Term Rewriting System R:
[y, x, n, m]
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

LE(s(x), s(y)) -> LE(x, y)
APP(add(n, x), y) -> APP(x, y)
LOW(n, add(m, x)) -> LE(m, n)
IFLOW(true, n, add(m, x)) -> LOW(n, x)
IFLOW(false, n, add(m, x)) -> LOW(n, x)
HIGH(n, add(m, x)) -> LE(m, n)
IFHIGH(true, n, add(m, x)) -> HIGH(n, x)
IFHIGH(false, n, add(m, x)) -> HIGH(n, x)

Furthermore, R contains five SCCs.

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

Dependency Pair:

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

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

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

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

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

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

LE(s(s(s(x''''))), s(s(s(y'''')))) -> LE(s(s(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(LE(x1, x2)) =  1 + x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

APP(add(n, x), y) -> APP(x, y)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

APP(add(n, x), y) -> APP(x, y)
one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 9`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 9`
`             ↳FwdInst`
`             ...`
`               →DP Problem 10`
`                 ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(APP(x1, x2)) =  1 + x1 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 9`
`             ↳FwdInst`
`             ...`
`               →DP Problem 11`
`                 ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Narrowing Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, n, add(m, x)) -> LOW(n, x)
IFLOW(true, n, add(m, x)) -> LOW(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Narrowing Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(true, n, add(m, x)) -> LOW(n, x)
IFLOW(false, n, add(m, x)) -> LOW(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 13`
`                 ↳Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(true, n, add(m, x)) -> LOW(n, x)
IFLOW(false, n, add(m, x)) -> LOW(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(true, n, add(m, x)) -> LOW(n, x)
three new Dependency Pairs are created:

IFLOW(true, n', add(0, x'')) -> LOW(n', x'')
IFLOW(true, s(y''''), add(s(0), x'')) -> LOW(s(y''''), x'')
IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 14`
`                 ↳Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
IFLOW(true, s(y''''), add(s(0), x'')) -> LOW(s(y''''), x'')
IFLOW(true, n', add(0, x'')) -> LOW(n', x'')
IFLOW(false, n, add(m, x)) -> LOW(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(false, n, add(m, x)) -> LOW(n, x)
three new Dependency Pairs are created:

IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 15`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(true, s(y''''), add(s(0), x'')) -> LOW(s(y''''), x'')
IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
IFLOW(true, n', add(0, x'')) -> LOW(n', x'')
IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(true, n', add(0, x'')) -> LOW(n', x'')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 16`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(true, s(y''''), add(s(0), x'')) -> LOW(s(y''''), x'')
IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 17`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(true, s(y''''), add(s(0), x'')) -> LOW(s(y''''), x'')
IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(true, s(y''''), add(s(0), x'')) -> LOW(s(y''''), x'')
seven new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 18`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(true, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 19`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(false, 0, add(s(x''''), x'')) -> LOW(0, x'')
three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 20`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 21`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(false, s(0), add(s(s(x''''')), x'')) -> LOW(s(0), x'')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 22`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFLOW(false, s(s(y'''')), add(s(s(x''''')), x')) -> LOW(s(s(y'''')), x')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 23`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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) =  1 POL(LOW(x1, x2)) =  x2 POL(false) =  0 POL(IF_LOW(x1, x2, x3)) =  x3 POL(true) =  0 POL(s(x1)) =  0 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 24`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 25`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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) =  1 POL(LOW(x1, x2)) =  x2 POL(false) =  0 POL(IF_LOW(x1, x2, x3)) =  x3 POL(true) =  0 POL(s(x1)) =  x1 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 27`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 29`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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(LOW(x1, x2)) =  x2 POL(false) =  0 POL(IF_LOW(x1, x2, x3)) =  x3 POL(true) =  0 POL(s(x1)) =  0 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 31`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 30`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(LOW(x1, x2)) =  x2 POL(false) =  0 POL(IF_LOW(x1, x2, x3)) =  x3 POL(s(x1)) =  1 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 32`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 26`
`                 ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(LOW(x1, x2)) =  x2 POL(false) =  0 POL(IF_LOW(x1, x2, x3)) =  x3 POL(s(x1)) =  1 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 12`
`             ↳Nar`
`             ...`
`               →DP Problem 28`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Narrowing Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, n, add(m, x)) -> HIGH(n, x)
IFHIGH(true, n, add(m, x)) -> HIGH(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Narrowing Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(true, n, add(m, x)) -> HIGH(n, x)
IFHIGH(false, n, add(m, x)) -> HIGH(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 34`
`                 ↳Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(true, n, add(m, x)) -> HIGH(n, x)
IFHIGH(false, n, add(m, x)) -> HIGH(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(true, n, add(m, x)) -> HIGH(n, x)
three new Dependency Pairs are created:

