Term Rewriting System R:
[X, Y, Z]
plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

PLUS(plus(X, Y), Z) -> PLUS(X, plus(Y, Z))
PLUS(plus(X, Y), Z) -> PLUS(Y, Z)
TIMES(X, s(Y)) -> PLUS(X, times(Y, X))
TIMES(X, s(Y)) -> TIMES(Y, X)

Furthermore, R contains two SCCs.

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

Dependency Pair:

PLUS(plus(X, Y), Z) -> PLUS(Y, Z)

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Strategy:

innermost

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

PLUS(plus(X, Y), Z) -> PLUS(Y, Z)
one new Dependency Pair is created:

PLUS(plus(X, plus(X'', Y'')), Z'') -> PLUS(plus(X'', Y''), Z'')

The transformation is resulting in one new DP problem:

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

Dependency Pair:

PLUS(plus(X, plus(X'', Y'')), Z'') -> PLUS(plus(X'', Y''), Z'')

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PLUS(plus(X, plus(X'', Y'')), Z'') -> PLUS(plus(X'', Y''), Z'')

Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(plus(x1, x2)) =  1 + x1 + x2 POL(PLUS(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳Polo`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

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 Pair:

TIMES(X, s(Y)) -> TIMES(Y, X)

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Strategy:

innermost

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

TIMES(X, s(Y)) -> TIMES(Y, X)
one new Dependency Pair is created:

TIMES(s(Y''), s(Y0)) -> TIMES(Y0, s(Y''))

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 Pair:

TIMES(s(Y''), s(Y0)) -> TIMES(Y0, s(Y''))

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Strategy:

innermost

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

TIMES(s(Y''), s(Y0)) -> TIMES(Y0, s(Y''))
one new Dependency Pair is created:

TIMES(s(Y''''), s(s(Y'''''))) -> TIMES(s(Y'''''), s(Y''''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 5`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pair:

TIMES(s(Y''''), s(s(Y'''''))) -> TIMES(s(Y'''''), s(Y''''))

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TIMES(s(Y''''), s(s(Y'''''))) -> TIMES(s(Y'''''), 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(TIMES(x1, x2)) =  1 + x1 + x2 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 5`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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