(0) Obligation:

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

app(id, x) → x
app(plus, 0) → id
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))

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(id, x) → x
app(plus, 0) → id
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))

The set Q consists of the following terms:

app(id, x0)
app(plus, 0)
app(app(plus, app(s, 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(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) → APP(plus, x)

The TRS R consists of the following rules:

app(id, x) → x
app(plus, 0) → id
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))

The set Q consists of the following terms:

app(id, x0)
app(plus, 0)
app(app(plus, app(s, x0)), x1)

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 2 less nodes.

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)

The TRS R consists of the following rules:

app(id, x) → x
app(plus, 0) → id
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))

The set Q consists of the following terms:

app(id, x0)
app(plus, 0)
app(app(plus, app(s, x0)), x1)

We have to consider all minimal (P,Q,R)-chains.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
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)
plus  =  plus
s  =  s
0  =  0
id  =  id

Lexicographic path order with status [LPO].
Quasi-Precedence:
plus > [app1, id] > [APP2, s]
0 > [app1, id] > [APP2, s]

Status:
APP2: [1,2]
app1: [1]
plus: []
s: []
0: []
id: []


The following usable rules [FROCOS05] were oriented:

app(plus, 0) → id

(8) Obligation:

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

app(id, x) → x
app(plus, 0) → id
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))

The set Q consists of the following terms:

app(id, x0)
app(plus, 0)
app(app(plus, app(s, x0)), x1)

We have to consider all minimal (P,Q,R)-chains.

(9) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(10) TRUE