(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) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
(4) 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)))
The set Q consists of the following terms:
f(x0)
We have to consider all minimal (P,Q,R)-chains.
(5) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
F(s(s(x))) → F(f(x))
F(s(s(x))) → F(x)
Used ordering: Polynomial interpretation [POLO]:
POL(F(x1)) = 3·x1
POL(f(x1)) = 3 + x1
POL(s(x1)) = 3 + x1
(6) 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)))
The set Q consists of the following terms:
f(x0)
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) YES