(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(x), y, y) → g(f(x, 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(g(x), y, y) → g(f(x, x, y))
The set Q consists of the following terms:
f(g(x0), x1, 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(g(x), y, y) → F(x, x, y)
The TRS R consists of the following rules:
f(g(x), y, y) → g(f(x, x, y))
The set Q consists of the following terms:
f(g(x0), x1, 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(g(x), y, y) → F(x, x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2,
x3) =
F(
x1,
x3)
g(
x1) =
g(
x1)
f(
x1,
x2,
x3) =
f(
x1,
x3)
Recursive path order with status [RPO].
Quasi-Precedence:
f2 > [F2, g1]
Status:
F2: multiset
g1: multiset
f2: multiset
The following usable rules [FROCOS05] were oriented:
f(g(x), y, y) → g(f(x, x, y))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(g(x), y, y) → g(f(x, x, y))
The set Q consists of the following terms:
f(g(x0), x1, 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