(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

app(app(apply, f), x) → app(f, 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(apply, f), x) → app(f, x)

The set Q consists of the following terms:

app(app(apply, x0), x1)

(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(apply, f), x) → APP(f, x)

The TRS R consists of the following rules:

app(app(apply, f), x) → app(f, x)

The set Q consists of the following terms:

app(app(apply, x0), x1)

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(apply, f), 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)  =  APP(x1, x2)
app(x1, x2)  =  app(x2)
apply  =  apply

Recursive path order with status [RPO].
Precedence:
app1 > APP2
apply > APP2

Status:
APP2: multiset
app1: [1]
apply: multiset

The following usable rules [FROCOS05] were oriented:

app(app(apply, f), x) → app(f, x)

(6) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app(app(apply, f), x) → app(f, x)

The set Q consists of the following terms:

app(app(apply, x0), x1)

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