(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(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:
F(s(s(x))) → F(f(x))
F(s(s(x))) → F(x)
The TRS R consists of the following rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(s(s(x))) → F(f(x))
F(s(s(x))) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
x1
s(
x1) =
s(
x1)
f(
x1) =
f(
x1)
Recursive Path Order [RPO].
Precedence:
[s1, f1]
The following usable rules [FROCOS05] were oriented:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) TRUE