(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
mark1 > af3 > f3 > [a, ab, b]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
a__b → b
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
[af3, a, ab] > b
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__b → b
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
Q is empty.