(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
The TRS R 2 is
g(0, x) → g(f(x, x), x)
The signature Sigma is {
g}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
(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:
F(s(x), s(y)) → F(x, y)
G(0, x) → G(f(x, x), x)
G(0, x) → F(x, x)
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
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:
F(s(x), s(y)) → F(x, y)
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
We have to consider all minimal (P,Q,R)-chains.
(7) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), s(y)) → F(x, y)
R is empty.
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
We have to consider all minimal (P,Q,R)-chains.
(9) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), s(y)) → F(x, y)
R is empty.
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(s(x), s(y)) → F(x, y)
The graph contains the following edges 1 > 1, 2 > 2
(12) TRUE