(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x
The set Q consists of the following terms:
a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(d(x)) → B(a(x))
A(d(x)) → A(x)
B(c(x)) → A(b(x))
B(c(x)) → B(x)
The TRS R consists of the following rules:
a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x
The set Q consists of the following terms:
a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))
We have to consider all minimal (P,Q,R)-chains.