Term Rewriting System R:
[x, y, z, l, l1, l2]
0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(0(x), 0(y)) -> 0'(+(x, y))
+'(0(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(1(x), 1(y)) -> 0'(+(+(x, y), 1(#)))
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 1(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)
*'(0(x), y) -> 0'(*(x, y))
*'(0(x), y) -> *'(x, y)
*'(1(x), y) -> +'(0(*(x, y)), y)
*'(1(x), y) -> 0'(*(x, y))
*'(1(x), y) -> *'(x, y)
*'(*(x, y), z) -> *'(x, *(y, z))
*'(*(x, y), z) -> *'(y, z)
*'(x, +(y, z)) -> +'(*(x, y), *(x, z))
*'(x, +(y, z)) -> *'(x, y)
*'(x, +(y, z)) -> *'(x, z)
APP(cons(x, l1), l2) -> APP(l1, l2)
SUM(nil) -> 0'(#)
SUM(cons(x, l)) -> +'(x, sum(l))
SUM(cons(x, l)) -> SUM(l)
SUM(app(l1, l2)) -> +'(sum(l1), sum(l2))
SUM(app(l1, l2)) -> SUM(l1)
SUM(app(l1, l2)) -> SUM(l2)
PROD(cons(x, l)) -> *'(x, prod(l))
PROD(cons(x, l)) -> PROD(l)
PROD(app(l1, l2)) -> *'(prod(l1), prod(l2))
PROD(app(l1, l2)) -> PROD(l1)
PROD(app(l1, l2)) -> PROD(l2)

Furthermore, R contains five SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(0(x), 0(y)) -> +'(x, y)


Rules:


0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))


Strategy:

innermost




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

+'(+(x, y), z) -> +'(x, +(y, z))
six new Dependency Pairs are created:

+'(+(x, y'), #) -> +'(x, y')
+'(+(x, 0(x'')), 0(y'')) -> +'(x, 0(+(x'', y'')))
+'(+(x, 0(x'')), 1(y'')) -> +'(x, 1(+(x'', y'')))
+'(+(x, 1(x'')), 0(y'')) -> +'(x, 1(+(x'', y'')))
+'(+(x, 1(x'')), 1(y'')) -> +'(x, 0(+(+(x'', y''), 1(#))))
+'(+(x, +(x'', y'')), z'') -> +'(x, +(x'', +(y'', z'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 6
             ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

+'(+(x, +(x'', y'')), z'') -> +'(x, +(x'', +(y'', z'')))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(0(x), 0(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(y, z)


Rules:


0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))


Strategy:

innermost




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

+'(0(x), 0(y)) -> +'(x, y)
six new Dependency Pairs are created:

+'(0(0(x'')), 0(0(y''))) -> +'(0(x''), 0(y''))
+'(0(0(x'')), 0(1(y''))) -> +'(0(x''), 1(y''))
+'(0(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(0(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(0(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(0(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

+'(0(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(0(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(0(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(0(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(0(0(x'')), 0(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(0(y''))) -> +'(0(x''), 0(y''))
+'(+(x, y), z) -> +'(y, z)
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(+(x, +(x'', y'')), z'') -> +'(x, +(x'', +(y'', z'')))


Rules:


0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))


Strategy:

innermost




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

+'(0(x), 1(y)) -> +'(x, y)
11 new Dependency Pairs are created:

+'(0(0(x'')), 1(1(y''))) -> +'(0(x''), 1(y''))
+'(0(1(x'')), 1(0(y''))) -> +'(1(x''), 0(y''))
+'(0(1(x'')), 1(1(y''))) -> +'(1(x''), 1(y''))
+'(0(+(x'', y'')), 1(y0)) -> +'(+(x'', y''), y0)
+'(0(+(x'', +(x'''', y''''))), 1(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(0(0(0(x''''))), 1(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(0(0(0(x''''))), 1(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(0(0(1(x''''))), 1(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(0(0(1(x''''))), 1(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(0(0(+(x'''', y''''))), 1(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(0(0(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

+'(0(0(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(0(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(0(0(+(x'''', y''''))), 1(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(0(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(0(0(1(x''''))), 1(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(0(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(0(0(1(x''''))), 1(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(0(0(0(x''''))), 1(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(0(0(0(x''''))), 1(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(0(+(x'', +(x'''', y''''))), 1(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(0(+(x'', y'')), 1(y0)) -> +'(+(x'', y''), y0)
+'(0(1(x'')), 1(1(y''))) -> +'(1(x''), 1(y''))
+'(0(1(x'')), 1(0(y''))) -> +'(1(x''), 0(y''))
+'(0(0(x'')), 1(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(0(y''))) -> +'(0(x''), 0(y''))
+'(+(x, +(x'', y'')), z'') -> +'(x, +(x'', +(y'', z'')))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(1(x), 0(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(y, z)
+'(0(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')


