(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