Term Rewriting System R:
[x, y, z]
flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FLATTEN(unit(x)) -> FLATTEN(x)
FLATTEN(++(x, y)) -> ++'(flatten(x), flatten(y))
FLATTEN(++(x, y)) -> FLATTEN(x)
FLATTEN(++(x, y)) -> FLATTEN(y)
FLATTEN(++(unit(x), y)) -> ++'(flatten(x), flatten(y))
FLATTEN(++(unit(x), y)) -> FLATTEN(x)
FLATTEN(++(unit(x), y)) -> FLATTEN(y)
REV(++(x, y)) -> ++'(rev(y), rev(x))
REV(++(x, y)) -> REV(y)
REV(++(x, y)) -> REV(x)
++'(++(x, y), z) -> ++'(x, ++(y, z))
++'(++(x, y), z) -> ++'(y, z)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

++'(++(x, y), z) -> ++'(y, z)


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

++'(++(x, y), z) -> ++'(y, z)
one new Dependency Pair is created:

++'(++(x, ++(x'', y'')), z'') -> ++'(++(x'', y''), z'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

++'(++(x, ++(x'', y'')), z'') -> ++'(++(x'', y''), z'')


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

++'(++(x, ++(x'', y'')), z'') -> ++'(++(x'', y''), z'')


The following usable rules for innermost can be oriented:

++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(++'(x1, x2))=  1 + x1 + x2  
  POL(++(x1, x2))=  1 + x1 + x2  
  POL(nil)=  0  

resulting in one new DP problem.
Used Argument Filtering System:
++'(x1, x2) -> ++'(x1, x2)
++(x1, x2) -> ++(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳AFS
             ...
               →DP Problem 5
                 ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst


Dependency Pair:


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), y)) -> FLATTEN(y)
FLATTEN(++(unit(x), y)) -> FLATTEN(x)
FLATTEN(++(x, y)) -> FLATTEN(y)
FLATTEN(++(x, y)) -> FLATTEN(x)
FLATTEN(unit(x)) -> FLATTEN(x)


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(unit(x)) -> FLATTEN(x)
three new Dependency Pairs are created:

FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(x), y)) -> FLATTEN(x)
FLATTEN(++(x, y)) -> FLATTEN(y)
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
FLATTEN(++(x, y)) -> FLATTEN(x)
FLATTEN(++(unit(x), y)) -> FLATTEN(y)


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(x, y)) -> FLATTEN(x)
three new Dependency Pairs are created:

FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
FLATTEN(++(unit(x), y)) -> FLATTEN(y)
FLATTEN(++(unit(x), y)) -> FLATTEN(x)
FLATTEN(++(x, y)) -> FLATTEN(y)
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(x, y)) -> FLATTEN(y)
eight new Dependency Pairs are created:

FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
FLATTEN(++(unit(x), y)) -> FLATTEN(y)
FLATTEN(++(unit(x), y)) -> FLATTEN(x)
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(unit(x), y)) -> FLATTEN(x)
15 new Dependency Pairs are created:

FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
FLATTEN(++(unit(x), y)) -> FLATTEN(y)
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(unit(x), y)) -> FLATTEN(y)
30 new Dependency Pairs are created:

FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 10
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(unit(unit(x''))) -> FLATTEN(unit(x''))
three new Dependency Pairs are created:

FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 11
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(unit(unit(x'''')), y)) -> FLATTEN(unit(unit(x'''')))
three new Dependency Pairs are created:

FLATTEN(++(unit(unit(unit(x''''''))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(++(x'''''', y''''''))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 12
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(++(x'''''', y''''''))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(x''''''))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(x, unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
three new Dependency Pairs are created:

FLATTEN(++(x, unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(x, unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(x, unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 13
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(x, unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(x, unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(x, unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(++(x'''''', y''''''))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(x''''''))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(unit(unit(unit(x''''))), y)) -> FLATTEN(unit(unit(x'''')))
three new Dependency Pairs are created:

FLATTEN(++(unit(unit(unit(unit(x'''''')))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(unit(++(x'''''', y'''''')))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 14
                 ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(unit(++(x'''''', y'''''')))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(unit(x'''''')))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(x, unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(x, unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(x, unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(++(x'''''', y''''''))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(x''''''))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

FLATTEN(++(unit(x), unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
three new Dependency Pairs are created:

FLATTEN(++(unit(x), unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(x), unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 15
                 ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(unit(++(x'''''', y'''''')))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(unit(x'''''')))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(x, unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(x, unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(x, unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(++(x'''''', y''''''))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(++(unit(unit(unit(x''''''))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(x'''''''''', y'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(unit(x'''''''''')), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(unit(x''''''''), y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(++(x'''''''', y'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', unit(unit(x'''''''')))), y''))) -> FLATTEN(++(unit(++(x'''', unit(unit(x'''''''')))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(unit(x''''''''), y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(++(x'''''''', y'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(unit(x'''''''')), y'''')), y''))) -> FLATTEN(++(unit(++(unit(unit(x'''''''')), y'''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(unit(x''''''), y''''''))), y''))) -> FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(++(x'''''', y''''''))), y''))) -> FLATTEN(++(unit(unit(++(x'''''', y''''''))), y''))
FLATTEN(++(unit(x), ++(unit(unit(unit(x''''''))), y''))) -> FLATTEN(++(unit(unit(unit(x''''''))), y''))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''), y'''')), y''))) -> FLATTEN(++(unit(++(unit(x''''), y'''')), y''))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(unit(unit(x'''''''')), y'''')))) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(x), ++(x'', unit(++(unit(x''''''), y''''''))))) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), ++(x'', unit(++(x'''''', y''''''))))) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(x), unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(x), unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(unit(++(x'''''', y'''''')))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(unit(x'''''')))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(x, unit(unit(++(unit(x''''''), y''''''))))) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(x, unit(unit(++(x'''''', y''''''))))) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(x, unit(unit(unit(x''''''))))) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(unit(++(unit(x''''''), y''''''))), y)) -> FLATTEN(unit(unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(unit(++(x'''''', y''''''))), y)) -> FLATTEN(unit(unit(++(x'''''', y''''''))))
FLATTEN(++(unit(unit(unit(x''''''))), y)) -> FLATTEN(unit(unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', unit(unit(x''''''))))) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(x), ++(x'', ++(unit(x''''), y'''')))) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(x), ++(x'', ++(x'''', y'''')))) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(x), unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(x), unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(unit(x''''''''), y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(++(x'''''''', y'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(++(x'''''''', y'''''''')), y'''')))
FLATTEN(++(unit(++(x'', ++(unit(unit(x'''''''')), y''''))), y)) -> FLATTEN(++(x'', ++(unit(unit(x'''''''')), y'''')))
FLATTEN(++(unit(++(x'', unit(++(unit(x''''''), y'''''')))), y)) -> FLATTEN(++(x'', unit(++(unit(x''''''), y''''''))))
FLATTEN(++(unit(++(x'', unit(++(x'''''', y'''''')))), y)) -> FLATTEN(++(x'', unit(++(x'''''', y''''''))))
FLATTEN(++(unit(++(x'', unit(unit(x'''''')))), y)) -> FLATTEN(++(x'', unit(unit(x''''''))))
FLATTEN(++(unit(++(x'', ++(unit(x''''), y''''))), y)) -> FLATTEN(++(x'', ++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'', ++(x'''', y''''))), y)) -> FLATTEN(++(x'', ++(x'''', y'''')))
FLATTEN(++(unit(++(unit(++(unit(x''''''), y'''''')), y'')), y)) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(unit(++(unit(++(x'''''', y'''''')), y'')), y)) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(unit(x), ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(unit(++(unit(unit(x'''''')), y'')), y)) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(unit(unit(++(unit(x''''), y''''))), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(unit(++(x'''', y''''))), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(++(unit(x''), y'')), y)) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(unit(++(unit(x''''''), y'''''')), y''))) -> FLATTEN(++(unit(++(unit(x''''''), y'''''')), y''))
FLATTEN(++(x, ++(unit(++(x'''''', y'''''')), y''))) -> FLATTEN(++(unit(++(x'''''', y'''''')), y''))
FLATTEN(++(x, ++(unit(unit(x'''''')), y''))) -> FLATTEN(++(unit(unit(x'''''')), y''))
FLATTEN(++(x, unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(x, unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(x, ++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(++(x, ++(x'', y''))) -> FLATTEN(++(x'', y''))
FLATTEN(++(unit(++(unit(x''''), y'''')), y)) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(++(unit(++(x'''', y'''')), y)) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(++(unit(x), ++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))) -> FLATTEN(++(unit(++(x'''', ++(unit(++(unit(x''''''''''), y'''''''''')), y''''''))), y''))


The following usable rules for innermost can be oriented:

++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(unit(x1))=  x1  
  POL(FLATTEN(x1))=  1 + x1  
  POL(++(x1, x2))=  1 + x1 + x2  
  POL(nil)=  1  

resulting in one new DP problem.
Used Argument Filtering System:
FLATTEN(x1) -> FLATTEN(x1)
++(x1, x2) -> ++(x1, x2)
unit(x1) -> unit(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 16
                 ↳Dependency Graph
       →DP Problem 3
         ↳FwdInst


Dependency Pairs:

FLATTEN(unit(unit(++(unit(x''''), y'''')))) -> FLATTEN(unit(++(unit(x''''), y'''')))
FLATTEN(unit(unit(++(x'''', y'''')))) -> FLATTEN(unit(++(x'''', y'''')))
FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))
FLATTEN(unit(++(unit(x''), y''))) -> FLATTEN(++(unit(x''), y''))
FLATTEN(unit(++(x'', y''))) -> FLATTEN(++(x'', y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 17
                 ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

FLATTEN(unit(unit(unit(x'''')))) -> FLATTEN(unit(unit(x'''')))


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(unit(x1))=  1 + x1  
  POL(FLATTEN(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
FLATTEN(x1) -> FLATTEN(x1)
unit(x1) -> unit(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 18
                 ↳Dependency Graph
       →DP Problem 3
         ↳FwdInst


Dependency Pair:


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Forward Instantiation Transformation


Dependency Pair:

REV(++(x, y)) -> REV(y)


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




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

REV(++(x, y)) -> REV(y)
one new Dependency Pair is created:

REV(++(x, ++(x'', y''))) -> REV(++(x'', y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst
           →DP Problem 19
             ↳Argument Filtering and Ordering


Dependency Pair:

REV(++(x, ++(x'', y''))) -> REV(++(x'', y''))


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

REV(++(x, ++(x'', y''))) -> REV(++(x'', y''))


The following usable rules for innermost can be oriented:

++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(REV(x1))=  1 + x1  
  POL(++(x1, x2))=  1 + x1 + x2  
  POL(nil)=  0  

resulting in one new DP problem.
Used Argument Filtering System:
REV(x1) -> REV(x1)
++(x1, x2) -> ++(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst
           →DP Problem 19
             ↳AFS
             ...
               →DP Problem 20
                 ↳Dependency Graph


Dependency Pair:


Rules:


flatten(nil) -> nil
flatten(unit(x)) -> flatten(x)
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
rev(nil) -> nil
rev(unit(x)) -> unit(x)
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x
++(x, nil) -> x
++(nil, y) -> y
++(++(x, y), z) -> ++(x, ++(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:19 minutes