(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(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)
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__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) =
x1
activate(
x1) =
x1
U13(
x1) =
x1
U21(
x1,
x2) =
U21(
x1,
x2)
U22(
x1) =
x1
U31(
x1,
x2) =
U31(
x1,
x2)
U41(
x1,
x2,
x3) =
U41(
x1,
x2,
x3)
s(
x1) =
s(
x1)
plus(
x1,
x2) =
plus(
x1,
x2)
and(
x1,
x2) =
and(
x1,
x2)
n__0 =
n__0
n__plus(
x1,
x2) =
n__plus(
x1,
x2)
isNatKind(
x1) =
x1
n__isNatKind(
x1) =
x1
n__s(
x1) =
n__s(
x1)
0 =
0
n__and(
x1,
x2) =
n__and(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[U413, plus2, nplus2] > U113 > U122
[U413, plus2, nplus2] > U312
[U413, plus2, nplus2] > [s1, ns1] > U212
[U413, plus2, nplus2] > [s1, ns1] > [and2, nand2]
[n0, 0] > tt
Status:
U113: multiset
tt: multiset
U122: [2,1]
U212: multiset
U312: multiset
U413: [3,2,1]
s1: multiset
plus2: [1,2]
and2: [1,2]
n0: multiset
nplus2: [1,2]
ns1: multiset
0: multiset
nand2: [1,2]
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, 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(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
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)
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__and(X1, X2)) → and(X1, X2)
activate(X) → X
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
U221 > tt > activate1 > and2 > nand2 > s1 > plus2 > isNatKind1 > ns1 > nisNatKind1 > nplus2 > n0 > 0 > U131
and weight map:
tt=1
0=2
n__0=1
U13_1=1
U22_1=0
isNatKind_1=2
n__isNatKind_1=1
s_1=1
n__s_1=1
activate_1=1
plus_2=1
n__plus_2=0
and_2=0
n__and_2=0
The variable weight is 1With 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
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)
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__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) YES