(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) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) 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))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) 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))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Polynomial interpretation [POLO]:
POL(0) = 0
POL(app(x1, x2)) = 1 + x2
POL(id) = 0
POL(plus) = 1
POL(s) = 1
From the DPs we obtained the following set of size-change graphs:
- APP(app(plus, app(s, x)), y) → APP(app(plus, x), y) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
app(plus, 0) → id
(6) TRUE