(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) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) 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))
The set Q consists of the following terms:
app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) 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))
The set Q consists of the following terms:
app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(6) 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))
The set Q consists of the following terms:
app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)
We have to consider all minimal (P,Q,R)-chains.
(7) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(8) 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)
R is empty.
The set Q consists of the following terms:
app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)
We have to consider all minimal (P,Q,R)-chains.
(9) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be 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)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(APP(x1, x2)) = 2·x1 + x2
POL(app(x1, x2)) = 2·x1 + 2·x2
POL(fcons) = 0
POL(fmap) = 0
(10) Obligation:
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
app(app(fmap, fnil), x0)
app(app(fmap, app(app(fcons, x0), x1)), x2)
We have to consider all minimal (P,Q,R)-chains.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) TRUE