(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(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:
APP(app(F, app(app(F, f), x)), x) → APP(app(F, app(G, app(app(F, f), x))), app(f, x))
APP(app(F, app(app(F, f), x)), x) → APP(F, app(G, app(app(F, f), x)))
APP(app(F, app(app(F, f), x)), x) → APP(G, app(app(F, f), x))
APP(app(F, app(app(F, f), x)), x) → APP(f, x)
The TRS R consists of the following rules:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(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 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(F, app(app(F, f), x)), x) → APP(f, x)
The TRS R consists of the following rules:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(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.
APP(app(F, app(app(F, f), x)), x) → APP(f, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(
x1,
x2) =
x1
app(
x1,
x2) =
app(
x1,
x2)
F =
F
Recursive Path Order [RPO].
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:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(f, x))
Q is empty.
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