(0) Obligation:

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

app(app(app(compose, f), g), x) → app(f, app(g, 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(app(compose, f), g), x) → app(f, app(g, x))

The set Q consists of the following terms:

app(app(app(compose, 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(app(compose, f), g), x) → APP(f, app(g, x))
APP(app(app(compose, f), g), x) → APP(g, x)

The TRS R consists of the following rules:

app(app(app(compose, f), g), x) → app(f, app(g, x))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)

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(app(compose, f), g), x) → APP(f, app(g, x))
APP(app(app(compose, f), g), x) → APP(g, 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)
compose  =  compose

Lexicographic path order with status [LPO].
Precedence:
compose > app2

Status:
app2: [1,2]
compose: []

The following usable rules [FROCOS05] were oriented:

app(app(app(compose, f), g), x) → app(f, app(g, x))

(6) Obligation:

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

app(app(app(compose, f), g), x) → app(f, app(g, x))

The set Q consists of the following terms:

app(app(app(compose, x0), x1), x2)

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