(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) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) 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))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) 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)
H(x, y) → G(x, f(y))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
G(
x0,
x1,
x2) =
G(
x1)
H(
x0,
x1,
x2) =
H(
x1)
Tags:
G has argument tags [1,0,1] and root tag 1
H has argument tags [0,6,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
G(
x1,
x2) =
G
f(
x1) =
f(
x1)
H(
x1,
x2) =
H(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
[f1, H2] > G
Status:
G: multiset
f1: multiset
H2: [2,1]
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) TRUE