Rules:


0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))


Strategy:

innermost




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

+'(1(x), 0(y)) -> +'(x, y)
21 new Dependency Pairs are created:

+'(1(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(1(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(1(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(1(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(0(0(x''''))), 0(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(0(0(x''''))), 0(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(1(x''''))), 0(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(1(x''''))), 0(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(1(0(+(x'''', y''''))), 0(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 0(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(1(0(0(x''''))), 0(1(1(y'''')))) -> +'(0(0(x'''')), 1(1(y'''')))
+'(1(0(1(x''''))), 0(1(0(y'''')))) -> +'(0(1(x'''')), 1(0(y'''')))
+'(1(0(1(x''''))), 0(1(1(y'''')))) -> +'(0(1(x'''')), 1(1(y'''')))
+'(1(0(+(x'''', y''''))), 0(1(y0''))) -> +'(0(+(x'''', y'''')), 1(y0''))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 0(1(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(1(0(0(0(x'''''')))), 0(1(0(0(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(0(y''''''))))
+'(1(0(0(0(x'''''')))), 0(1(0(1(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(1(y''''''))))
+'(1(0(0(1(x'''''')))), 0(1(0(0(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(0(y''''''))))
+'(1(0(0(1(x'''''')))), 0(1(0(1(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(1(y''''''))))
+'(1(0(0(+(x'''''', y'''''')))), 0(1(0(y0'''')))) -> +'(0(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(1(0(0(+(x'''''', +(x'''''''', y''''''''))))), 0(1(0(y''''')))) -> +'(0(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

+'(1(0(0(+(x'''''', +(x'''''''', y''''''''))))), 0(1(0(y''''')))) -> +'(0(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))
+'(0(0(+(x'''', y''''))), 1(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(1(0(0(+(x'''''', y'''''')))), 0(1(0(y0'''')))) -> +'(0(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(0(0(1(x''''))), 1(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(1(0(0(1(x'''''')))), 0(1(0(1(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(1(y''''''))))
+'(0(0(1(x''''))), 1(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(0(1(x'''''')))), 0(1(0(0(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(0(y''''''))))
+'(0(0(0(x''''))), 1(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(0(0(x'''''')))), 0(1(0(1(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(1(y''''''))))
+'(0(0(0(x''''))), 1(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(0(0(0(x'''''')))), 0(1(0(0(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(0(y''''''))))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 0(1(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(0(+(x'', +(x'''', y''''))), 1(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(0(+(x'', y'')), 1(y0)) -> +'(+(x'', y''), y0)
+'(1(0(+(x'''', y''''))), 0(1(y0''))) -> +'(0(+(x'''', y'''')), 1(y0''))
+'(0(1(x'')), 1(1(y''))) -> +'(1(x''), 1(y''))
+'(1(0(1(x''''))), 0(1(1(y'''')))) -> +'(0(1(x'''')), 1(1(y'''')))
+'(1(0(1(x''''))), 0(1(0(y'''')))) -> +'(0(1(x'''')), 1(0(y'''')))
+'(1(0(0(x''''))), 0(1(1(y'''')))) -> +'(0(0(x'''')), 1(1(y'''')))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 0(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(0(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(0(+(x'''', y''''))), 0(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(0(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(1(0(1(x''''))), 0(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(0(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(1(0(1(x''''))), 0(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(0(x''''))), 0(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(0(x''''))), 0(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(1(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(1(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(0(1(x'')), 1(0(y''))) -> +'(1(x''), 0(y''))
+'(0(0(x'')), 1(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(0(y''))) -> +'(0(x''), 0(y''))
+'(+(x, +(x'', y'')), z'') -> +'(x, +(x'', +(y'', z'')))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(+(x, y), z) -> +'(y, z)
+'(0(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(0(0(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))


Rules:


0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))


Strategy:

innermost




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

+'(1(x), 1(y)) -> +'(x, y)
41 new Dependency Pairs are created:

