(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__ba
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__bb

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > [af3, a, ab] > f3 > b

Status:
af3: multiset
a: multiset
ab: multiset
b: multiset
mark1: multiset
f3: multiset

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)
a__bb


(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__ba

Q is empty.