(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
(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(a, a) → F(a, b)
F(a, b) → F(s(a), c)
F(s(X), c) → F(X, c)
F(c, c) → F(a, a)
The TRS R consists of the following rules:
f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
We have to consider all minimal (P,Q,R)-chains.
(5) 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.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, a) → F(a, b)
F(a, b) → F(s(a), c)
F(s(X), c) → F(X, c)
F(c, c) → F(a, a)
R is empty.
The set Q consists of the following terms:
f(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
We have to consider all minimal (P,Q,R)-chains.
(7) 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(a, a)
f(a, b)
f(s(x0), c)
f(c, c)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, a) → F(a, b)
F(a, b) → F(s(a), c)
F(s(X), c) → F(X, c)
F(c, c) → F(a, a)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(c, c) → F(a, a)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(F(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
none
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, a) → F(a, b)
F(a, b) → F(s(a), c)
F(s(X), c) → F(X, c)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(X), c) → F(X, c)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) 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), c) → F(X, c)
The graph contains the following edges 1 > 1, 2 >= 2
(14) TRUE