(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
ap(ap(g, x), y) → y
ap(f, x) → ap(f, app(g, x))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(ap(x1, x2)) = 1 + 2·x1 + 2·x2
POL(app(x1, x2)) = x1 + x2
POL(f) = 0
POL(g) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
ap(ap(g, x), y) → y
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
ap(f, x) → ap(f, app(g, x))
Q is empty.
(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:
AP(f, x) → AP(f, app(g, x))
The TRS R consists of the following rules:
ap(f, x) → ap(f, app(g, x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x0 / app(g, x0)] on the rule
AP(f, app(g, x0))[ ]n[ ] → AP(f, app(g, x0))[ ]n[x0 / app(g, x0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
AP(f, x)[ ]n[ ] → AP(f, app(g, x))[ ]n[ ]
by OriginalRule from TRS P
(6) NO