+'(1(1(x'')), 1(1(y''))) -> +'(1(x''), 1(y''))
+'(1(+(x'', y'')), 1(y0)) -> +'(+(x'', y''), y0)
+'(1(+(x'', +(x'''', y''''))), 1(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(0(0(x''''))), 1(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(0(0(x''''))), 1(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(1(x''''))), 1(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(1(x''''))), 1(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(1(0(+(x'''', y''''))), 1(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(1(0(0(x''''))), 1(1(1(y'''')))) -> +'(0(0(x'''')), 1(1(y'''')))
+'(1(0(1(x''''))), 1(1(0(y'''')))) -> +'(0(1(x'''')), 1(0(y'''')))
+'(1(0(1(x''''))), 1(1(1(y'''')))) -> +'(0(1(x'''')), 1(1(y'''')))
+'(1(0(+(x'''', y''''))), 1(1(y0''))) -> +'(0(+(x'''', y'''')), 1(y0''))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 1(1(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(1(0(0(0(x'''''')))), 1(1(0(0(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(0(y''''''))))
+'(1(0(0(0(x'''''')))), 1(1(0(1(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(1(y''''''))))
+'(1(0(0(1(x'''''')))), 1(1(0(0(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(0(y''''''))))
+'(1(0(0(1(x'''''')))), 1(1(0(1(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(1(y''''''))))
+'(1(0(0(+(x'''''', y'''''')))), 1(1(0(y0'''')))) -> +'(0(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(1(0(0(+(x'''''', +(x'''''''', y''''''''))))), 1(1(0(y''''')))) -> +'(0(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))
+'(1(1(1(x''''))), 1(0(0(y'''')))) -> +'(1(1(x'''')), 0(0(y'''')))
+'(1(1(1(x''''))), 1(0(1(y'''')))) -> +'(1(1(x'''')), 0(1(y'''')))
+'(1(1(+(x'''', y''''))), 1(0(y0''))) -> +'(1(+(x'''', y'''')), 0(y0''))
+'(1(1(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(1(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(1(1(0(0(x'''''')))), 1(0(0(0(y''''''))))) -> +'(1(0(0(x''''''))), 0(0(0(y''''''))))
+'(1(1(0(0(x'''''')))), 1(0(0(1(y''''''))))) -> +'(1(0(0(x''''''))), 0(0(1(y''''''))))
+'(1(1(0(1(x'''''')))), 1(0(0(0(y''''''))))) -> +'(1(0(1(x''''''))), 0(0(0(y''''''))))
+'(1(1(0(1(x'''''')))), 1(0(0(1(y''''''))))) -> +'(1(0(1(x''''''))), 0(0(1(y''''''))))
+'(1(1(0(+(x'''''', y'''''')))), 1(0(0(y0'''')))) -> +'(1(0(+(x'''''', y''''''))), 0(0(y0'''')))
+'(1(1(0(+(x'''''', +(x'''''''', y''''''''))))), 1(0(0(y''''')))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 0(0(y''''')))
+'(1(1(0(0(x'''''')))), 1(0(1(1(y''''''))))) -> +'(1(0(0(x''''''))), 0(1(1(y''''''))))
+'(1(1(0(1(x'''''')))), 1(0(1(0(y''''''))))) -> +'(1(0(1(x''''''))), 0(1(0(y''''''))))
+'(1(1(0(1(x'''''')))), 1(0(1(1(y''''''))))) -> +'(1(0(1(x''''''))), 0(1(1(y''''''))))
+'(1(1(0(+(x'''''', y'''''')))), 1(0(1(y0'''')))) -> +'(1(0(+(x'''''', y''''''))), 0(1(y0'''')))
+'(1(1(0(+(x'''''', +(x'''''''', y''''''''))))), 1(0(1(y''''')))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 0(1(y''''')))
+'(1(1(0(0(0(x''''''''))))), 1(0(1(0(0(y'''''''')))))) -> +'(1(0(0(0(x'''''''')))), 0(1(0(0(y'''''''')))))
+'(1(1(0(0(0(x''''''''))))), 1(0(1(0(1(y'''''''')))))) -> +'(1(0(0(0(x'''''''')))), 0(1(0(1(y'''''''')))))
+'(1(1(0(0(1(x''''''''))))), 1(0(1(0(0(y'''''''')))))) -> +'(1(0(0(1(x'''''''')))), 0(1(0(0(y'''''''')))))
+'(1(1(0(0(1(x''''''''))))), 1(0(1(0(1(y'''''''')))))) -> +'(1(0(0(1(x'''''''')))), 0(1(0(1(y'''''''')))))
+'(1(1(0(0(+(x'''''''', y''''''''))))), 1(0(1(0(y0''''''))))) -> +'(1(0(0(+(x'''''''', y'''''''')))), 0(1(0(y0''''''))))
+'(1(1(0(0(+(x'''''''', +(x'''''''''', y'''''''''')))))), 1(0(1(0(y'''''''))))) -> +'(1(0(0(+(x'''''''', +(x'''''''''', y''''''''''))))), 0(1(0(y'''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 6
             ↳FwdInst
             ...
               →DP Problem 10
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining


