(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(X), Y) → f(X, f(g(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) → f(X, f(g(X), Y))
The set Q consists of the following terms:
f(g(x0), 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) → F(X, f(g(X), Y))
F(g(X), Y) → F(g(X), Y)
The TRS R consists of the following rules:
f(g(X), Y) → f(X, f(g(X), Y))
The set Q consists of the following terms:
f(g(x0), 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) → F(X, f(g(X), Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1)
g(
x1) =
g(
x1)
f(
x1,
x2) =
f
Lexicographic Path Order [LPO].
Precedence:
F1 > f
g1 > f
The following usable rules [FROCOS05] were oriented:
f(g(X), Y) → f(X, f(g(X), Y))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(X), Y) → F(g(X), Y)
The TRS R consists of the following rules:
f(g(X), Y) → f(X, f(g(X), Y))
The set Q consists of the following terms:
f(g(x0), x1)
We have to consider all minimal (P,Q,R)-chains.