Term Rewriting System R:
[x, y, z]
cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

INF(x) -> CONS(x, inf(s(x)))
INF(x) -> INF(s(x))

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Instantiation Transformation`

Dependency Pair:

INF(x) -> INF(s(x))

Rules:

cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Strategy:

innermost

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

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

INF(s(x'')) -> INF(s(s(x'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Instantiation Transformation`

Dependency Pair:

INF(s(x'')) -> INF(s(s(x'')))

Rules:

cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Strategy:

innermost

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

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

INF(s(s(x''''))) -> INF(s(s(s(x''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 3`
`                 ↳Instantiation Transformation`

Dependency Pair:

INF(s(s(x''''))) -> INF(s(s(s(x''''))))

Rules:

cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Strategy:

innermost

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

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

INF(s(s(s(x'''''')))) -> INF(s(s(s(s(x'''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 4`
`                 ↳Instantiation Transformation`

Dependency Pair:

INF(s(s(s(x'''''')))) -> INF(s(s(s(s(x'''''')))))

Rules:

cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Strategy:

innermost

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

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

INF(s(s(s(s(x''''''''))))) -> INF(s(s(s(s(s(x''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Instantiation Transformation`

Dependency Pair:

INF(s(s(s(s(x''''''''))))) -> INF(s(s(s(s(s(x''''''''))))))

Rules:

cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Strategy:

innermost

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

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

INF(s(s(s(s(s(x'''''''''')))))) -> INF(s(s(s(s(s(s(x'''''''''')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 6`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

INF(s(s(s(s(s(x'''''''''')))))) -> INF(s(s(s(s(s(s(x'''''''''')))))))

Rules:

cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes