Term Rewriting System R:
[x, y, n, m]
minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

MINUS(s(x), s(y)) -> MINUS(x, y)
QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
QUOT(s(x), s(y)) -> MINUS(x, y)
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 seven 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
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
quot > s
{true, le} > false

resulting in one new DP problem.
Used Argument Filtering System:
MINUS(x1, x2) -> MINUS(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
quot(x1, x2) -> quot(x1, x2)
le(x1, x2) -> le(x1, x2)
app(x1, x2) -> app(x1, x2)
low(x1, x2) -> x2
iflow(x1, x2, x3) -> x3
high(x1, x2) -> x2
ifhigh(x1, x2, x3) -> x3
quicksort(x1) -> quicksort(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 8
Dependency Graph
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
quot > {false, s}
{true, le}

resulting in one new DP problem.
Used Argument Filtering System:
LE(x1, x2) -> LE(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
quot(x1, x2) -> quot(x1, x2)
le(x1, x2) -> le(x1, x2)
app(x1, x2) -> app(x1, x2)
low(x1, x2) -> x2
iflow(x1, x2, x3) -> x3
high(x1, x2) -> x2
ifhigh(x1, x2, x3) -> x3
quicksort(x1) -> quicksort(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 9
Dependency Graph
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
high > {0, false}
quot > {0, false}
APP > {0, false}
low > {0, false}
nil > {0, false}
iflow > {0, false}
quicksort > {add, app} > {0, false}
minus > {0, false}
ifhigh > {0, false}
s > {0, false}
le > true > {0, false}

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
minus(x1, x2) -> minus(x1, x2)
s(x1) -> x1
quot(x1, x2) -> x2
le(x1, x2) -> le(x1, x2)
app(x1, x2) -> app(x1, x2)
low(x1, x2) -> x2
iflow(x1, x2, x3) -> x3
high(x1, x2) -> x2
ifhigh(x1, x2, x3) -> x3
quicksort(x1) -> quicksort(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 10
Dependency Graph
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

The following dependency pair can be strictly oriented:

QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

The following rules can be oriented:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
quot > s > false
{true, le}

resulting in one new DP problem.
Used Argument Filtering System:
QUOT(x1, x2) -> QUOT(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
quot(x1, x2) -> quot(x1, x2)
le(x1, x2) -> le(x1, x2)
app(x1, x2) -> app(x1, x2)
low(x1, x2) -> x2
iflow(x1, x2, x3) -> x3
high(x1, x2) -> x2
ifhigh(x1, x2, x3) -> x3
quicksort(x1) -> quicksort(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 11
Dependency Graph
→DP Problem 5
AFS
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pairs:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

The following dependency pairs can be strictly oriented:

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

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
high > false
quot > s > false
0 > true > false
{LOW, IFLOW} > false
low > false
nil > false
iflow > false
quicksort > app > add > false
minus > false
ifhigh > false
le > false

resulting in one new DP problem.
Used Argument Filtering System:
LOW(x1, x2) -> LOW(x1, x2)
IFLOW(x1, x2, x3) -> IFLOW(x2, x3)
le(x1, x2) -> le(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
quot(x1, x2) -> quot(x1, x2)
app(x1, x2) -> app(x1, x2)
low(x1, x2) -> x2
iflow(x1, x2, x3) -> x3
high(x1, x2) -> x2
ifhigh(x1, x2, x3) -> x3
quicksort(x1) -> quicksort(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 12
Dependency Graph
→DP Problem 6
AFS
→DP Problem 7
AFS

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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
AFS
→DP Problem 6
Argument Filtering and Ordering
→DP Problem 7
AFS

Dependency Pairs:

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

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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 rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
quot > s
0 > true
{false, le}

resulting in one new DP problem.
Used Argument Filtering System:
IFHIGH(x1, x2, x3) -> x3
HIGH(x1, x2) -> x2
le(x1, x2) -> le(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
quot(x1, x2) -> quot(x1, x2)
app(x1, x2) -> app(x1, x2)
low(x1, x2) -> x2
iflow(x1, x2, x3) -> x3
high(x1, x2) -> x2
ifhigh(x1, x2, x3) -> x3
quicksort(x1) -> quicksort(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 6
AFS
→DP Problem 13
Dependency Graph
→DP Problem 7
AFS

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

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
AFS
→DP Problem 6
AFS
→DP Problem 7
Argument Filtering and Ordering

Dependency Pairs:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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

The following dependency pairs can be strictly oriented:

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
quot > s
{0, false}
le > true

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

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 3
AFS
→DP Problem 4
AFS
→DP Problem 5
AFS
→DP Problem 6
AFS
→DP Problem 7
AFS
→DP Problem 14
Dependency Graph

Dependency Pair:

Rules:

minus(x, 0) -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0, s(y)) -> 0
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)