(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), 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(fmap, app(app(fcons, f), t)), x) → APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(fmap, t)
The TRS R consists of the following rules:
app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), 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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)
The TRS R consists of the following rules:
app(app(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), 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(fmap, app(app(fcons, f), t)), x) → APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) → APP(f, x)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(
x0,
x1,
x2) =
APP(
x1)
Tags:
APP has argument tags [1,3,0] 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.
APP(
x1,
x2) =
APP
app(
x1,
x2) =
app(
x1,
x2)
fmap =
fmap
fcons =
fcons
Recursive path order with status [RPO].
Quasi-Precedence:
APP > [app2, fmap]
Status:
APP: multiset
app2: [1,2]
fmap: multiset
fcons: multiset
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(fmap, fnil), x) → nil
app(app(fmap, app(app(fcons, f), t)), x) → app(app(cons, app(f, x)), app(app(fmap, t), 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