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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

LE(s(x), s(y)) -> LE(x, y)
APP(add(n, x), y) -> APP(x, y)
LOW(n, add(m, x)) -> IFLOW(le(m, n), n, add(m, x))
LOW(n, add(m, x)) -> LE(m, n)
IFLOW(true, n, add(m, x)) -> LOW(n, x)
IFLOW(false, n, add(m, x)) -> LOW(n, x)
HIGH(n, add(m, x)) -> IFHIGH(le(m, n), n, add(m, x))
HIGH(n, add(m, x)) -> LE(m, n)
IFHIGH(true, n, add(m, x)) -> HIGH(n, x)
IFHIGH(false, n, add(m, x)) -> HIGH(n, x)
QUICKSORT(add(n, x)) -> APP(quicksort(low(n, x)), add(n, quicksort(high(n, x))))
QUICKSORT(add(n, x)) -> QUICKSORT(low(n, x))
QUICKSORT(add(n, x)) -> LOW(n, x)
QUICKSORT(add(n, x)) -> QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) -> HIGH(n, x)

Furthermore, R contains five SCCs. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules


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


Strategy:

innermost




As we are in the innermost case, we can delete all 15 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 6
                 ↳Size-Change Principle
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. LE(s(x), s(y)) -> LE(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳Usable Rules (Innermost)
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules


Dependency Pair:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 15 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
             ...
               →DP Problem 7
                 ↳Size-Change Principle
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(add(n, x), y) -> APP(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳Usable Rules (Innermost)
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 12 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
             ...
               →DP Problem 8
                 ↳Size-Change Principle
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




We number the DPs as follows:
  1. IFLOW(false, n, add(m, x)) -> LOW(n, x)
  2. IFLOW(true, n, add(m, x)) -> LOW(n, x)
  3. LOW(n, add(m, x)) -> IFLOW(le(m, n), n, add(m, x))
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
2=1
3>2
{3} , {3}
1=2
2=3

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {3}
2=2
3>3
{3} , {1, 2}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳Usable Rules (Innermost)
           →DP Problem 5
             ↳UsableRules


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 12 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
             ...
               →DP Problem 9
                 ↳Size-Change Principle
           →DP Problem 5
             ↳UsableRules


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




We number the DPs as follows:
  1. IFHIGH(false, n, add(m, x)) -> HIGH(n, x)
  2. IFHIGH(true, n, add(m, x)) -> HIGH(n, x)
  3. HIGH(n, add(m, x)) -> IFHIGH(le(m, n), n, add(m, x))
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
2=1
3>2
{3} , {3}
1=2
2=3

which lead(s) to this/these maximal multigraph(s):
{3} , {1, 2}
1=1
2>2
{1, 2} , {3}
2=2
3>3

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳Usable Rules (Innermost)


Dependency Pairs:

QUICKSORT(add(n, x)) -> QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) -> QUICKSORT(low(n, x))


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules
             ...
               →DP Problem 10
                 ↳Negative Polynomial Order


Dependency Pairs:

QUICKSORT(add(n, x)) -> QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) -> QUICKSORT(low(n, x))


Rules:


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


Strategy:

innermost




The following Dependency Pairs can be strictly oriented using the given order.

QUICKSORT(add(n, x)) -> QUICKSORT(high(n, x))
QUICKSORT(add(n, x)) -> QUICKSORT(low(n, x))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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


Used ordering:
Polynomial Order with Interpretation:

POL( QUICKSORT(x1) ) = x1

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

POL( high(x1, x2) ) = x2

POL( low(x1, x2) ) = x2

POL( le(x1, x2) ) = 0

POL( true ) = 0

POL( false ) = 0

POL( ifhigh(x1, ..., x3) ) = x3

POL( nil ) = 0

POL( iflow(x1, ..., x3) ) = x3


This results in one new DP problem. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳UsableRules
           →DP Problem 4
             ↳UsableRules
           →DP Problem 5
             ↳UsableRules
             ...
               →DP Problem 11
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:01 minutes