(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
[af2, ab, mark1, f2] > [a, b]

Status:
af2: [1,2]
a: []
b: []
ab: []
mark1: [1]
f2: [1,2]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__bb


(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 with status [LPO].
Quasi-Precedence:
af2 > [a, b, f2]

Status:
af2: [1,2]
a: []
b: []
f2: [1,2]

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.