Termination Proof using AProVE

Term Rewriting System R:
[Y, X, N, L, M]
le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

LE(s(X), s(Y)) -> LE(X, Y)
APP(cons(N, L), Y) -> APP(L, Y)
LOW(N, cons(M, L)) -> IFLOW(le(M, N), N, cons(M, L))
LOW(N, cons(M, L)) -> LE(M, N)
IFLOW(true, N, cons(M, L)) -> LOW(N, L)
IFLOW(false, N, cons(M, L)) -> LOW(N, L)
HIGH(N, cons(M, L)) -> IFHIGH(le(M, N), N, cons(M, L))
HIGH(N, cons(M, L)) -> LE(M, N)
IFHIGH(true, N, cons(M, L)) -> HIGH(N, L)
IFHIGH(false, N, cons(M, L)) -> HIGH(N, L)
QUICKSORT(cons(N, L)) -> APP(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))
QUICKSORT(cons(N, L)) -> QUICKSORT(low(N, L))
QUICKSORT(cons(N, L)) -> LOW(N, L)
QUICKSORT(cons(N, L)) -> QUICKSORT(high(N, L))
QUICKSORT(cons(N, L)) -> HIGH(N, L)

Furthermore, R contains five SCCs.

     ↳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
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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:

     ↳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
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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:

     ↳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
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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.

     ↳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
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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(cons(N, L), Y) -> APP(L, Y)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

APP(cons(N, L), Y) -> APP(L, Y)

one new Dependency Pair is created:

APP(cons(N, cons(N'', L'')), Y'') -> APP(cons(N'', L''), Y'')

The transformation is resulting in one new DP problem:

     ↳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:

APP(cons(N, cons(N'', L'')), Y'') -> APP(cons(N'', L''), Y'')

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

APP(cons(N, cons(N'', L'')), Y'') -> APP(cons(N'', L''), Y'')

one new Dependency Pair is created:

APP(cons(N, cons(N'''', cons(N''''', L''''))), Y'''') -> APP(cons(N'''', cons(N''''', L'''')), Y'''')

The transformation is resulting in one new DP problem:

     ↳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:

APP(cons(N, cons(N'''', cons(N''''', L''''))), Y'''') -> APP(cons(N'''', cons(N''''', L'''')), Y'''')

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(cons(N, cons(N'''', cons(N''''', L''''))), Y'''') -> APP(cons(N'''', cons(N''''', L'''')), 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(cons(x₁, x₂)) = 1 + x₂

POL(APP(x₁, x₂)) = 1 + x₁

resulting in one new DP problem.

     ↳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
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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, cons(M, L)) -> LOW(N, L)
IFLOW(true, N, cons(M, L)) -> LOW(N, L)
LOW(N, cons(M, L)) -> IFLOW(le(M, N), N, cons(M, L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

LOW(N, cons(M, L)) -> IFLOW(le(M, N), N, cons(M, L))

three new Dependency Pairs are created:

LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
LOW(s(Y'), cons(s(X'), L)) -> IFLOW(le(X', Y'), s(Y'), cons(s(X'), L))

The transformation is resulting in one new DP problem:

     ↳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:

LOW(s(Y'), cons(s(X'), L)) -> IFLOW(le(X', Y'), s(Y'), cons(s(X'), L))
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, N, cons(M, L)) -> LOW(N, L)
LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))
IFLOW(false, N, cons(M, L)) -> LOW(N, L)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

LOW(s(Y'), cons(s(X'), L)) -> IFLOW(le(X', Y'), s(Y'), cons(s(X'), L))

three new Dependency Pairs are created:

LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))

The transformation is resulting in one new DP problem:

     ↳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:

LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, N, cons(M, L)) -> LOW(N, L)
LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))
IFLOW(false, N, cons(M, L)) -> LOW(N, L)
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFLOW(true, N, cons(M, L)) -> LOW(N, L)

three new Dependency Pairs are created:

IFLOW(true, N', cons(0, L'')) -> LOW(N', L'')
IFLOW(true, s(Y''''), cons(s(0), L'')) -> LOW(s(Y''''), L'')
IFLOW(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(Y''''), cons(s(0), L'')) -> LOW(s(Y''''), L'')
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, N', cons(0, L'')) -> LOW(N', L'')
LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))
IFLOW(false, N, cons(M, L)) -> LOW(N, L)
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFLOW(false, N, cons(M, L)) -> LOW(N, L)

three new Dependency Pairs are created:

