(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
f1 > b > e
f1 > d > e
f1 > [g2, h2] > e
a > b > e
c > d > e
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(x, x) → h(e, x)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
[g2, e] > h2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
g(x, x) → h(e, 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) TRUE