Term Rewriting System R:
[x, y, f, xs]
app(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

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(add, app(s, x)), y) -> APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)
APP(app(add, app(s, x)), y) -> APP(add, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)


Rules:


app(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




As we are in the innermost case, we can delete all 5 non-usable-rules.


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


Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
  2. APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1=1
2>2
{1, 2} , {1, 2}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1
2>2
{1, 2} , {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:
app(x1, x2) -> app(x1, x2)

We obtain no new DP problems. 


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


Dependency Pair:

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


Rules:


app(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), y))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


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 4
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rule:


app(add, 0) -> id


Strategy:

innermost




We have the following set of usable rules:

app(add, 0) -> id
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(0)=  0  
  POL(s)=  0  
  POL(APP(x1, x2))=  1 + x1 + x2  
  POL(app(x1, x2))=  x1 + x2  
  POL(id)=  0  
  POL(add)=  0  

We have the following set D of usable symbols: {APP, app, id, add}
The following Dependency Pairs can be deleted as they contain symbols in their lhs which do not occur in D:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.


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