Dependency Pairs:

+'(1(1(0(0(+(x'''''''', +(x'''''''''', y'''''''''')))))), 1(0(1(0(y'''''''))))) -> +'(1(0(0(+(x'''''''', +(x'''''''''', y''''''''''))))), 0(1(0(y'''''''))))
+'(1(0(0(+(x'''''', y'''''')))), 0(1(0(y0'''')))) -> +'(0(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(1(1(0(0(+(x'''''''', y''''''''))))), 1(0(1(0(y0''''''))))) -> +'(1(0(0(+(x'''''''', y'''''''')))), 0(1(0(y0''''''))))
+'(1(0(0(1(x'''''')))), 0(1(0(1(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(1(y''''''))))
+'(1(1(0(0(1(x''''''''))))), 1(0(1(0(1(y'''''''')))))) -> +'(1(0(0(1(x'''''''')))), 0(1(0(1(y'''''''')))))
+'(1(0(0(1(x'''''')))), 0(1(0(0(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(0(y''''''))))
+'(1(1(0(0(1(x''''''''))))), 1(0(1(0(0(y'''''''')))))) -> +'(1(0(0(1(x'''''''')))), 0(1(0(0(y'''''''')))))
+'(1(0(0(0(x'''''')))), 0(1(0(1(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(1(y''''''))))
+'(1(1(0(0(0(x''''''''))))), 1(0(1(0(1(y'''''''')))))) -> +'(1(0(0(0(x'''''''')))), 0(1(0(1(y'''''''')))))
+'(1(0(0(0(x'''''')))), 0(1(0(0(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(0(y''''''))))
+'(1(1(0(0(0(x''''''''))))), 1(0(1(0(0(y'''''''')))))) -> +'(1(0(0(0(x'''''''')))), 0(1(0(0(y'''''''')))))
+'(1(1(0(+(x'''''', +(x'''''''', y''''''''))))), 1(0(1(y''''')))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 0(1(y''''')))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 0(1(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(1(0(+(x'''', y''''))), 0(1(y0''))) -> +'(0(+(x'''', y'''')), 1(y0''))
+'(1(1(0(+(x'''''', y'''''')))), 1(0(1(y0'''')))) -> +'(1(0(+(x'''''', y''''''))), 0(1(y0'''')))
+'(1(0(1(x''''))), 0(1(1(y'''')))) -> +'(0(1(x'''')), 1(1(y'''')))
+'(1(1(0(1(x'''''')))), 1(0(1(1(y''''''))))) -> +'(1(0(1(x''''''))), 0(1(1(y''''''))))
+'(1(0(1(x''''))), 0(1(0(y'''')))) -> +'(0(1(x'''')), 1(0(y'''')))
+'(1(1(0(1(x'''''')))), 1(0(1(0(y''''''))))) -> +'(1(0(1(x''''''))), 0(1(0(y''''''))))
+'(1(0(0(x''''))), 0(1(1(y'''')))) -> +'(0(0(x'''')), 1(1(y'''')))
+'(1(1(0(0(x'''''')))), 1(0(1(1(y''''''))))) -> +'(1(0(0(x''''''))), 0(1(1(y''''''))))
+'(1(1(0(+(x'''''', +(x'''''''', y''''''''))))), 1(0(0(y''''')))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 0(0(y''''')))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 0(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(1(0(+(x'''', y''''))), 0(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(1(1(0(+(x'''''', y'''''')))), 1(0(0(y0'''')))) -> +'(1(0(+(x'''''', y''''''))), 0(0(y0'''')))
+'(1(1(0(1(x'''''')))), 1(0(0(1(y''''''))))) -> +'(1(0(1(x''''''))), 0(0(1(y''''''))))
+'(1(1(0(1(x'''''')))), 1(0(0(0(y''''''))))) -> +'(1(0(1(x''''''))), 0(0(0(y''''''))))
+'(1(1(0(0(x'''''')))), 1(0(0(1(y''''''))))) -> +'(1(0(0(x''''''))), 0(0(1(y''''''))))
+'(1(1(0(0(x'''''')))), 1(0(0(0(y''''''))))) -> +'(1(0(0(x''''''))), 0(0(0(y''''''))))
+'(1(1(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(1(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(1(1(+(x'''', y''''))), 1(0(y0''))) -> +'(1(+(x'''', y'''')), 0(y0''))
+'(1(1(1(x''''))), 1(0(1(y'''')))) -> +'(1(1(x'''')), 0(1(y'''')))
+'(1(1(1(x''''))), 1(0(0(y'''')))) -> +'(1(1(x'''')), 0(0(y'''')))
+'(1(0(0(+(x'''''', +(x'''''''', y''''''''))))), 1(1(0(y''''')))) -> +'(0(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))
