(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(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:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))
The set Q consists of the following terms:
fib(0)
fib(s(0))
fib(s(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:
FIB(s(s(x))) → FIB(s(x))
FIB(s(s(x))) → FIB(x)
The TRS R consists of the following rules:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))
The set Q consists of the following terms:
fib(0)
fib(s(0))
fib(s(s(x0)))
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.
FIB(s(s(x))) → FIB(s(x))
FIB(s(s(x))) → FIB(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIB(
x1) =
x1
s(
x1) =
s(
x1)
fib(
x1) =
fib(
x1)
0 =
0
+(
x1,
x2) =
+
Lexicographic path order with status [LPO].
Quasi-Precedence:
0 > [s1, fib1, +]
Status:
s1: [1]
fib1: [1]
0: []
+: []
The following usable rules [FROCOS05] were oriented:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))
The set Q consists of the following terms:
fib(0)
fib(s(0))
fib(s(s(x0)))
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