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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`

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

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(LE(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`

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

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(APP(x1, x2)) =  x1 + x2 POL(add(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 7`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`

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

The following dependency pair can be strictly oriented:

The following usable rules for innermost can be oriented:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(LOW(x1, x2)) =  1 + x1 + x2 POL(false) =  0 POL(IF_LOW(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(le) =  0 POL(add(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
LOW(x1, x2) -> LOW(x1, x2)
IFLOW(x1, x2, x3) -> IFLOW(x1, x2, x3)
le(x1, x2) -> le

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 8`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`

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

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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 5`
`         ↳AFS`

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

The following dependency pairs can be strictly oriented:

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

The following usable rules for innermost can be oriented:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(IF_HIGH(x1, x2, x3)) =  x1 + x2 + x3 POL(false) =  0 POL(HIGH(x1, x2)) =  x1 + x2 POL(true) =  0 POL(le) =  0 POL(add(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
IFHIGH(x1, x2, x3) -> IFHIGH(x1, x2, x3)
HIGH(x1, x2) -> HIGH(x1, x2)
le(x1, x2) -> le

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`           →DP Problem 9`
`             ↳Dependency Graph`
`       →DP Problem 5`
`         ↳AFS`

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳Argument Filtering and 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:

The following usable rules for innermost can be oriented:

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

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

resulting in one new DP problem.
Used Argument Filtering System:
QUICKSORT(x1) -> QUICKSORT(x1)
high(x1, x2) -> high(x1, x2)
low(x1, x2) -> low(x1, x2)
ifhigh(x1, x2, x3) -> ifhigh(x1, x2, x3)
le(x1, x2) -> le
iflow(x1, x2, x3) -> iflow(x1, x2, x3)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`
`           →DP Problem 10`
`             ↳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.

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