Term Rewriting System R:
[x]
not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

EVENODD(x, 0) -> NOT(evenodd(x, s(0)))
EVENODD(x, 0) -> EVENODD(x, s(0))
EVENODD(s(x), s(0)) -> EVENODD(x, 0)

Furthermore, R contains one SCC.

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

Dependency Pairs:

EVENODD(s(x), s(0)) -> EVENODD(x, 0)
EVENODD(x, 0) -> EVENODD(x, s(0))

Rules:

not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Strategy:

innermost

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

EVENODD(x, 0) -> EVENODD(x, s(0))
one new Dependency Pair is created:

EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))
EVENODD(s(x), s(0)) -> EVENODD(x, 0)

Rules:

not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Strategy:

innermost

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

EVENODD(s(x), s(0)) -> EVENODD(x, 0)
one new Dependency Pair is created:

EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)

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:

EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)
EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))

Rules:

not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Strategy:

innermost

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

EVENODD(s(x''), 0) -> EVENODD(s(x''), s(0))
one new Dependency Pair is created:

EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))

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:

EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))
EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)

Rules:

not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Strategy:

innermost

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

EVENODD(s(s(x'''')), s(0)) -> EVENODD(s(x''''), 0)
one new Dependency Pair is created:

EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`

Dependency Pairs:

EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)
EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))

Rules:

not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

EVENODD(s(s(s(x''''''''))), s(0)) -> EVENODD(s(s(x'''''''')), 0)

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(0) =  0 POL(EVENODD(x1, x2)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`

Dependency Pair:

EVENODD(s(s(x'''''')), 0) -> EVENODD(s(s(x'''''')), s(0))

Rules:

not(true) -> false
not(false) -> true
evenodd(x, 0) -> not(evenodd(x, s(0)))
evenodd(0, s(0)) -> false
evenodd(s(x), s(0)) -> evenodd(x, 0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes