(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(h(g(x))) → g(x)
g(g(x)) → g(h(g(x)))
h(h(x)) → h(f(h(x), 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:
G(g(x)) → G(h(g(x)))
G(g(x)) → H(g(x))
H(h(x)) → H(f(h(x), x))
The TRS R consists of the following rules:
g(h(g(x))) → g(x)
g(g(x)) → g(h(g(x)))
h(h(x)) → h(f(h(x), x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(g(x)) → G(h(g(x)))
The TRS R consists of the following rules:
g(h(g(x))) → g(x)
g(g(x)) → g(h(g(x)))
h(h(x)) → h(f(h(x), x))
Q is empty.
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(g(x)) → G(h(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)
g(
x1) =
g
h(
x1) =
h
f(
x1,
x2) =
f(
x1,
x2)
Lexicographic path order with status [LPO].
Precedence:
G1 > g > h
Status:
f2: [1,2]
g: []
G1: [1]
h: []
The following usable rules [FROCOS05] were oriented:
h(h(x)) → h(f(h(x), x))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
g(h(g(x))) → g(x)
g(g(x)) → g(h(g(x)))
h(h(x)) → h(f(h(x), x))
Q is empty.
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