(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(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:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))
The set Q consists of the following terms:
g(f(x0), x1)
h(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:
G(f(x), y) → H(x, y)
H(x, y) → G(x, f(y))
The TRS R consists of the following rules:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))
The set Q consists of the following terms:
g(f(x0), x1)
h(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.
G(f(x), y) → H(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
G(
x1,
x2) =
x1
f(
x1) =
f(
x1)
H(
x1,
x2) =
x1
g(
x1,
x2) =
x1
h(
x1,
x2) =
x1
Lexicographic path order with status [LPO].
Precedence:
trivial
Status:
f1: [1]
The following usable rules [FROCOS05] were oriented:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(x, y) → G(x, f(y))
The TRS R consists of the following rules:
g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))
The set Q consists of the following terms:
g(f(x0), x1)
h(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(8) TRUE