(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)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(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)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
U11(
x1,
x2) =
U11(
x1,
x2)
tt =
tt
activate(
x1) =
x1
U21(
x1,
x2,
x3) =
U21(
x1,
x2,
x3)
s(
x1) =
s(
x1)
plus(
x1,
x2) =
plus(
x1,
x2)
U31(
x1) =
U31(
x1)
0 =
0
U41(
x1,
x2,
x3) =
U41(
x1,
x2,
x3)
x(
x1,
x2) =
x(
x1,
x2)
and(
x1,
x2) =
and(
x1,
x2)
isNat(
x1) =
x1
n__0 =
n__0
n__plus(
x1,
x2) =
n__plus(
x1,
x2)
n__isNat(
x1) =
x1
n__s(
x1) =
n__s(
x1)
n__x(
x1,
x2) =
n__x(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[U413, x2, nx2] > [U213, plus2, nplus2] > U112
[U413, x2, nx2] > [U213, plus2, nplus2] > [s1, ns1]
[U413, x2, nx2] > [U213, plus2, nplus2] > and2
[U413, x2, nx2] > U311 > [0, n0] > U112
[U413, x2, nx2] > U311 > [0, n0] > tt
Status:
U112: [2,1]
tt: multiset
U213: [3,2,1]
s1: multiset
plus2: [1,2]
U311: multiset
0: multiset
U413: [2,3,1]
x2: [2,1]
and2: multiset
n0: multiset
nplus2: [1,2]
ns1: multiset
nx2: [2,1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
activate1 > isNat1 > x2 > nx2 > s1 > ns1 > nisNat1 > plus2 > 0 > nplus2 > n0
and weight map:
0=1
n__0=1
isNat_1=1
n__isNat_1=1
s_1=1
n__s_1=1
activate_1=0
plus_2=0
n__plus_2=0
x_2=0
n__x_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:
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(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