(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x, y) → b(x, b(0, c(y)))
c(b(y, c(x))) → c(c(b(a(0, 0), y)))
b(y, 0) → y
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(x, y) → B(x, b(0, c(y)))
A(x, y) → B(0, c(y))
A(x, y) → C(y)
C(b(y, c(x))) → C(c(b(a(0, 0), y)))
C(b(y, c(x))) → C(b(a(0, 0), y))
C(b(y, c(x))) → B(a(0, 0), y)
C(b(y, c(x))) → A(0, 0)
The TRS R consists of the following rules:
a(x, y) → b(x, b(0, c(y)))
c(b(y, c(x))) → c(c(b(a(0, 0), y)))
b(y, 0) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(x, y) → C(y)
C(b(y, c(x))) → C(c(b(a(0, 0), y)))
C(b(y, c(x))) → C(b(a(0, 0), y))
C(b(y, c(x))) → A(0, 0)
The TRS R consists of the following rules:
a(x, y) → b(x, b(0, c(y)))
c(b(y, c(x))) → c(c(b(a(0, 0), y)))
b(y, 0) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.