(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(x0, x1)

Tags:
APP has argument tags [0,0,3] 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.
APP(x1, x2)  =  x2
app(x1, x2)  =  app(x1, x2)
fmap  =  fmap
fcons  =  fcons

Recursive path order with status [RPO].
Quasi-Precedence:
app2 > fmap

Status:
app2: multiset
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