(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:
Lexicographic path order with status [LPO].
Quasi-Precedence:
f1 > b > h2
f1 > d > h2
f1 > g2 > e > h2
a > h2
c > h2
Status:
f1: [1]
a: []
b: []
c: []
d: []
g2: [2,1]
h2: [2,1]
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)))
g(x, x) → h(e, x)
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) TRUE