IFLOW(false, 0, cons(s(X'''), L'')) -> LOW(0, L'')
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
IFLOW(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(Y''''), cons(s(0), L'')) -> LOW(s(Y''''), L'')
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(false, 0, cons(s(X'''), L'')) -> LOW(0, L'')
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, N', cons(0, L'')) -> LOW(N', L'')
LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))
IFLOW(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFLOW(true, N', cons(0, L'')) -> LOW(N', L'')

five new Dependency Pairs are created:

IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(Y''''), cons(s(0), L'')) -> LOW(s(Y''''), L'')
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
IFLOW(false, 0, cons(s(X'''), L'')) -> LOW(0, L'')
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))
IFLOW(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

LOW(N', cons(0, L)) -> IFLOW(true, N', cons(0, L))

five new Dependency Pairs are created:

LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(false, 0, cons(s(X'''), L'')) -> LOW(0, L'')
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(Y''''), cons(s(0), L'')) -> LOW(s(Y''''), L'')
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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''''), cons(s(0), L'')) -> LOW(s(Y''''), L'')

seven new Dependency Pairs are created:

IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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:

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')

five new Dependency Pairs are created:

IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(false, 0, cons(s(X'''), L'')) -> LOW(0, L'')
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFLOW(false, 0, cons(s(X'''), L'')) -> LOW(0, L'')

three new Dependency Pairs are created:

IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> LOW(0, cons(0, cons(0, L''''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> LOW(0, cons(0, cons(s(X'''''''), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> LOW(0, cons(0, cons(s(X'''''''), L''''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> LOW(0, cons(0, cons(0, L''''''')))
IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

LOW(0, cons(s(X'), L)) -> IFLOW(false, 0, cons(s(X'), L))

three new Dependency Pairs are created:

LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
LOW(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
LOW(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> LOW(0, cons(0, cons(s(X'''''''), L''''''')))
LOW(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> LOW(0, cons(0, cons(0, L''''''')))
LOW(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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), cons(s(s(X'''')), L'')) -> LOW(s(0), L'')

five new Dependency Pairs are created:

IFLOW(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(0), cons(s(0), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(0), cons(0, cons(0, L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(0), cons(0, cons(s(0), L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(0), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> LOW(0, cons(0, cons(s(X'''''''), L''''''')))
LOW(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> LOW(0, cons(0, cons(0, L''''''')))
LOW(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(0), cons(0, cons(0, L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(0), cons(s(0), L'''))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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'''')), cons(s(s(X'''')), L'')) -> LOW(s(s(Y'''')), L'')

five new Dependency Pairs are created:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(0), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> LOW(0, cons(0, cons(s(X'''''''), L''''''')))
LOW(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> LOW(0, cons(0, cons(0, L''''''')))
LOW(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(0), cons(0, cons(0, L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(0), cons(s(0), L'''))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IFLOW(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y''''), cons(0, cons(s(0), L'''))) -> LOW(s(Y''''), cons(s(0), L'''))
IFLOW(true, 0, cons(0, cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
IFLOW(true, N''', cons(0, cons(0, L'''))) -> LOW(N''', cons(0, L'''))
IFLOW(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> LOW(s(s(Y'''')), cons(s(s(X'''')), L'''))

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(cons(x₁, x₂)) = x₁ + x₂

POL(true) = 0

POL(s(x₁)) = 0

POL(le(x₁, x₂)) = 0

POL(IFLOW(x₁, x₂, x₃)) = x₃

resulting in one new DP problem.

     ↳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:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> LOW(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFLOW(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> LOW(s(Y'''''), cons(0, cons(0, L''''''')))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> LOW(s(0), cons(0, cons(s(0), L''''''')))
LOW(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
LOW(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFLOW(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> LOW(0, cons(0, cons(s(X'''''''), L''''''')))
LOW(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> LOW(0, cons(0, cons(0, L''''''')))
LOW(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFLOW(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))
LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
LOW(0, cons(0, cons(s(X'''''), L'''''))) -> IFLOW(true, 0, cons(0, cons(s(X'''''), L''''')))
LOW(N'', cons(0, cons(0, L'''''))) -> IFLOW(true, N'', cons(0, cons(0, L''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> LOW(s(0), cons(0, cons(0, L''''''')))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(0), cons(s(0), L'''))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
LOW(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFLOW(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> LOW(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

     ↳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:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(0), cons(s(0), L'''))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IFLOW(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> LOW(s(Y'''''), cons(s(0), L'''))

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(cons(x₁, x₂)) = x₁ + x₂

POL(true) = 0

POL(s(x₁)) = x₁

POL(le(x₁, x₂)) = 0

POL(IFLOW(x₁, x₂, x₃)) = x₃

resulting in one new DP problem.

