Term Rewriting System R:
[X]
f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))
F(s(0)) -> P(s(0))
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(n0) -> 0'

Furthermore, R contains one SCC.

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

Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> ACTIVATE(X)

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(X)) -> ACTIVATE(X)
two new Dependency Pairs are created:

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(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:

ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
two new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(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:

ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(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:

ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(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:

ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
two new Dependency Pairs are created:

ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nf(ns(ns(X''''''))))) -> ACTIVATE(nf(ns(ns(X''''''))))
ACTIVATE(ns(nf(ns(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ns(nf(nf(X'''''''')))))
ACTIVATE(ns(nf(ns(nf(ns(X'''''''')))))) -> ACTIVATE(nf(ns(nf(ns(X'''''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`

Dependency Pairs:

ACTIVATE(ns(nf(ns(nf(ns(X'''''''')))))) -> ACTIVATE(nf(ns(nf(ns(X'''''''')))))
ACTIVATE(ns(nf(ns(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ns(nf(nf(X'''''''')))))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(ns(X''''''))))) -> ACTIVATE(nf(ns(ns(X''''''))))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(ns(nf(ns(nf(ns(X'''''''')))))) -> ACTIVATE(nf(ns(nf(ns(X'''''''')))))
ACTIVATE(ns(nf(ns(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ns(nf(nf(X'''''''')))))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(ns(X''''''))))) -> ACTIVATE(nf(ns(ns(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(n__f(x1)) =  x1 POL(n__s(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Dependency Graph`

Dependency Pairs:

ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Polynomial Ordering`

Dependency Pair:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(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(n__f(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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