(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(f, x))
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(F, app(app(F, f), x)), x) → APP(app(F, app(G, app(app(F, f), x))), app(f, x))
APP(app(F, app(app(F, f), x)), x) → APP(F, app(G, app(app(F, f), x)))
APP(app(F, app(app(F, f), x)), x) → APP(G, app(app(F, f), x))
APP(app(F, app(app(F, f), x)), x) → APP(f, x)
The TRS R consists of the following rules:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(f, x))
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 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(F, app(app(F, f), x)), x) → APP(f, x)
The TRS R consists of the following rules:
app(app(F, app(app(F, f), x)), x) → app(app(F, app(G, app(app(F, f), x))), app(f, x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(F, app(app(F, f), x)), x) → APP(f, x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- APP(app(F, app(app(F, f), x)), x) → APP(f, x)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2
(8) TRUE