(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)
app(
x1,
x2) =
app(
x1,
x2)
apply =
apply
Lexicographic path order with status [LPO].
Precedence:
app2 > APP1
apply > APP1
Status:
APP1: [1]
apply: []
app2: [1,2]
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(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