(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(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:
F(x, c(y)) → F(x, s(f(y, y)))
F(x, c(y)) → F(y, y)
F(s(x), y) → F(x, s(c(y)))
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(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 2 SCCs with 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(c(y)))
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) 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.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(c(y)))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) 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:
- F(s(x), y) → F(x, s(c(y)))
The graph contains the following edges 1 > 1
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, c(y)) → F(y, y)
The TRS R consists of the following rules:
f(x, c(y)) → f(x, s(f(y, y)))
f(s(x), y) → f(x, s(c(y)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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:
- F(x, c(y)) → F(y, y)
The graph contains the following edges 2 > 1, 2 > 2
(12) TRUE