(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(f(x1, x2, x3)) = 2·x1 + x2 + x3
POL(g(x1, x2)) = 2 + x1 + x2
POL(s(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
g(x, y) → x
g(x, y) → y
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x), y, y) → f(y, x, s(x))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(f(x1, x2, x3)) = 2·x1 + 2·x2 + x3
POL(s(x1)) = 1 + 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(s(x), y, y) → f(y, x, s(x))
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES