Term Rewriting System R:
[x, y]
fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FAC(0) -> 1'
FAC(s(x)) -> *'(s(x), fac(x))
FAC(s(x)) -> FAC(x)
FLOOP(s(x), y) -> FLOOP(x, *(s(x), y))
FLOOP(s(x), y) -> *'(s(x), y)
*'(x, s(y)) -> +'(*(x, y), x)
*'(x, s(y)) -> *'(x, y)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains four SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 5`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 5`
`             ↳FwdInst`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(+'(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 5`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 8`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 8`
`             ↳FwdInst`
`             ...`
`               →DP Problem 9`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  1 + x2 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 8`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

FAC(s(x)) -> FAC(x)

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

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

FAC(s(x)) -> FAC(x)
one new Dependency Pair is created:

FAC(s(s(x''))) -> FAC(s(x''))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 11`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

FAC(s(s(x''))) -> FAC(s(x''))

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

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

FAC(s(s(x''))) -> FAC(s(x''))
one new Dependency Pair is created:

FAC(s(s(s(x'''')))) -> FAC(s(s(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 11`
`             ↳FwdInst`
`             ...`
`               →DP Problem 12`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

FAC(s(s(s(x'''')))) -> FAC(s(s(x'''')))

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

FAC(s(s(s(x'''')))) -> FAC(s(s(x'''')))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(FAC(x1)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`           →DP Problem 11`
`             ↳FwdInst`
`             ...`
`               →DP Problem 13`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Polo`

Dependency Pair:

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(*(x1, x2)) =  0 POL(s(x1)) =  1 + x1 POL(FLOOP(x1, x2)) =  x1 POL(+(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳FwdInst`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 14`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

fac(0) -> 1
fac(s(x)) -> *(s(x), fac(x))
fac(0) -> s(0)
floop(0, y) -> y
floop(s(x), y) -> floop(x, *(s(x), y))
*(x, 0) -> 0
*(x, s(y)) -> +(*(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
1 -> s(0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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