(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, x) → a
f(g(x), y) → f(x, y)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
f(x, x) → a
f(g(x), y) → f(x, y)
The signature Sigma is {
a,
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, x) → a
f(g(x), y) → f(x, y)
The set Q consists of the following terms:
f(x0, x0)
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, y)
The TRS R consists of the following rules:
f(x, x) → a
f(g(x), y) → f(x, y)
The set Q consists of the following terms:
f(x0, x0)
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, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x1
g(
x1) =
g(
x1)
Lexicographic Path Order [LPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x, x) → a
f(g(x), y) → f(x, y)
The set Q consists of the following terms:
f(x0, x0)
f(g(x0), 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