(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
a__b → a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__b → b
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Lexicographic Path Order [LPO].
Precedence:
[af2, ab, mark1, f2] > [a, b]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__b → a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__b → b
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
a__f(X1, X2) → f(X1, X2)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Lexicographic Path Order [LPO].
Precedence:
af2 > [a, b, f2]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__f(X1, X2) → f(X1, X2)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
Q is empty.