Term Rewriting System R:
[f, x, xs]
app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(mapt, f), app(leaf, x)) -> APP(leaf, app(f, x))
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(mapt, f), app(node, xs)) -> APP(node, app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(node, xs)) -> APP(maptlist, f)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(cons, app(app(mapt, f), x))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(mapt, f)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)


Rules:


app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Strategy:

innermost




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

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
four new Dependency Pairs are created:

APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, x), nil)) -> APP(app(cons, app(app(mapt, f''), x)), nil)
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))

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(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)


Rules:


app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Strategy:

innermost




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

APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
six new Dependency Pairs are created:

APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(leaf, x'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(leaf, x'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(node, xs'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(node, xs'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), xs'''')))) -> APP(app(maptlist, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))

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(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), xs'''')))) -> APP(app(maptlist, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(node, xs'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(node, xs'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(leaf, x'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(leaf, x'''')), xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))


Rules:


app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))


Additionally, the following usable rules for innermost can be oriented:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(mapt)=  1  
  POL(maptlist)=  1  
  POL(nil)=  0  
  POL(node)=  0  
  POL(leaf)=  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
Polynomial Ordering


Dependency Pairs:

APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), xs'''')))) -> APP(app(maptlist, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(node, xs'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(node, xs'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(leaf, x'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(leaf, x'''')), xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)


Rules:


app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), xs'''')))) -> APP(app(maptlist, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(node, xs'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(node, xs'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(leaf, x'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(leaf, x'''')), xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))


Additionally, the following usable rules for innermost can be oriented:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(mapt)=  0  
  POL(maptlist)=  0  
  POL(nil)=  0  
  POL(node)=  0  
  POL(leaf)=  0  
  POL(app(x1, x2))=  1 + x2  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
FwdInst
             ...
               →DP Problem 5
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)


Rules:


app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:03 minutes