(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)

Lexicographic Path Order [LPO].
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