Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
+'(s(x), y) -> +'(x, y)
*'(x, s(y)) -> +'(x, *(x, y))
*'(x, s(y)) -> *'(x, y)
F(s(x)) -> F(-(p(*(s(x), s(x))), *(s(x), s(x))))
F(s(x)) -> -'(p(*(s(x), s(x))), *(s(x), s(x)))
F(s(x)) -> P(*(s(x), s(x)))
F(s(x)) -> *'(s(x), s(x))

Furthermore, R contains four SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`

Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

-'(s(x), s(y)) -> -'(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
-'(x1, x2) -> -'(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`

Dependency Pair:

+'(s(x), y) -> +'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

+'(s(x), y) -> +'(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 4`
`         ↳Rw`

Dependency Pair:

*'(x, s(y)) -> *'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

*'(x, s(y)) -> *'(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
*'(x1, x2) -> *'(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 7`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Rw`

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rewriting Transformation`

Dependency Pair:

F(s(x)) -> F(-(p(*(s(x), s(x))), *(s(x), s(x))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(x)) -> F(-(p(*(s(x), s(x))), *(s(x), s(x))))
one new Dependency Pair is created:

F(s(x)) -> F(-(p(+(s(x), *(s(x), x))), *(s(x), s(x))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rewriting Transformation`

Dependency Pair:

F(s(x)) -> F(-(p(+(s(x), *(s(x), x))), *(s(x), s(x))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(x)) -> F(-(p(+(s(x), *(s(x), x))), *(s(x), s(x))))
one new Dependency Pair is created:

F(s(x)) -> F(-(p(s(+(x, *(s(x), x)))), *(s(x), s(x))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 9`
`                 ↳Rewriting Transformation`

Dependency Pair:

F(s(x)) -> F(-(p(s(+(x, *(s(x), x)))), *(s(x), s(x))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(x)) -> F(-(p(s(+(x, *(s(x), x)))), *(s(x), s(x))))
one new Dependency Pair is created:

F(s(x)) -> F(-(+(x, *(s(x), x)), *(s(x), s(x))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 10`
`                 ↳Rewriting Transformation`

Dependency Pair:

F(s(x)) -> F(-(+(x, *(s(x), x)), *(s(x), s(x))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(x)) -> F(-(+(x, *(s(x), x)), *(s(x), s(x))))
one new Dependency Pair is created:

F(s(x)) -> F(-(+(x, *(s(x), x)), +(s(x), *(s(x), x))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 11`
`                 ↳Rewriting Transformation`

Dependency Pair:

F(s(x)) -> F(-(+(x, *(s(x), x)), +(s(x), *(s(x), x))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(x)) -> F(-(+(x, *(s(x), x)), +(s(x), *(s(x), x))))
one new Dependency Pair is created:

F(s(x)) -> F(-(+(x, *(s(x), x)), s(+(x, *(s(x), x)))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 12`
`                 ↳Narrowing Transformation`

Dependency Pair:

F(s(x)) -> F(-(+(x, *(s(x), x)), s(+(x, *(s(x), x)))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(x)) -> F(-(+(x, *(s(x), x)), s(+(x, *(s(x), x)))))
eight new Dependency Pairs are created:

