(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(b) = 0
POL(f(x1, x2)) = 2 + x1 + x2
POL(g(x1)) = 2·x1
POL(h(x1, x2, x3)) = x1 + 2·x2 + 2·x3
POL(i(x1)) = 2 + 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(x, y) → x
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(b) = 0
POL(f(x1, x2)) = 1 + x1 + x2
POL(g(x1)) = x1
POL(h(x1, x2, x3)) = x1 + x2 + 2·x3
POL(i(x1)) = 2 + 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
i(x) → f(x, x)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(a) → h(a, b, a)
h(x, x, y) → g(x)
Q is empty.
(5) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
g(a) → h(a, b, a)
h(x, x, y) → g(x)
The signature Sigma is {
h,
g}
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(a) → h(a, b, a)
h(x, x, y) → g(x)
The set Q consists of the following terms:
g(a)
h(x0, x0, x1)
(7) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(a) → H(a, b, a)
H(x, x, y) → G(x)
The TRS R consists of the following rules:
g(a) → h(a, b, a)
h(x, x, y) → g(x)
The set Q consists of the following terms:
g(a)
h(x0, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(10) TRUE