(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y, z) → g(x, y, z)
g(0, 1, x) → 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 none
The TRS R 2 is
f(x, y, z) → g(x, y, z)
g(0, 1, x) → f(x, x, x)
The signature Sigma is {
f,
g}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y, z) → g(x, y, z)
g(0, 1, x) → f(x, x, x)
The set Q consists of the following terms:
f(x0, x1, x2)
g(0, 1, 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(x, y, z) → G(x, y, z)
G(0, 1, x) → F(x, x, x)
The TRS R consists of the following rules:
f(x, y, z) → g(x, y, z)
g(0, 1, x) → f(x, x, x)
The set Q consists of the following terms:
f(x0, x1, x2)
g(0, 1, x0)
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(x, y, z) → G(x, y, z)
G(0, 1, x) → F(x, x, x)
R is empty.
The set Q consists of the following terms:
f(x0, x1, x2)
g(0, 1, x0)
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(x0, x1, x2)
g(0, 1, x0)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, y, z) → G(x, y, z)
G(0, 1, x) → F(x, x, x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
x,
y,
z) →
G(
x,
y,
z) we obtained the following new rules [LPAR04]:
F(z0, z0, z0) → G(z0, z0, z0) → F(z0, z0, z0) → G(z0, z0, z0)
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(0, 1, x) → F(x, x, x)
F(z0, z0, z0) → G(z0, z0, z0)
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 0 SCCs with 2 less nodes.
(12) TRUE