     ↳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:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(0), cons(s(0), L'''))
LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
LOW(s(Y''), cons(s(0), L)) -> IFLOW(true, s(Y''), cons(s(0), L))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> LOW(s(s(Y''''')), cons(s(0), L'''))
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

     ↳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:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IFLOW(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFLOW(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(s(Y''''')), cons(s(s(X'''')), L'''))

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(cons(x₁, x₂)) = 1 + x₂

POL(true) = 0

POL(s(x₁)) = 0

POL(le(x₁, x₂)) = 0

POL(IFLOW(x₁, x₂, x₃)) = x₃

resulting in one new DP problem.

     ↳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:

LOW(s(s(Y'')), cons(s(s(X'')), L)) -> IFLOW(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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:

LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))
IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFLOW(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> LOW(s(0), cons(s(s(X'''')), L'''))

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(cons(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(s(x₁)) = 1

POL(IFLOW(x₁, x₂, x₃)) = x₃

resulting in one new DP problem.

     ↳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:

LOW(s(0), cons(s(s(X'')), L)) -> IFLOW(false, s(0), cons(s(s(X'')), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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:

LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFLOW(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> LOW(0, cons(s(X'''), L'''))

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(cons(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(s(x₁)) = 1

POL(IFLOW(x₁, x₂, x₃)) = x₃

resulting in one new DP problem.

     ↳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:

LOW(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFLOW(false, 0, cons(s(X''), cons(s(X''''''), L''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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, cons(M, L)) -> HIGH(N, L)
IFHIGH(true, N, cons(M, L)) -> HIGH(N, L)
HIGH(N, cons(M, L)) -> IFHIGH(le(M, N), N, cons(M, L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

HIGH(N, cons(M, L)) -> IFHIGH(le(M, N), N, cons(M, L))

three new Dependency Pairs are created:

HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
HIGH(s(Y'), cons(s(X'), L)) -> IFHIGH(le(X', Y'), s(Y'), cons(s(X'), L))

The transformation is resulting in one new DP problem:

     ↳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:

HIGH(s(Y'), cons(s(X'), L)) -> IFHIGH(le(X', Y'), s(Y'), cons(s(X'), L))
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, N, cons(M, L)) -> HIGH(N, L)
HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))
IFHIGH(false, N, cons(M, L)) -> HIGH(N, L)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

HIGH(s(Y'), cons(s(X'), L)) -> IFHIGH(le(X', Y'), s(Y'), cons(s(X'), L))

three new Dependency Pairs are created:

HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))

The transformation is resulting in one new DP problem:

     ↳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:

HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, N, cons(M, L)) -> HIGH(N, L)
HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))
IFHIGH(false, N, cons(M, L)) -> HIGH(N, L)
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFHIGH(true, N, cons(M, L)) -> HIGH(N, L)

three new Dependency Pairs are created:

IFHIGH(true, N', cons(0, L'')) -> HIGH(N', L'')
IFHIGH(true, s(Y''''), cons(s(0), L'')) -> HIGH(s(Y''''), L'')
IFHIGH(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(Y''''), cons(s(0), L'')) -> HIGH(s(Y''''), L'')
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, N', cons(0, L'')) -> HIGH(N', L'')
HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))
IFHIGH(false, N, cons(M, L)) -> HIGH(N, L)
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFHIGH(false, N, cons(M, L)) -> HIGH(N, L)

three new Dependency Pairs are created:

IFHIGH(false, 0, cons(s(X'''), L'')) -> HIGH(0, L'')
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
IFHIGH(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(Y''''), cons(s(0), L'')) -> HIGH(s(Y''''), L'')
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(false, 0, cons(s(X'''), L'')) -> HIGH(0, L'')
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, N', cons(0, L'')) -> HIGH(N', L'')
HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))
IFHIGH(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFHIGH(true, N', cons(0, L'')) -> HIGH(N', L'')

five new Dependency Pairs are created:

IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(Y''''), cons(s(0), L'')) -> HIGH(s(Y''''), L'')
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
IFHIGH(false, 0, cons(s(X'''), L'')) -> HIGH(0, L'')
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))
IFHIGH(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

HIGH(N', cons(0, L)) -> IFHIGH(true, N', cons(0, L))

five new Dependency Pairs are created:

HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))

The transformation is resulting in one new DP problem:

     ↳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'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(false, 0, cons(s(X'''), L'')) -> HIGH(0, L'')
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(Y''''), cons(s(0), L'')) -> HIGH(s(Y''''), L'')
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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''''), cons(s(0), L'')) -> HIGH(s(Y''''), L'')

seven new Dependency Pairs are created:

IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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:

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')

five new Dependency Pairs are created:

IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(false, 0, cons(s(X'''), L'')) -> HIGH(0, L'')
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

IFHIGH(false, 0, cons(s(X'''), L'')) -> HIGH(0, L'')

three new Dependency Pairs are created:

IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> HIGH(0, cons(0, cons(0, L''''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> HIGH(0, cons(0, cons(s(X'''''''), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> HIGH(0, cons(0, cons(s(X'''''''), L''''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> HIGH(0, cons(0, cons(0, L''''''')))
IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

HIGH(0, cons(s(X'), L)) -> IFHIGH(false, 0, cons(s(X'), L))

three new Dependency Pairs are created:

HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
HIGH(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
HIGH(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> HIGH(0, cons(0, cons(s(X'''''''), L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> HIGH(0, cons(0, cons(0, L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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), cons(s(s(X'''')), L'')) -> HIGH(s(0), L'')

five new Dependency Pairs are created:

IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(0), cons(s(0), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(0), cons(0, cons(0, L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(0), cons(0, cons(s(0), L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(0), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> HIGH(0, cons(0, cons(s(X'''''''), L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> HIGH(0, cons(0, cons(0, L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(0), cons(0, cons(0, L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(0), cons(s(0), L'''))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(false, s(s(Y'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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'''')), cons(s(s(X'''')), L'')) -> HIGH(s(s(Y'''')), L'')

five new Dependency Pairs are created:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

The transformation is resulting in one new DP problem:

     ↳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:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(0), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> HIGH(0, cons(0, cons(s(X'''''''), L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> HIGH(0, cons(0, cons(0, L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(0), cons(0, cons(0, L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(0), cons(s(0), L'''))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IFHIGH(true, s(0), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y''''), cons(0, cons(s(0), L'''))) -> HIGH(s(Y''''), cons(s(0), L'''))
IFHIGH(true, 0, cons(0, cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
IFHIGH(true, N''', cons(0, cons(0, L'''))) -> HIGH(N''', cons(0, L'''))
IFHIGH(true, s(s(Y'''')), cons(0, cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y'''')), cons(s(s(X'''')), L'''))

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(IFHIGH(x₁, x₂, x₃)) = x₃

POL(false) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(HIGH(x₁, x₂)) = x₂

POL(true) = 0

POL(s(x₁)) = 0

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳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:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y'''''''')), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y'''''''')), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(0), cons(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(s(0), L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(s(0), L''''''')))
IFHIGH(true, s(Y'''''), cons(s(0), cons(0, cons(0, L''''''')))) -> HIGH(s(Y'''''), cons(0, cons(0, L''''''')))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(0), cons(0, cons(s(s(X'''''''')), L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(s(0), L''''''')))) -> HIGH(s(0), cons(0, cons(s(0), L''''''')))
HIGH(s(0), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(0), cons(0, cons(s(s(X'''''')), L''''')))
HIGH(s(Y''''''), cons(0, cons(s(0), L'''''))) -> IFHIGH(true, s(Y''''''), cons(0, cons(s(0), L''''')))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(s(X'''''''), L''''''')))) -> HIGH(0, cons(0, cons(s(X'''''''), L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(s(X'''''''''), L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(s(X'''''''''), L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(0, cons(0, L''''''')))) -> HIGH(0, cons(0, cons(0, L''''''')))
HIGH(0, cons(s(X''), cons(0, cons(0, L''''''''')))) -> IFHIGH(false, 0, cons(s(X''), cons(0, cons(0, L'''''''''))))
IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))
HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
HIGH(0, cons(0, cons(s(X'''''), L'''''))) -> IFHIGH(true, 0, cons(0, cons(s(X'''''), L''''')))
HIGH(N'', cons(0, cons(0, L'''''))) -> IFHIGH(true, N'', cons(0, cons(0, L''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(0, cons(0, L''''''')))) -> HIGH(s(0), cons(0, cons(0, L''''''')))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(0), cons(s(0), L'''))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
HIGH(s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L'''''))) -> IFHIGH(true, s(s(Y'''''')), cons(0, cons(s(s(X'''''')), L''''')))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(0, cons(s(s(X'''''''')), L''''''')))) -> HIGH(s(s(Y''''')), cons(0, cons(s(s(X'''''''')), L''''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

