Term Rewriting System R:
[x, y]
times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) -> x
times(x, 0) -> 0
plus(x, 0) -> x

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

TIMES(x, plus(y, 1)) -> PLUS(times(x, plus(y, times(1, 0))), x)
TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, times(1, 0)))
TIMES(x, plus(y, 1)) -> PLUS(y, times(1, 0))
TIMES(x, plus(y, 1)) -> TIMES(1, 0)

Furthermore, R contains one SCC.

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

Dependency Pair:

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, times(1, 0)))

Rules:

times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) -> x
times(x, 0) -> 0
plus(x, 0) -> x

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

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, times(1, 0)))
one new Dependency Pair is created:

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, 0))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Argument Filtering and Ordering`

Dependency Pair:

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, 0))

Rules:

times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) -> x
times(x, 0) -> 0
plus(x, 0) -> x

The following dependency pair can be strictly oriented:

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, 0))

The following usable rule using the Ce-refinement can be oriented:

plus(x, 0) -> x

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(TIMES(x1, x2)) =  1 + x1 + x2 POL(plus(x1, x2)) =  x1 + x2 POL(0) =  0 POL(1) =  1

resulting in one new DP problem.
Used Argument Filtering System:
TIMES(x1, x2) -> TIMES(x1, x2)
plus(x1, x2) -> plus(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) -> x
times(x, 0) -> 0
plus(x, 0) -> x

Using the Dependency Graph resulted in no new DP problems.

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