Term Rewriting System R:
[f, g, x, y, z]
app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, x), y)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) -> app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) -> y
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), y) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(asort, z) -> app(app(app(sort, min), max), z)
app(dsort, z) -> app(app(app(sort, max), min), z)

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:

APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(insert, f), g), app(app(app(sort, f), g), y))
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(insert, f), g)
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(insert, f)
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(sort, f), g), y)
APP(app(app(app(insert, f), g), nil), y) -> APP(app(cons, y), nil)
APP(app(app(app(insert, f), g), nil), y) -> APP(cons, y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(cons, app(app(f, x), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(f, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(app(insert, f), g), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(g, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(g, x)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(max, x)
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(min, app(s, x)), app(s, y)) -> APP(min, x)
APP(asort, z) -> APP(app(app(sort, min), max), z)
APP(asort, z) -> APP(app(sort, min), max)
APP(asort, z) -> APP(sort, min)
APP(dsort, z) -> APP(app(app(sort, max), min), z)
APP(dsort, z) -> APP(app(sort, max), min)
APP(dsort, z) -> APP(sort, max)

Furthermore, R contains three SCCs. 


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


Dependency Pair:

APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)


Rules:


app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, x), y)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) -> app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) -> y
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), y) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(asort, z) -> app(app(app(sort, min), max), z)
app(dsort, z) -> app(app(app(sort, max), min), z)


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 4
                 ↳A-Transformation
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳Nar


Dependency Pair:

APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


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


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. MAX(s(x), s(y)) -> MAX(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
             ↳Nar


Dependency Pair:

APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)


Rules:


app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, x), y)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) -> app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) -> y
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), y) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(asort, z) -> app(app(app(sort, min), max), z)
app(dsort, z) -> app(app(app(sort, max), min), z)


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 6
                 ↳A-Transformation
           →DP Problem 3
             ↳Nar


Dependency Pair:

APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


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


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. MIN(s(x), s(y)) -> MIN(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
             ↳UsableRules
           →DP Problem 3
             ↳Narrowing Transformation


Dependency Pairs:

APP(dsort, z) -> APP(app(app(sort, max), min), z)
APP(asort, z) -> APP(app(app(sort, min), max), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(g, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(g, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(f, x)
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(sort, f), g), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(f, x), y)
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)


Rules:


app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, x), y)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) -> app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) -> y
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), y) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(asort, z) -> app(app(app(sort, min), max), z)
app(dsort, z) -> app(app(app(sort, max), min), z)


Strategy:

innermost




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

APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
two new Dependency Pairs are created:

APP(app(app(sort, f''), g''), app(app(cons, x), nil)) -> APP(app(app(app(insert, f''), g''), nil), x)
APP(app(app(sort, f''), g''), app(app(cons, x), app(app(cons, x''), y''))) -> APP(app(app(app(insert, f''), g''), app(app(app(app(insert, f''), g''), app(app(app(sort, f''), g''), y'')), x'')), x)

The transformation is resulting in one new DP problem: 



   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
           →DP Problem 3
             ↳Nar
             ...
               →DP Problem 8
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(asort, z) -> APP(app(app(sort, min), max), z)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(g, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(g, x), y)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(app(app(insert, f), g), z), app(app(g, x), y))
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> APP(app(f, x), y)
APP(app(app(sort, f''), g''), app(app(cons, x), app(app(cons, x''), y''))) -> APP(app(app(app(insert, f''), g''), app(app(app(app(insert, f''), g''), app(app(app(sort, f''), g''), y'')), x'')), x)
APP(app(app(sort, f), g), app(app(cons, x), y)) -> APP(app(app(sort, f), g), y)
APP(dsort, z) -> APP(app(app(sort, max), min), z)


Rules:


app(app(app(sort, f), g), nil) -> nil
app(app(app(sort, f), g), app(app(cons, x), y)) -> app(app(app(app(insert, f), g), app(app(app(sort, f), g), y)), x)
app(app(app(app(insert, f), g), nil), y) -> app(app(cons, y), nil)
app(app(app(app(insert, f), g), app(app(cons, x), z)), y) -> app(app(cons, app(app(f, x), y)), app(app(app(app(insert, f), g), z), app(app(g, x), y)))
app(app(max, 0), y) -> y
app(app(max, x), 0) -> x
app(app(max, app(s, x)), app(s, y)) -> app(app(max, x), y)
app(app(min, 0), y) -> 0
app(app(min, x), 0) -> 0
app(app(min, app(s, x)), app(s, y)) -> app(app(min, x), y)
app(asort, z) -> app(app(app(sort, min), max), z)
app(dsort, z) -> app(app(app(sort, max), min), z)


Strategy:

innermost



The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes