(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