     ↳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:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(0), cons(s(0), L'''))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IFHIGH(true, s(s(Y''''')), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(0), cons(s(0), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(true, s(Y'''''), cons(s(0), cons(s(0), L'''))) -> HIGH(s(Y'''''), cons(s(0), L'''))

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(IFHIGH(x₁, x₂, x₃)) = x₃

POL(false) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(HIGH(x₁, x₂)) = x₂

POL(true) = 0

POL(s(x₁)) = x₁

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳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:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(0), cons(s(0), L'''))
HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
HIGH(s(Y''), cons(s(0), L)) -> IFHIGH(true, s(Y''), cons(s(0), L))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(0), L'''))) -> HIGH(s(s(Y''''')), cons(s(0), L'''))
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

     ↳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:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IFHIGH(false, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))
IFHIGH(true, s(s(Y''''')), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(s(Y''''')), cons(s(s(X'''')), L'''))

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(IFHIGH(x₁, x₂, x₃)) = x₃

POL(false) = 0

POL(cons(x₁, x₂)) = 1 + x₂

POL(HIGH(x₁, x₂)) = x₂

POL(true) = 0

POL(s(x₁)) = 0

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳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:

HIGH(s(s(Y'')), cons(s(s(X'')), L)) -> IFHIGH(le(X'', Y''), s(s(Y'')), cons(s(s(X'')), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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:

HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))
IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFHIGH(false, s(0), cons(s(s(X'''')), cons(s(s(X'''')), L'''))) -> HIGH(s(0), cons(s(s(X'''')), L'''))

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(IFHIGH(x₁, x₂, x₃)) = x₃

POL(cons(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(HIGH(x₁, x₂)) = x₂

POL(s(x₁)) = 1

resulting in one new DP problem.

     ↳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:

HIGH(s(0), cons(s(s(X'')), L)) -> IFHIGH(false, s(0), cons(s(s(X'')), L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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:

HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))
IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFHIGH(false, 0, cons(s(X'''), cons(s(X'''), L'''))) -> HIGH(0, cons(s(X'''), L'''))

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(IFHIGH(x₁, x₂, x₃)) = x₃

POL(cons(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(HIGH(x₁, x₂)) = x₂

POL(s(x₁)) = 1

resulting in one new DP problem.

     ↳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:

HIGH(0, cons(s(X''), cons(s(X''''''), L'''''))) -> IFHIGH(false, 0, cons(s(X''), cons(s(X''''''), L''''')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Nar

       →DP Problem 5

         ↳Narrowing Transformation

Dependency Pairs:

QUICKSORT(cons(N, L)) -> QUICKSORT(high(N, L))
QUICKSORT(cons(N, L)) -> QUICKSORT(low(N, L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(N, L)) -> QUICKSORT(low(N, L))

two new Dependency Pairs are created:

QUICKSORT(cons(N'', nil)) -> QUICKSORT(nil)
QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(iflow(le(M', N''), N'', cons(M', L'')))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(iflow(le(M', N''), N'', cons(M', L'')))
QUICKSORT(cons(N, L)) -> QUICKSORT(high(N, L))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(N, L)) -> QUICKSORT(high(N, L))

two new Dependency Pairs are created:

