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))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Termination of R to be shown.

`   R`
`     ↳Overlay and local confluence Check`

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳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(-(*(s(s(0)), s(x)), s(s(x))))
F(s(x)) -> -'(*(s(s(0)), s(x)), s(s(x)))
F(s(x)) -> *'(s(s(0)), s(x))

Furthermore, R contains four SCCs.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Usable Rules (Innermost)`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`

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))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 7 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`             ...`
`               →DP Problem 5`
`                 ↳Size-Change Principle`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. -'(s(x), s(y)) -> -'(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳Usable Rules (Innermost)`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`

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))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 7 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`             ...`
`               →DP Problem 6`
`                 ↳Size-Change Principle`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. +'(s(x), y) -> +'(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳Usable Rules (Innermost)`
`           →DP Problem 4`
`             ↳UsableRules`

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))
f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

Strategy:

innermost

As we are in the innermost case, we can delete all 7 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 7`
`                 ↳Size-Change Principle`
`           →DP Problem 4`
`             ↳UsableRules`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. *'(x, s(y)) -> *'(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳Usable Rules (Innermost)`

Dependency Pair:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 8`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 9`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 10`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 11`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 12`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 13`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 14`
`                 ↳Narrowing Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

F(s(x)) -> F(-(*(s(s(0)), x), x))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 15`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

Rules:

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

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(s(0)), *(s(s(0)), y')), s(y')))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 16`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

Rules:

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

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))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 17`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

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(+(s(0), *(s(s(0)), y'))), s(y')))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 18`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

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(0), *(s(s(0)), y')), y'))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 19`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rules:

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

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(+(0, *(s(s(0)), y'))), y'))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`           →DP Problem 4`
`             ↳UsableRules`
`             ...`
`               →DP Problem 20`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

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

Rules:

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

Strategy:

innermost

Termination of R could not be shown.
Duration:
0:27 minutes