F(s(0)) -> F(-(*(s(0), 0), s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(+(s(x''), *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(+(0, 0), s(+(0, *(s(0), 0)))))
F(s(s(y'))) -> F(-(+(s(y'), +(s(s(y')), *(s(s(y')), y'))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 13`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(s(y'), +(s(s(y')), *(s(s(y')), y'))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(+(0, 0), s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(+(s(x''), *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, *(s(0), 0)))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(*(s(0), 0), s(+(0, *(s(0), 0)))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 14`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(s(y'), +(s(s(y')), *(s(s(y')), y'))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(+(0, 0), s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(+(s(x''), *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(+(s(x''), *(s(s(x'')), s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 15`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(s(y'), +(s(s(y')), *(s(s(y')), y'))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(+(0, 0), s(+(0, *(s(0), 0)))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(+(0, 0), s(+(0, *(s(0), 0)))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 16`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(s(y'), +(s(s(y')), *(s(s(y')), y'))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(s(y'), +(s(s(y')), *(s(s(y')), y'))), s(+(s(y'), *(s(s(y')), s(y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 17`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(+(0, *(s(0), 0)), s(*(s(0), 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 18`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(s(x''), *(s(s(x'')), s(x''))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 19`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(+(0, *(s(0), 0)), s(+(0, 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 20`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(s(y'), *(s(s(y')), s(y'))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 21`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(*(s(0), 0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 22`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), +(s(x''), *(s(s(x'')), s(x'')))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 23`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(+(0, *(s(0), 0)))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(*(s(0), 0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 24`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(s(+(y', +(s(s(y')), *(s(s(y')), y')))), s(+(s(y'), *(s(s(y')), s(y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 25`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(*(s(0), 0), s(*(s(0), 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(*(s(0), 0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 26`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(s(+(x'', *(s(s(x'')), s(x'')))), s(s(+(x'', *(s(s(x'')), s(x'')))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 27`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(*(s(0), 0), s(+(0, 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(+(0, 0))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 28`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(s(+(y', *(s(s(y')), s(y')))), s(+(s(y'), +(s(s(y')), *(s(s(y')), y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 29`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(+(0, 0))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(*(s(0), 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 30`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), +(s(x''), *(s(s(x'')), s(x'')))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 31`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(0)))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(*(s(0), 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 32`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), *(s(s(y')), s(y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 33`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(*(s(0), 0))))
F(s(0)) -> F(-(0, s(0)))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(*(s(0), 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 34`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', *(s(s(x'')), s(x''))), s(+(x'', *(s(s(x'')), s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 35`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(+(0, 0))))
F(s(0)) -> F(-(0, s(0)))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(+(0, 0))))
one new Dependency Pair is created:

F(s(0)) -> F(-(0, s(0)))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 36`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', *(s(s(y')), s(y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 37`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), +(s(x''), *(s(s(x'')), s(x'')))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), +(s(x''), *(s(s(x'')), s(x'')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 38`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), *(s(s(y')), s(y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), *(s(s(y')), s(y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 39`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), *(s(s(y')), s(y')))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), +(s(x''), *(s(s(x'')), s(x'')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', +(s(s(x'')), *(s(s(x'')), x''))), s(+(x'', *(s(s(x'')), s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), s(+(x'', *(s(s(x'')), s(x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 40`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), *(s(s(y')), s(y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', +(s(s(y')), *(s(s(y')), y'))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 41`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), *(s(s(y')), s(y')))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), +(s(x''), *(s(s(x'')), s(x'')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), s(+(x'', *(s(s(x'')), s(x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), +(s(x''), *(s(s(x'')), s(x'')))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 42`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), *(s(s(y')), s(y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), *(s(s(y')), s(y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', *(s(s(y')), s(y'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 43`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', *(s(s(y')), s(y'))))))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(+(s(x''), *(s(s(x'')), x'')))), s(+(x'', *(s(s(x'')), s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 44`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', *(s(s(y')), s(y'))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(+(s(y'), *(s(s(y')), y')))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 45`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', *(s(s(y')), s(y'))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 46`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', *(s(s(y')), s(y'))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', *(s(s(y')), s(y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 47`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', *(s(s(x'')), s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 48`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), +(s(y'), +(s(s(y')), *(s(s(y')), y')))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 49`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 50`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 51`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', +(s(s(x'')), *(s(s(x'')), x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 52`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', +(s(s(y')), *(s(s(y')), y'))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 53`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 54`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 55`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(+(s(x''), *(s(s(x'')), x'')))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 56`
`                 ↳Rewriting Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(+(s(y'), *(s(s(y')), y')))))))
one new Dependency Pair is created:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 57`
`                 ↳Narrowing Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(0)))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 58`
`                 ↳Narrowing Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(0)))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 59`
`                 ↳Narrowing Transformation`

Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(0)) -> F(-(0, s(0)))
F(s(0)) -> F(-(0, s(0)))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(0)))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 60`
`                 ↳Narrowing Transformation`

Dependency Pairs:

F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(0)) -> F(-(0, s(0)))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

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

F(s(0)) -> F(-(0, s(0)))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Rw`
`           →DP Problem 8`
`             ↳Rw`
`             ...`
`               →DP Problem 61`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(y'))) -> F(-(+(y', s(s(+(y', *(s(s(y')), y'))))), s(+(y', s(s(+(y', *(s(s(y')), y'))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))
F(s(s(x''))) -> F(-(+(x'', s(s(+(x'', *(s(s(x'')), x''))))), s(+(x'', s(s(+(x'', *(s(s(x'')), x''))))))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
*(x, 0) -> 0
*(x, s(y)) -> +(x, *(x, y))
p(s(x)) -> x
f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x))))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:05 minutes