(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)) = x1 + x2
POL(g(x1)) = x1
POL(h(x1, x2, x3)) = 2·x1 + 2·x2 + 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)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y) → x
g(a) → h(a, b, a)
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)) = 2 + x1 + x2
POL(g(x1)) = x1
POL(h(x1, x2, x3)) = 2·x1 + 2·x2 + x3
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(x, y) → 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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
a > [g1, h3, b]
Status:
g1: [1]
a: multiset
h3: [2,3,1]
b: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
g(a) → h(a, b, a)
h(x, x, y) → g(x)
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES