Term Rewriting System R:
[X, Y, Z]
div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

DIV(X, e) -> I(X)
DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
DIV(div(X, Y), Z) -> DIV(i(X), Z)
DIV(div(X, Y), Z) -> I(X)
I(div(X, Y)) -> DIV(Y, X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

DIV(div(X, Y), Z) -> DIV(i(X), Z)
DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
I(div(X, Y)) -> DIV(Y, X)
DIV(X, e) -> I(X)

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

DIV(X, e) -> I(X)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

i(div(X, Y)) -> div(Y, X)
div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(I(x1)) =  x1 POL(i(x1)) =  x1 POL(e) =  1 POL(DIV(x1, x2)) =  x1 + x2 POL(div(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
DIV(x1, x2) -> DIV(x1, x2)
div(x1, x2) -> div(x1, x2)
i(x1) -> i(x1)
I(x1) -> I(x1)

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

Dependency Pairs:

DIV(div(X, Y), Z) -> DIV(i(X), Z)
DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
I(div(X, Y)) -> DIV(Y, X)

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 3`
`                 ↳Argument Filtering and Ordering`

Dependency Pairs:

DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
DIV(div(X, Y), Z) -> DIV(i(X), Z)

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

DIV(div(X, Y), Z) -> DIV(i(X), Z)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

i(div(X, Y)) -> div(Y, X)
div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(i(x1)) =  x1 POL(e) =  0 POL(DIV(x1, x2)) =  1 + x1 + x2 POL(div(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
DIV(x1, x2) -> DIV(x1, x2)
div(x1, x2) -> div(x1, x2)
i(x1) -> i(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 4`
`                 ↳Argument Filtering and Ordering`

Dependency Pair:

DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(div(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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