(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
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:
A(f, a(f, x)) → A(x, x)
A(h, x) → A(f, a(g, a(f, x)))
A(h, x) → A(g, a(f, x))
A(h, x) → A(f, x)
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(h, x) → A(f, x)
A(f, a(f, x)) → A(x, x)
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
Q is empty.
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.
A(f, a(f, x)) → A(x, x)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A(
x1,
x2) =
A(
x1,
x2)
Tags:
A has tags [1,0]
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
h =
h
f =
f
a(
x1,
x2) =
a(
x2)
Lexicographic path order with status [LPO].
Quasi-Precedence:
a1 > [h, f]
Status:
h: []
f: []
a1: [1]
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(h, x) → A(f, x)
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
Q is empty.
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