QUICKSORT(cons(N'', nil)) -> QUICKSORT(nil)
QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(ifhigh(le(M', N''), N'', cons(M', L'')))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(ifhigh(le(M', N''), N'', cons(M', L'')))
QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(iflow(le(M', N''), N'', cons(M', L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(iflow(le(M', N''), N'', cons(M', L'')))

three new Dependency Pairs are created:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(iflow(true, N''', cons(0, L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(iflow(false, 0, cons(s(X'), L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(iflow(false, 0, cons(s(X'), L'')))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(iflow(true, N''', cons(0, L'')))
QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(ifhigh(le(M', N''), N'', cons(M', L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(iflow(true, N''', cons(0, L'')))

one new Dependency Pair is created:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(iflow(false, 0, cons(s(X'), L'')))
QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(ifhigh(le(M', N''), N'', cons(M', L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(iflow(false, 0, cons(s(X'), L'')))

one new Dependency Pair is created:

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(low(0, L''))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(low(0, L''))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(ifhigh(le(M', N''), N'', cons(M', L'')))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(N'', cons(M', L''))) -> QUICKSORT(ifhigh(le(M', N''), N'', cons(M', L'')))

three new Dependency Pairs are created:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(ifhigh(true, N''', cons(0, L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(ifhigh(false, 0, cons(s(X'), L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(ifhigh(false, 0, cons(s(X'), L'')))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(ifhigh(true, N''', cons(0, L'')))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(low(0, L''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(ifhigh(true, N''', cons(0, L'')))

one new Dependency Pair is created:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(high(N''', L''))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(high(N''', L''))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(ifhigh(false, 0, cons(s(X'), L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(low(0, L''))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(ifhigh(false, 0, cons(s(X'), L'')))

one new Dependency Pair is created:

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(cons(s(X'), high(0, L'')))

The transformation is resulting in one new DP problem:

     ↳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:

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(cons(s(X'), high(0, L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(low(0, L''))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(high(N''', L''))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(cons(s(X'), high(0, L'')))
QUICKSORT(cons(0, cons(s(X'), L''))) -> QUICKSORT(low(0, L''))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(high(N''', L''))

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

high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUICKSORT(x₁)) = 1 + x₁

POL(iflow(x₁, x₂, x₃)) = x₃

POL(0) = 1

POL(cons(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(high(x₁, x₂)) = x₂

POL(low(x₁, x₂)) = x₂

POL(nil) = 0

POL(true) = 0

POL(s(x₁)) = 0

POL(ifhigh(x₁, x₂, x₃)) = x₃

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳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:

QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

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

     ↳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:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

QUICKSORT(cons(N''', cons(0, L''))) -> QUICKSORT(cons(0, low(N''', L'')))

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

iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUICKSORT(x₁)) = 1 + x₁

POL(iflow(x₁, x₂, x₃)) = x₃

POL(0) = 0

POL(cons(x₁, x₂)) = 1 + x₂

POL(false) = 0

POL(low(x₁, x₂)) = x₂

POL(true) = 0

POL(nil) = 0

POL(s(x₁)) = 0

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳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
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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:

QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
app(nil, Y) -> Y
app(cons(N, L), Y) -> cons(N, app(L, Y))
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
quicksort(nil) -> nil
quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(iflow(le(X', Y'), s(Y'), cons(s(X'), L'')))
QUICKSORT(cons(s(Y'), cons(s(X'), L''))) -> QUICKSORT(ifhigh(le(X', Y'), s(Y'), cons(s(X'), L'')))

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

iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
iflow(false, N, cons(M, L)) -> low(N, L)
low(N, nil) -> nil
low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
high(N, nil) -> nil
high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
ifhigh(true, N, cons(M, L)) -> high(N, L)
ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUICKSORT(x₁)) = 1 + x₁

POL(iflow(x₁, x₂, x₃)) = x₃

POL(0) = 0

POL(cons(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(high(x₁, x₂)) = x₂

POL(low(x₁, x₂)) = x₂

POL(true) = 0

POL(nil) = 0

POL(s(x₁)) = 1

POL(ifhigh(x₁, x₂, x₃)) = x₃

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

Innermost Termination of R successfully shown.
Duration:
0:17 minutes

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

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

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

POL(QUICKSORT(x₁))	= 1 + x₁
POL(iflow(x₁, x₂, x₃))	= x₃
POL(0)	= 1
POL(cons(x₁, x₂))	= x₁ + x₂
POL(false)	= 0
POL(high(x₁, x₂))	= x₂
POL(low(x₁, x₂))	= x₂
POL(nil)	= 0
POL(true)	= 0
POL(s(x₁))	= 0
POL(ifhigh(x₁, x₂, x₃))	= x₃
POL(le(x₁, x₂))	= 0