+'(0(0(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(1(0(0(+(x'''''', y'''''')))), 1(1(0(y0'''')))) -> +'(0(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(0(0(1(x''''))), 1(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(1(0(0(1(x'''''')))), 1(1(0(1(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(1(y''''''))))
+'(0(0(1(x''''))), 1(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(0(1(x'''''')))), 1(1(0(0(y''''''))))) -> +'(0(0(1(x''''''))), 1(0(0(y''''''))))
+'(0(0(0(x''''))), 1(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(0(0(x'''''')))), 1(1(0(1(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(1(y''''''))))
+'(0(0(0(x''''))), 1(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(0(0(0(x'''''')))), 1(1(0(0(y''''''))))) -> +'(0(0(0(x''''''))), 1(0(0(y''''''))))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 1(1(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(0(+(x'', +(x'''', y''''))), 1(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(0(+(x'', y'')), 1(y0)) -> +'(+(x'', y''), y0)
+'(1(0(+(x'''', y''''))), 1(1(y0''))) -> +'(0(+(x'''', y'''')), 1(y0''))
+'(0(1(x'')), 1(1(y''))) -> +'(1(x''), 1(y''))
+'(1(0(1(x''''))), 1(1(1(y'''')))) -> +'(0(1(x'''')), 1(1(y'''')))
+'(1(0(1(x''''))), 1(1(0(y'''')))) -> +'(0(1(x'''')), 1(0(y'''')))
+'(1(0(0(x''''))), 1(1(1(y'''')))) -> +'(0(0(x'''')), 1(1(y'''')))
+'(1(0(+(x'''', +(x'''''', y'''''')))), 1(0(y'''))) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(0(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(0(+(x'''', y''''))), 1(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(1(0(1(x''''))), 1(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(0(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(1(0(1(x''''))), 0(0(1(y'''')))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(1(0(1(x''''))), 0(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(0(x''''))), 0(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(0(x''''))), 0(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(+(x'', +(x'''', y''''))), 0(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(0(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(1(0(1(x''''))), 1(0(0(y'''')))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(1(0(0(x''''))), 1(0(1(y'''')))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(1(0(0(x''''))), 1(0(0(y'''')))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(1(+(x'', +(x'''', y''''))), 1(y')) -> +'(+(x'', +(x'''', y'''')), y')
+'(1(+(x'', y'')), 1(y0)) -> +'(+(x'', y''), y0)
+'(1(1(x'')), 1(1(y''))) -> +'(1(x''), 1(y''))
+'(1(1(x'')), 0(1(y''))) -> +'(1(x''), 1(y''))
+'(1(1(x'')), 0(0(y''))) -> +'(1(x''), 0(y''))
+'(0(1(x'')), 1(0(y''))) -> +'(1(x''), 0(y''))
+'(0(0(x'')), 1(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(1(y''))) -> +'(0(x''), 1(y''))
+'(0(0(x'')), 0(0(y''))) -> +'(0(x''), 0(y''))
+'(+(x, +(x'', y'')), z'') -> +'(x, +(x'', +(y'', z'')))
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(+(x, y), z) -> +'(y, z)
+'(0(+(x'', y'')), 0(y0)) -> +'(+(x'', y''), y0)
+'(0(0(+(x'''', y''''))), 1(0(y0''))) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(1(0(0(+(x'''''', +(x'''''''', y''''''''))))), 0(1(0(y''''')))) -> +'(0(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))


