(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__F(X) → A__IF(mark(X), c, f(true))
A__F(X) → MARK(X)
A__IF(true, X, Y) → MARK(X)
A__IF(false, X, Y) → MARK(Y)
MARK(f(X)) → A__F(mark(X))
MARK(f(X)) → MARK(X)
MARK(if(X1, X2, X3)) → A__IF(mark(X1), mark(X2), X3)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(if(X1, X2, X3)) → MARK(X2)
The TRS R consists of the following rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.