Term Rewriting System R:
[x]
fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

FIB(s(s(x))) -> FIB(s(x))
FIB(s(s(x))) -> FIB(x)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Forward Instantiation Transformation

Dependency Pairs:

FIB(s(s(x))) -> FIB(x)
FIB(s(s(x))) -> FIB(s(x))

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FIB(s(s(x))) -> FIB(s(x))
one new Dependency Pair is created:

FIB(s(s(s(x'')))) -> FIB(s(s(x'')))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
Forward Instantiation Transformation

Dependency Pairs:

FIB(s(s(s(x'')))) -> FIB(s(s(x'')))
FIB(s(s(x))) -> FIB(x)

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FIB(s(s(x))) -> FIB(x)
two new Dependency Pairs are created:

FIB(s(s(s(s(x''))))) -> FIB(s(s(x'')))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(x''''))))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
FwdInst
...
→DP Problem 3
Forward Instantiation Transformation

Dependency Pairs:

FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(x''''))))
FIB(s(s(s(s(x''))))) -> FIB(s(s(x'')))
FIB(s(s(s(x'')))) -> FIB(s(s(x'')))

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FIB(s(s(s(x'')))) -> FIB(s(s(x'')))
three new Dependency Pairs are created:

FIB(s(s(s(s(x''''))))) -> FIB(s(s(s(x''''))))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(s(x''''''))))))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
FwdInst
...
→DP Problem 4
Forward Instantiation Transformation

Dependency Pairs:

FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(x''''))))) -> FIB(s(s(s(x''''))))
FIB(s(s(s(s(x''))))) -> FIB(s(s(x'')))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(x''''))))

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FIB(s(s(s(s(x''))))) -> FIB(s(s(x'')))
four new Dependency Pairs are created:

FIB(s(s(s(s(s(s(x''''))))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(s(s(s(x'''''')))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(x'''''')))))
FIB(s(s(s(s(s(s(s(s(x''''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''''')))))))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
FwdInst
...
→DP Problem 5
Forward Instantiation Transformation

Dependency Pairs:

FIB(s(s(s(s(s(s(s(s(x''''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''''')))))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(x'''''')))))
FIB(s(s(s(s(s(s(s(x'''''')))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(s(x''''))))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(x''''))))) -> FIB(s(s(s(x''''))))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(x''''))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(s(x''''''))))))

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(x''''))))
six new Dependency Pairs are created:

FIB(s(s(s(s(s(s(s(x'''''')))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(x'''''')))))
FIB(s(s(s(s(s(s(s(s(x''''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''''')))))))
FIB(s(s(s(s(s(s(s(s(x''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''')))))))
FIB(s(s(s(s(s(s(s(s(s(x'''''''')))))))))) -> FIB(s(s(s(s(s(s(s(x''''''''))))))))
FIB(s(s(s(s(s(s(s(s(s(s(x''''''''''))))))))))) -> FIB(s(s(s(s(s(s(s(s(x'''''''''')))))))))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
FwdInst
...
→DP Problem 6
Polynomial Ordering

Dependency Pairs:

FIB(s(s(s(s(s(s(s(s(s(s(x''''''''''))))))))))) -> FIB(s(s(s(s(s(s(s(s(x'''''''''')))))))))
FIB(s(s(s(s(s(s(s(s(s(x'''''''')))))))))) -> FIB(s(s(s(s(s(s(s(x''''''''))))))))
FIB(s(s(s(s(s(s(s(s(x''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''')))))))
FIB(s(s(s(s(s(s(s(s(x''''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''''')))))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(x'''''')))))
FIB(s(s(s(s(s(s(s(x'''''')))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(x'''''')))))
FIB(s(s(s(s(s(s(s(x'''''')))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(s(x''''))))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(x''''))))) -> FIB(s(s(s(x''''))))
FIB(s(s(s(s(s(s(s(s(x''''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''''')))))))

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

FIB(s(s(s(s(s(s(s(s(s(s(x''''''''''))))))))))) -> FIB(s(s(s(s(s(s(s(s(x'''''''''')))))))))
FIB(s(s(s(s(s(s(s(s(s(x'''''''')))))))))) -> FIB(s(s(s(s(s(s(s(x''''''''))))))))
FIB(s(s(s(s(s(s(s(s(x''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''')))))))
FIB(s(s(s(s(s(s(s(s(x''''''''))))))))) -> FIB(s(s(s(s(s(s(x'''''''')))))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(x'''''')))))
FIB(s(s(s(s(s(s(s(x'''''')))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(s(x''''))))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(s(s(x''''''))))))) -> FIB(s(s(s(s(s(x''''''))))))
FIB(s(s(s(s(s(x'''')))))) -> FIB(s(s(s(s(x'''')))))
FIB(s(s(s(s(x''''))))) -> FIB(s(s(s(x''''))))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(FIB(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
FwdInst
...
→DP Problem 7
Dependency Graph

Dependency Pair:

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(x))) -> +(fib(s(x)), fib(x))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes