isNatKind(x1)) = x1
POL(n__plus(x1, x2)) = x1 + x2
POL(n__s(x1)) = 1 + x1
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = 1 + x1
POL(tt) = 0
The following usable rules [FROCOS05] were oriented:
activate(n__isNat(X)) → isNat(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U41(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
0 → n__0
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
isNat(X) → n__isNat(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
(17) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(18) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(19) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(20) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- AND(tt, X) → ACTIVATE(X)
The graph contains the following edges 2 >= 1
- ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
The graph contains the following edges 1 > 1
- ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
The graph contains the following edges 1 > 2
(21) TRUE
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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.
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(PLUS(x1, x2)) = x2
POL(U11(x1, x2, x3)) = 1
POL(U12(x1, x2)) = 0
POL(U13(x1)) = 0
POL(U21(x1, x2)) = 1
POL(U22(x1)) = 1
POL(U31(x1, x2)) = x2
POL(U41(x1, x2, x3)) = 1 + x2 + x3
POL(U411(x1, x2, x3)) = x2
POL(activate(x1)) = x1
POL(and(x1, x2)) = x2
POL(isNat(x1)) = 1
POL(isNatKind(x1)) = 0
POL(n__0) = 0
POL(n__and(x1, x2)) = x2
POL(n__isNat(x1)) = 1
POL(n__isNatKind(x1)) = 0
POL(n__plus(x1, x2)) = x1 + x2
POL(n__s(x1)) = 1 + x1
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = 1 + x1
POL(tt) = 0
The following usable rules [FROCOS05] were oriented:
activate(n__isNat(X)) → isNat(X)
activate(X) → X
activate(n__s(X)) → s(activate(X))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U41(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
0 → n__0
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
and(X1, X2) → n__and(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
isNat(X) → n__isNat(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__isNatKind(X)) → isNatKind(X)
(24) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U411(tt, M, N) → PLUS(activate(N), activate(M))
The TRS R consists of the following rules:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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 0 SCCs with 1 less node.
(26) TRUE