Term Rewriting System R:
[y, x, f, ys]
app(app(neq, 0), 0) -> false
app(app(neq, 0), app(s, y)) -> true
app(app(neq, app(s, x)), 0) -> true
app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y)
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)
nonzero -> app(filter, app(neq, 0))

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(neq, app(s, x)), app(s, y)) -> APP(app(neq, x), y)
APP(app(neq, app(s, x)), app(s, y)) -> APP(neq, x)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(filtersub, app(f, y)), f)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(filtersub, app(f, y))
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(cons, y), app(app(filter, f), ys))
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(filter, f)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(filter, f)
NONZERO -> APP(filter, app(neq, 0))
NONZERO -> APP(neq, 0)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


app(app(neq, 0), 0) -> false
app(app(neq, 0), app(s, y)) -> true
app(app(neq, app(s, x)), 0) -> true
app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y)
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)
nonzero -> app(filter, app(neq, 0))


Strategy:

innermost




As we are in the innermost case, we can delete all 9 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 3
                 ↳A-Transformation
           →DP Problem 2
             ↳UsableRules


Dependency Pair:

APP(app(neq, app(s, x)), app(s, y)) -> APP(app(neq, 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 4
                 ↳Size-Change Principle
           →DP Problem 2
             ↳UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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


Dependency Pairs:

APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))


Rules:


app(app(neq, 0), 0) -> false
app(app(neq, 0), app(s, y)) -> true
app(app(neq, app(s, x)), 0) -> true
app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y)
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)
nonzero -> app(filter, app(neq, 0))


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


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


Dependency Pairs:

APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))


Rules:


app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y)
app(app(neq, 0), 0) -> false
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys))
app(app(neq, app(s, x)), 0) -> true
app(app(neq, 0), app(s, y)) -> true
app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys)


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(app(filtersub, false), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
  2. APP(app(filter, f), app(app(cons, y), ys)) -> APP(f, y)
  3. APP(app(app(filtersub, true), f), app(app(cons, y), ys)) -> APP(app(filter, f), ys)
  4. APP(app(filter, f), app(app(cons, y), ys)) -> APP(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys))
and get the following Size-Change Graph(s):
{1, 3} , {1, 3}
2>2
{2} , {2}
1>1
2>2
{4} , {4}
2=2

which lead(s) to this/these maximal multigraph(s):
{2} , {2}
1>1
2>2
{4} , {1, 3}
2>2
{1, 3} , {4}
2>2
{2} , {1, 3}
2>2
{1, 3} , {2}
2>2
{4} , {2}
2>2
{2} , {2}
2>2

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
trivial

We obtain no new DP problems.

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