Rules:


0(#) -> #
+(x, #) -> x
+(#, x) -> x
+(0(x), 0(y)) -> 0(+(x, y))
+(0(x), 1(y)) -> 1(+(x, y))
+(1(x), 0(y)) -> 1(+(x, y))
+(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
+(+(x, y), z) -> +(x, +(y, z))
*(#, x) -> #
*(0(x), y) -> 0(*(x, y))
*(1(x), y) -> +(0(*(x, y)), y)
*(*(x, y), z) -> *(x, *(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0(#)
sum(cons(x, l)) -> +(x, sum(l))
sum(app(l1, l2)) -> +(sum(l1), sum(l2))
prod(nil) -> 1(#)
prod(cons(x, l)) -> *(x, prod(l))
prod(app(l1, l2)) -> *(prod(l1), prod(l2))


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)
82 new Dependency Pairs are created:

+'(+(x, +(x'', y'')), z'') -> +'(+(x'', y''), z'')
+'(+(x, 1(x'')), 1(y'')) -> +'(1(x''), 1(y''))
+'(+(x, +(x'', +(x'''', y''''))), z') -> +'(+(x'', +(x'''', y'''')), z')
+'(+(x, 0(0(x''''))), 0(0(y''''))) -> +'(0(0(x'''')), 0(0(y'''')))
+'(+(x, 0(0(x''''))), 0(1(y''''))) -> +'(0(0(x'''')), 0(1(y'''')))
+'(+(x, 0(1(x''''))), 0(0(y''''))) -> +'(0(1(x'''')), 0(0(y'''')))
+'(+(x, 0(1(x''''))), 0(1(y''''))) -> +'(0(1(x'''')), 0(1(y'''')))
+'(+(x, 0(+(x'''', y''''))), 0(y0'')) -> +'(0(+(x'''', y'''')), 0(y0''))
+'(+(x, 0(+(x'''', +(x'''''', y'''''')))), 0(y''')) -> +'(0(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(+(x, 0(0(x''''))), 1(1(y''''))) -> +'(0(0(x'''')), 1(1(y'''')))
+'(+(x, 0(1(x''''))), 1(0(y''''))) -> +'(0(1(x'''')), 1(0(y'''')))
+'(+(x, 0(1(x''''))), 1(1(y''''))) -> +'(0(1(x'''')), 1(1(y'''')))
+'(+(x, 0(+(x'''', y''''))), 1(y0'')) -> +'(0(+(x'''', y'''')), 1(y0''))
+'(+(x, 0(+(x'''', +(x'''''', y'''''')))), 1(y''')) -> +'(0(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(+(x, 0(0(0(x'''''')))), 1(0(0(y'''''')))) -> +'(0(0(0(x''''''))), 1(0(0(y''''''))))
+'(+(x, 0(0(0(x'''''')))), 1(0(1(y'''''')))) -> +'(0(0(0(x''''''))), 1(0(1(y''''''))))
+'(+(x, 0(0(1(x'''''')))), 1(0(0(y'''''')))) -> +'(0(0(1(x''''''))), 1(0(0(y''''''))))
+'(+(x, 0(0(1(x'''''')))), 1(0(1(y'''''')))) -> +'(0(0(1(x''''''))), 1(0(1(y''''''))))
+'(+(x, 0(0(+(x'''''', y'''''')))), 1(0(y0''''))) -> +'(0(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(+(x, 0(0(+(x'''''', +(x'''''''', y''''''''))))), 1(0(y'''''))) -> +'(0(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))
+'(+(x, 1(1(x''''))), 0(0(y''''))) -> +'(1(1(x'''')), 0(0(y'''')))
+'(+(x, 1(1(x''''))), 0(1(y''''))) -> +'(1(1(x'''')), 0(1(y'''')))
+'(+(x, 1(+(x'''', y''''))), 0(y0'')) -> +'(1(+(x'''', y'''')), 0(y0''))
+'(+(x, 1(+(x'''', +(x'''''', y'''''')))), 0(y''')) -> +'(1(+(x'''', +(x'''''', y''''''))), 0(y'''))
+'(+(x, 1(0(0(x'''''')))), 0(0(0(y'''''')))) -> +'(1(0(0(x''''''))), 0(0(0(y''''''))))
+'(+(x, 1(0(0(x'''''')))), 0(0(1(y'''''')))) -> +'(1(0(0(x''''''))), 0(0(1(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 0(0(0(y'''''')))) -> +'(1(0(1(x''''''))), 0(0(0(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 0(0(1(y'''''')))) -> +'(1(0(1(x''''''))), 0(0(1(y''''''))))
+'(+(x, 1(0(+(x'''''', y'''''')))), 0(0(y0''''))) -> +'(1(0(+(x'''''', y''''''))), 0(0(y0'''')))
