(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, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(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)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
U11(
x1,
x2,
x3) =
U11(
x1,
x2,
x3)
tt =
tt
U12(
x1,
x2) =
U12(
x1,
x2)
isNat(
x1) =
isNat(
x1)
activate(
x1) =
x1
U13(
x1) =
x1
U21(
x1,
x2) =
U21(
x1,
x2)
U22(
x1) =
x1
U31(
x1,
x2,
x3) =
U31(
x1,
x2,
x3)
U32(
x1,
x2) =
U32(
x1,
x2)
U33(
x1) =
x1
U41(
x1,
x2) =
U41(
x1,
x2)
U51(
x1,
x2,
x3) =
U51(
x1,
x2,
x3)
s(
x1) =
s(
x1)
plus(
x1,
x2) =
plus(
x1,
x2)
U61(
x1) =
x1
0 =
0
U71(
x1,
x2,
x3) =
U71(
x1,
x2,
x3)
x(
x1,
x2) =
x(
x1,
x2)
and(
x1,
x2) =
and(
x1,
x2)
n__0 =
n__0
n__plus(
x1,
x2) =
n__plus(
x1,
x2)
isNatKind(
x1) =
isNatKind(
x1)
n__isNatKind(
x1) =
n__isNatKind(
x1)
n__s(
x1) =
n__s(
x1)
n__x(
x1,
x2) =
n__x(
x1,
x2)
n__and(
x1,
x2) =
n__and(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[U713, x2, nx2] > U313 > U322 > [tt, isNat1, 0, n0]
[U713, x2, nx2] > [U513, plus2, nplus2] > U113 > U122 > [tt, isNat1, 0, n0]
[U713, x2, nx2] > [U513, plus2, nplus2] > U412 > [tt, isNat1, 0, n0]
[U713, x2, nx2] > [U513, plus2, nplus2] > [s1, ns1] > U212 > [tt, isNat1, 0, n0]
[U713, x2, nx2] > [U513, plus2, nplus2] > [s1, ns1] > [isNatKind1, nisNatKind1] > [and2, nand2] > [tt, isNat1, 0, n0]
Status:
nplus2: [2,1]
plus2: [2,1]
nisNatKind1: multiset
U322: multiset
U113: multiset
U122: multiset
x2: [2,1]
and2: multiset
isNatKind1: multiset
ns1: [1]
0: multiset
U212: multiset
nand2: multiset
U313: multiset
tt: multiset
U412: [1,2]
n0: multiset
nx2: [2,1]
s1: [1]
isNat1: [1]
U513: [2,3,1]
U713: [2,3,1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U21(tt, V1) → U22(isNat(activate(V1)))
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
U13(tt) → tt
U22(tt) → tt
U33(tt) → tt
U61(tt) → 0
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 1
POL(U13(x1)) = 1 + x1
POL(U22(x1)) = 1 + x1
POL(U33(x1)) = 1 + x1
POL(U61(x1)) = x1
POL(activate(x1)) = 2 + x1
POL(and(x1, x2)) = 1 + x1 + x2
POL(isNatKind(x1)) = 1 + x1
POL(n__0) = 0
POL(n__and(x1, x2)) = x1 + x2
POL(n__isNatKind(x1)) = x1
POL(n__plus(x1, x2)) = x1 + x2
POL(n__s(x1)) = x1
POL(n__x(x1, x2)) = x1 + x2
POL(plus(x1, x2)) = 1 + x1 + x2
POL(s(x1)) = 1 + x1
POL(tt) = 2
POL(x(x1, x2)) = 1 + x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
U13(tt) → tt
U22(tt) → tt
U33(tt) → tt
U61(tt) → 0
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) TRUE
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) TRUE
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE