Term Rewriting System R:
[x, f, t]
app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(cons, app(f, x))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(fmap, t)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost




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

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t), x))
four new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost




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

APP(app(fmap, app(app(fcons, f), t)), x) -> APP(f, x)
five new Dependency Pairs are created:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, f''), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t''''))), x')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, f''), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(fmap, app(app(fcons, f), fnil)), x'') -> APP(app(cons, app(f, x'')), nil)
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(fmap, t''), x''))), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, app(fmap, fnil)), t)), x'') -> APP(app(cons, nil), app(app(fmap, t), x''))
APP(app(fmap, app(app(fcons, f), app(app(fcons, f''), t''))), x'') -> APP(app(cons, app(f, x'')), app(app(cons, app(f'', x'')), app(app(fmap, t''), x'')))


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

app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(fcons)=  0  
  POL(cons)=  0  
  POL(fnil)=  0  
  POL(fmap)=  1  
  POL(nil)=  0  
  POL(app(x1, x2))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t'''')))), t)), x') -> APP(app(fmap, app(app(fcons, f''), app(app(fcons, f''''), t''''))), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), fnil))), t)), x') -> APP(app(fmap, app(app(fcons, f''), fnil)), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''''), t''''))), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, app(fmap, fnil)), t''))), t)), x') -> APP(app(fmap, app(app(fcons, app(fmap, fnil)), t'')), x')
APP(app(fmap, app(app(fcons, app(fmap, app(app(fcons, f''), t''))), t)), x'') -> APP(app(fmap, app(app(fcons, f''), t'')), x'')
APP(app(fmap, app(app(fcons, f), t)), x) -> APP(app(fmap, t), x)


Rules:


app(app(fmap, fnil), x) -> nil
app(app(fmap, app(app(fcons, f), t)), x) -> app(app(cons, app(f, x)), app(app(fmap, t), x))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:02 minutes