+'(+(x, 1(0(+(x'''''', +(x'''''''', y''''''''))))), 0(0(y'''''))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 0(0(y''''')))
+'(+(x, 1(0(0(x'''''')))), 0(1(1(y'''''')))) -> +'(1(0(0(x''''''))), 0(1(1(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 0(1(0(y'''''')))) -> +'(1(0(1(x''''''))), 0(1(0(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 0(1(1(y'''''')))) -> +'(1(0(1(x''''''))), 0(1(1(y''''''))))
+'(+(x, 1(0(+(x'''''', y'''''')))), 0(1(y0''''))) -> +'(1(0(+(x'''''', y''''''))), 0(1(y0'''')))
+'(+(x, 1(0(+(x'''''', +(x'''''''', y''''''''))))), 0(1(y'''''))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 0(1(y''''')))
+'(+(x, 1(0(0(0(x''''''''))))), 0(1(0(0(y''''''''))))) -> +'(1(0(0(0(x'''''''')))), 0(1(0(0(y'''''''')))))
+'(+(x, 1(0(0(0(x''''''''))))), 0(1(0(1(y''''''''))))) -> +'(1(0(0(0(x'''''''')))), 0(1(0(1(y'''''''')))))
+'(+(x, 1(0(0(1(x''''''''))))), 0(1(0(0(y''''''''))))) -> +'(1(0(0(1(x'''''''')))), 0(1(0(0(y'''''''')))))
+'(+(x, 1(0(0(1(x''''''''))))), 0(1(0(1(y''''''''))))) -> +'(1(0(0(1(x'''''''')))), 0(1(0(1(y'''''''')))))
+'(+(x, 1(0(0(+(x'''''''', y''''''''))))), 0(1(0(y0'''''')))) -> +'(1(0(0(+(x'''''''', y'''''''')))), 0(1(0(y0''''''))))
+'(+(x, 1(0(0(+(x'''''''', +(x'''''''''', y'''''''''')))))), 0(1(0(y''''''')))) -> +'(1(0(0(+(x'''''''', +(x'''''''''', y''''''''''))))), 0(1(0(y'''''''))))
+'(+(x, 1(1(x''''))), 1(1(y''''))) -> +'(1(1(x'''')), 1(1(y'''')))
+'(+(x, 1(+(x'''', y''''))), 1(y0'')) -> +'(1(+(x'''', y'''')), 1(y0''))
+'(+(x, 1(+(x'''', +(x'''''', y'''''')))), 1(y''')) -> +'(1(+(x'''', +(x'''''', y''''''))), 1(y'''))
+'(+(x, 1(0(0(x'''''')))), 1(0(0(y'''''')))) -> +'(1(0(0(x''''''))), 1(0(0(y''''''))))
+'(+(x, 1(0(0(x'''''')))), 1(0(1(y'''''')))) -> +'(1(0(0(x''''''))), 1(0(1(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 1(0(0(y'''''')))) -> +'(1(0(1(x''''''))), 1(0(0(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 1(0(1(y'''''')))) -> +'(1(0(1(x''''''))), 1(0(1(y''''''))))
+'(+(x, 1(0(+(x'''''', y'''''')))), 1(0(y0''''))) -> +'(1(0(+(x'''''', y''''''))), 1(0(y0'''')))
+'(+(x, 1(0(+(x'''''', +(x'''''''', y''''''''))))), 1(0(y'''''))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))
+'(+(x, 1(0(0(x'''''')))), 1(1(1(y'''''')))) -> +'(1(0(0(x''''''))), 1(1(1(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 1(1(0(y'''''')))) -> +'(1(0(1(x''''''))), 1(1(0(y''''''))))
+'(+(x, 1(0(1(x'''''')))), 1(1(1(y'''''')))) -> +'(1(0(1(x''''''))), 1(1(1(y''''''))))
+'(+(x, 1(0(+(x'''''', y'''''')))), 1(1(y0''''))) -> +'(1(0(+(x'''''', y''''''))), 1(1(y0'''')))
+'(+(x, 1(0(+(x'''''', +(x'''''''', y''''''''))))), 1(1(y'''''))) -> +'(1(0(+(x'''''', +(x'''''''', y'''''''')))), 1(1(y''''')))
+'(+(x, 1(0(0(0(x''''''''))))), 1(1(0(0(y''''''''))))) -> +'(1(0(0(0(x'''''''')))), 1(1(0(0(y'''''''')))))
+'(+(x, 1(0(0(0(x''''''''))))), 1(1(0(1(y''''''''))))) -> +'(1(0(0(0(x'''''''')))), 1(1(0(1(y'''''''')))))
+'(+(x, 1(0(0(1(x''''''''))))), 1(1(0(0(y''''''''))))) -> +'(1(0(0(1(x'''''''')))), 1(1(0(0(y'''''''')))))
