(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))
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:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))
The set Q consists of the following terms:
dfib(s(s(x0)), x1)
(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:
DFIB(s(s(x)), y) → DFIB(s(x), dfib(x, y))
DFIB(s(s(x)), y) → DFIB(x, y)
The TRS R consists of the following rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))
The set Q consists of the following terms:
dfib(s(s(x0)), x1)
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
DFIB(s(s(x)), y) → DFIB(s(x), dfib(x, y))
DFIB(s(s(x)), y) → DFIB(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DFIB(
x1,
x2) =
x1
s(
x1) =
s(
x1)
dfib(
x1,
x2) =
dfib
Lexicographic path order with status [LPO].
Precedence:
trivial
Status:
trivial
The following usable rules [FROCOS05] were oriented:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
dfib(s(s(x)), y) → dfib(s(x), dfib(x, y))
The set Q consists of the following terms:
dfib(s(s(x0)), x1)
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE