(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
The set Q consists of the following terms:
g(s(x0))
f(0)
f(s(x0))
g(0)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(s(x)) → F(x)
F(s(x)) → G(x)
The TRS R consists of the following rules:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
The set Q consists of the following terms:
g(s(x0))
f(0)
f(s(x0))
g(0)
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.
G(s(x)) → F(x)
F(s(x)) → G(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
G(
x1) =
G(
x1)
s(
x1) =
s(
x1)
F(
x1) =
F(
x1)
g(
x1) =
x1
f(
x1) =
f(
x1)
0 =
0
Lexicographic path order with status [LPO].
Quasi-Precedence:
[G1, s1, f1] > 0 > F1
Status:
G1: [1]
s1: [1]
F1: [1]
f1: [1]
0: []
The following usable rules [FROCOS05] were oriented:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
The set Q consists of the following terms:
g(s(x0))
f(0)
f(s(x0))
g(0)
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE