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

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(x))
FIB(s(s(x))) -> FIB(s(x))
FIB(s(s(x))) -> FIB(x)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs.

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

Dependency Pair:

+'(x, s(y)) -> +'(x, y)

Rules:

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

Strategy:

innermost

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

+'(x, s(y)) -> +'(x, y)
one new Dependency Pair is created:

+'(x'', s(s(y''))) -> +'(x'', s(y''))

The transformation is resulting in one new DP problem:

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

Dependency Pair:

+'(x'', s(s(y''))) -> +'(x'', s(y''))

Rules:

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

Strategy:

innermost

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

+'(x'', s(s(y''))) -> +'(x'', s(y''))
one new Dependency Pair is created:

+'(x'''', s(s(s(y'''')))) -> +'(x'''', s(s(y'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

+'(x'''', s(s(s(y'''')))) -> +'(x'''', s(s(y'''')))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

+'(x'''', s(s(s(y'''')))) -> +'(x'''', s(s(y'''')))

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(+'(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳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))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

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`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳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))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

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 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳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))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

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 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳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))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

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 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳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))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

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`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳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))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

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 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes