(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, 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:
H(f(x), y) → G(x, y)
G(x, y) → H(x, y)
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, 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(x, y) → H(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
H(
x0,
x1,
x2) =
H(
x0,
x1)
G(
x0,
x1,
x2) =
G(
x0,
x2)
Tags:
H has argument tags [6,1,5] and root tag 0
G has argument tags [1,0,6] and root tag 0
Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
H(
x1,
x2) =
x2
f(
x1) =
f(
x1)
G(
x1,
x2) =
G(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
[f1, G1]
Status:
f1: [1]
G1: [1]
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(f(x), y) → G(x, y)
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(6) TRUE