+'(+(x, 1(0(0(1(x''''''''))))), 1(1(0(1(y''''''''))))) -> +'(1(0(0(1(x'''''''')))), 1(1(0(1(y'''''''')))))
+'(+(x, 1(0(0(+(x'''''''', y''''''''))))), 1(1(0(y0'''''')))) -> +'(1(0(0(+(x'''''''', y'''''''')))), 1(1(0(y0''''''))))
+'(+(x, 1(0(0(+(x'''''''', +(x'''''''''', y'''''''''')))))), 1(1(0(y''''''')))) -> +'(1(0(0(+(x'''''''', +(x'''''''''', y''''''''''))))), 1(1(0(y'''''''))))
+'(+(x, 1(1(1(x'''''')))), 1(0(0(y'''''')))) -> +'(1(1(1(x''''''))), 1(0(0(y''''''))))
+'(+(x, 1(1(1(x'''''')))), 1(0(1(y'''''')))) -> +'(1(1(1(x''''''))), 1(0(1(y''''''))))
+'(+(x, 1(1(+(x'''''', y'''''')))), 1(0(y0''''))) -> +'(1(1(+(x'''''', y''''''))), 1(0(y0'''')))
+'(+(x, 1(1(+(x'''''', +(x'''''''', y''''''''))))), 1(0(y'''''))) -> +'(1(1(+(x'''''', +(x'''''''', y'''''''')))), 1(0(y''''')))
+'(+(x, 1(1(0(0(x''''''''))))), 1(0(0(0(y''''''''))))) -> +'(1(1(0(0(x'''''''')))), 1(0(0(0(y'''''''')))))
+'(+(x, 1(1(0(0(x''''''''))))), 1(0(0(1(y''''''''))))) -> +'(1(1(0(0(x'''''''')))), 1(0(0(1(y'''''''')))))
+'(+(x, 1(1(0(1(x''''''''))))), 1(0(0(0(y''''''''))))) -> +'(1(1(0(1(x'''''''')))), 1(0(0(0(y'''''''')))))
+'(+(x, 1(1(0(1(x''''''''))))), 1(0(0(1(y''''''''))))) -> +'(1(1(0(1(x'''''''')))), 1(0(0(1(y'''''''')))))
+'(+(x, 1(1(0(+(x'''''''', y''''''''))))), 1(0(0(y0'''''')))) -> +'(1(1(0(+(x'''''''', y'''''''')))), 1(0(0(y0''''''))))
+'(+(x, 1(1(0(+(x'''''''', +(x'''''''''', y'''''''''')))))), 1(0(0(y''''''')))) -> +'(1(1(0(+(x'''''''', +(x'''''''''', y''''''''''))))), 1(0(0(y'''''''))))
+'(+(x, 1(1(0(0(x''''''''))))), 1(0(1(1(y''''''''))))) -> +'(1(1(0(0(x'''''''')))), 1(0(1(1(y'''''''')))))
+'(+(x, 1(1(0(1(x''''''''))))), 1(0(1(0(y''''''''))))) -> +'(1(1(0(1(x'''''''')))), 1(0(1(0(y'''''''')))))
+'(+(x, 1(1(0(1(x''''''''))))), 1(0(1(1(y''''''''))))) -> +'(1(1(0(1(x'''''''')))), 1(0(1(1(y'''''''')))))
+'(+(x, 1(1(0(+(x'''''''', y''''''''))))), 1(0(1(y0'''''')))) -> +'(1(1(0(+(x'''''''', y'''''''')))), 1(0(1(y0''''''))))
+'(+(x, 1(1(0(+(x'''''''', +(x'''''''''', y'''''''''')))))), 1(0(1(y''''''')))) -> +'(1(1(0(+(x'''''''', +(x'''''''''', y''''''''''))))), 1(0(1(y'''''''))))
+'(+(x, 1(1(0(0(0(x'''''''''')))))), 1(0(1(0(0(y'''''''''')))))) -> +'(1(1(0(0(0(x''''''''''))))), 1(0(1(0(0(y''''''''''))))))
+'(+(x, 1(1(0(0(0(x'''''''''')))))), 1(0(1(0(1(y'''''''''')))))) -> +'(1(1(0(0(0(x''''''''''))))), 1(0(1(0(1(y''''''''''))))))
+'(+(x, 1(1(0(0(1(x'''''''''')))))), 1(0(1(0(0(y'''''''''')))))) -> +'(1(1(0(0(1(x''''''''''))))), 1(0(1(0(0(y''''''''''))))))
+'(+(x, 1(1(0(0(1(x'''''''''')))))), 1(0(1(0(1(y'''''''''')))))) -> +'(1(1(0(0(1(x''''''''''))))), 1(0(1(0(1(y''''''''''))))))
+'(+(x, 1(1(0(0(+(x'''''''''', y'''''''''')))))), 1(0(1(0(y0''''''''))))) -> +'(1(1(0(0(+(x'''''''''', y''''''''''))))), 1(0(1(0(y0'''''''')))))
+'(+(x, 1(1(0(0(+(x'''''''''', +(x'''''''''''', y''''''''''''))))))), 1(0(1(0(y'''''''''))))) -> +'(1(1(0(0(+(x'''''''''', +(x'''''''''''', y'''''''''''')))))), 1(0(1(0(y''''''''')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:50 minutes