(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, 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(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x
The set Q consists of the following terms:
a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(s, 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(f, 0) → A(s, 0)
A(d, a(s, x)) → A(s, a(s, a(d, a(p, a(s, x)))))
A(d, a(s, x)) → A(s, a(d, a(p, a(s, x))))
A(d, a(s, x)) → A(d, a(p, a(s, x)))
A(d, a(s, x)) → A(p, a(s, x))
A(f, a(s, x)) → A(d, a(f, a(p, a(s, x))))
A(f, a(s, x)) → A(f, a(p, a(s, x)))
A(f, a(s, x)) → A(p, a(s, x))
The TRS R consists of the following rules:
a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x
The set Q consists of the following terms:
a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(s, x0))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(d, a(s, x)) → A(d, a(p, a(s, x)))
The TRS R consists of the following rules:
a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x
The set Q consists of the following terms:
a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(s, x0))
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(f, a(s, x)) → A(f, a(p, a(s, x)))
The TRS R consists of the following rules:
a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x
The set Q consists of the following terms:
a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(s, x0))
We have to consider all minimal (P,Q,R)-chains.