IFHIGH(true, n', add(0, x'')) -> HIGH(n', x'')
IFHIGH(true, s(y''''), add(s(0), x'')) -> HIGH(s(y''''), x'')
IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 35`
`                 ↳Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
IFHIGH(true, s(y''''), add(s(0), x'')) -> HIGH(s(y''''), x'')
IFHIGH(true, n', add(0, x'')) -> HIGH(n', x'')
IFHIGH(false, n, add(m, x)) -> HIGH(n, x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(false, n, add(m, x)) -> HIGH(n, x)
three new Dependency Pairs are created:

IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 36`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(true, s(y''''), add(s(0), x'')) -> HIGH(s(y''''), x'')
IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
IFHIGH(true, n', add(0, x'')) -> HIGH(n', x'')
IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(true, n', add(0, x'')) -> HIGH(n', x'')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 37`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(true, s(y''''), add(s(0), x'')) -> HIGH(s(y''''), x'')
IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 38`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(true, s(y''''), add(s(0), x'')) -> HIGH(s(y''''), x'')
IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(true, s(y''''), add(s(0), x'')) -> HIGH(s(y''''), x'')
seven new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 39`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(true, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 40`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(false, 0, add(s(x''''), x'')) -> HIGH(0, x'')
three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 41`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 42`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(false, s(0), add(s(s(x''''')), x'')) -> HIGH(s(0), x'')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 43`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

IFHIGH(false, s(s(y'''')), add(s(s(x''''')), x')) -> HIGH(s(s(y'''')), x')
five new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 44`
`                 ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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) =  1 POL(false) =  0 POL(IF_HIGH(x1, x2, x3)) =  x3 POL(HIGH(x1, x2)) =  x2 POL(true) =  0 POL(s(x1)) =  0 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 45`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 46`
`                 ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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) =  1 POL(false) =  0 POL(IF_HIGH(x1, x2, x3)) =  x3 POL(HIGH(x1, x2)) =  x2 POL(true) =  0 POL(s(x1)) =  x1 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 48`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 50`
`                 ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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(false) =  0 POL(IF_HIGH(x1, x2, x3)) =  x3 POL(HIGH(x1, x2)) =  x2 POL(true) =  0 POL(s(x1)) =  0 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 52`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 51`
`                 ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(IF_HIGH(x1, x2, x3)) =  x3 POL(false) =  0 POL(HIGH(x1, x2)) =  x2 POL(s(x1)) =  1 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 53`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 47`
`                 ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(IF_HIGH(x1, x2, x3)) =  x3 POL(false) =  0 POL(HIGH(x1, x2)) =  x2 POL(s(x1)) =  1 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 33`
`             ↳Nar`
`             ...`
`               →DP Problem 49`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Nar`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Narrowing Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

two new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Narrowing Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

two new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 55`
`                 ↳Narrowing Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 56`
`                 ↳Rewriting Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 57`
`                 ↳Rewriting Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 58`
`                 ↳Narrowing Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

three new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 59`
`                 ↳Rewriting Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 60`
`                 ↳Rewriting Transformation`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

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

one new Dependency Pair is created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 61`
`                 ↳Polynomial Ordering`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

ifhigh(true, n, add(m, x)) -> high(n, x)
high(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
low(n, nil) -> nil

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(QUICKSORT(x1)) =  1 + x1 POL(0) =  1 POL(if_low(x1, x2, x3)) =  x3 POL(false) =  0 POL(if_high(x1, x2, x3)) =  x3 POL(high(x1, x2)) =  x2 POL(low(x1, x2)) =  x2 POL(true) =  0 POL(nil) =  0 POL(s(x1)) =  0 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  x1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 62`
`                 ↳Dependency Graph`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 63`
`                 ↳Polynomial Ordering`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

iflow(false, n, add(m, x)) -> low(n, x)
low(n, nil) -> nil

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(QUICKSORT(x1)) =  1 + x1 POL(0) =  0 POL(if_low(x1, x2, x3)) =  x3 POL(false) =  0 POL(low(x1, x2)) =  x2 POL(true) =  0 POL(nil) =  0 POL(s(x1)) =  0 POL(le(x1, x2)) =  0 POL(add(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 65`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

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`
`         ↳Nar`
`       →DP Problem 4`
`         ↳Nar`
`       →DP Problem 5`
`         ↳Nar`
`           →DP Problem 54`
`             ↳Nar`
`             ...`
`               →DP Problem 64`
`                 ↳Polynomial Ordering`

Dependency Pairs:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
low(n, nil) -> nil
iflow(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
ifhigh(true, n, add(m, x)) -> high(n, x)
quicksort(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

iflow(false, n, add(m, x)) -> low(n, x)
low(n, nil) -> nil
high(n, nil) -> nil