(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)
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:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)
The set Q consists of the following terms:
f(x0, g(x0))
f(x0, h(x1))
(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:
F(x, h(y)) → F(h(x), y)
The TRS R consists of the following rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)
The set Q consists of the following terms:
f(x0, g(x0))
f(x0, h(x1))
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.
F(x, h(y)) → F(h(x), y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x2
h(
x1) =
h(
x1)
f(
x1,
x2) =
f(
x1,
x2)
g(
x1) =
g
Lexicographic path order with status [LPO].
Precedence:
f2 > h1
g > h1
Status:
h1: [1]
f2: [2,1]
g: []
The following usable rules [FROCOS05] were oriented:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)
The set Q consists of the following terms:
f(x0, g(x0))
f(x0, h(x1))
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