(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, y) → 0
f(s(x), y) → f(f(x, y), y)
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, y) → 0
f(s(x), y) → f(f(x, y), y)
The set Q consists of the following terms:
f(0, x0)
f(s(x0), x1)
(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), y) → F(f(x, y), y)
F(s(x), y) → F(x, y)
The TRS R consists of the following rules:
f(0, y) → 0
f(s(x), y) → f(f(x, y), y)
The set Q consists of the following terms:
f(0, x0)
f(s(x0), x1)
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 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, y)
The TRS R consists of the following rules:
f(0, y) → 0
f(s(x), y) → f(f(x, y), y)
The set Q consists of the following terms:
f(0, x0)
f(s(x0), x1)
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), y) → F(x, y)
R is empty.
The set Q consists of the following terms:
f(0, x0)
f(s(x0), x1)
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(0, x0)
f(s(x0), x1)
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), 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), y) → F(x, y)
The graph contains the following edges 1 > 1, 2 >= 2
(12) TRUE