Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
F(s(x), y) -> F(p(-(s(x), y)), p(-(y, s(x))))
F(s(x), y) -> P(-(s(x), y))
F(s(x), y) -> -'(s(x), y)
F(s(x), y) -> P(-(y, s(x)))
F(s(x), y) -> -'(y, s(x))
F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))
F(x, s(y)) -> P(-(x, s(y)))
F(x, s(y)) -> -'(x, s(y))
F(x, s(y)) -> P(-(s(y), x))
F(x, s(y)) -> -'(s(y), x)

Furthermore, R contains two SCCs.


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


Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

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

-'(s(s(x'')), s(s(y''))) -> -'(s(x''), s(y''))

The transformation is resulting in one new DP problem:



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


Dependency Pair:

-'(s(s(x'')), s(s(y''))) -> -'(s(x''), s(y''))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

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

-'(s(s(s(x''''))), s(s(s(y'''')))) -> -'(s(s(x'''')), s(s(y'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
FwdInst
             ...
               →DP Problem 4
Polynomial Ordering
       →DP Problem 2
Nar


Dependency Pair:

-'(s(s(s(x''''))), s(s(s(y'''')))) -> -'(s(s(x'''')), s(s(y'''')))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

-'(s(s(s(x''''))), s(s(s(y'''')))) -> -'(s(s(x'''')), s(s(y'''')))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
FwdInst
             ...
               →DP Problem 5
Dependency Graph
       →DP Problem 2
Nar


Dependency Pair:


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Narrowing Transformation


Dependency Pairs:

F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))
F(s(x), y) -> F(p(-(s(x), y)), p(-(y, s(x))))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(x), y) -> F(p(-(s(x), y)), p(-(y, s(x))))
three new Dependency Pairs are created:

F(s(x''), 0) -> F(p(s(x'')), p(-(0, s(x''))))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
F(s(x0), s(x'')) -> F(p(-(s(x0), s(x''))), p(-(x'', x0)))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rewriting Transformation


Dependency Pairs:

F(s(x0), s(x'')) -> F(p(-(s(x0), s(x''))), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
F(s(x''), 0) -> F(p(s(x'')), p(-(0, s(x''))))
F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(x''), 0) -> F(p(s(x'')), p(-(0, s(x''))))
one new Dependency Pair is created:

F(s(x''), 0) -> F(x'', p(-(0, s(x''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 7
Rewriting Transformation


Dependency Pairs:

F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))
F(s(x0), s(x'')) -> F(p(-(s(x0), s(x''))), p(-(x'', x0)))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
one new Dependency Pair is created:

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 8
Rewriting Transformation


Dependency Pairs:

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x0), s(x'')) -> F(p(-(s(x0), s(x''))), p(-(x'', x0)))
F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))
F(s(x''), 0) -> F(x'', p(-(0, s(x''))))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(x0), s(x'')) -> F(p(-(s(x0), s(x''))), p(-(x'', x0)))
one new Dependency Pair is created:

F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 9
Narrowing Transformation


Dependency Pairs:

F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(x, s(y)) -> F(p(-(x, s(y))), p(-(s(y), x)))
three new Dependency Pairs are created:

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
F(0, s(y')) -> F(p(-(0, s(y'))), p(s(y')))
F(s(y''), s(y0)) -> F(p(-(s(y''), s(y0))), p(-(y0, y'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 10
Rewriting Transformation


Dependency Pairs:

F(s(y''), s(y0)) -> F(p(-(s(y''), s(y0))), p(-(y0, y'')))
F(0, s(y')) -> F(p(-(0, s(y'))), p(s(y')))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(s(y''), s(x''))))
one new Dependency Pair is created:

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 11
Rewriting Transformation


Dependency Pairs:

F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(0, s(y')) -> F(p(-(0, s(y'))), p(s(y')))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
F(s(y''), s(y0)) -> F(p(-(s(y''), s(y0))), p(-(y0, y'')))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(0, s(y')) -> F(p(-(0, s(y'))), p(s(y')))
one new Dependency Pair is created:

F(0, s(y')) -> F(p(-(0, s(y'))), y')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 12
Rewriting Transformation


Dependency Pairs:

F(0, s(y')) -> F(p(-(0, s(y'))), y')
F(s(y''), s(y0)) -> F(p(-(s(y''), s(y0))), p(-(y0, y'')))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(y''), s(y0)) -> F(p(-(s(y''), s(y0))), p(-(y0, y'')))
one new Dependency Pair is created:

F(s(y''), s(y0)) -> F(p(-(y'', y0)), p(-(y0, y'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 13
Narrowing Transformation


Dependency Pairs:

F(s(y''), s(y0)) -> F(p(-(y'', y0)), p(-(y0, y'')))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
F(0, s(y')) -> F(p(-(0, s(y'))), y')


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(s(x''), 0) -> F(x'', p(-(0, s(x''))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 14
Narrowing Transformation


Dependency Pairs:

F(0, s(y')) -> F(p(-(0, s(y'))), y')
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(y''), s(y0)) -> F(p(-(y'', y0)), p(-(y0, y'')))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




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

F(0, s(y')) -> F(p(-(0, s(y'))), y')
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 15
Polynomial Ordering


Dependency Pairs:

F(s(y''), s(y0)) -> F(p(-(y'', y0)), p(-(y0, y'')))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(s(y''), s(y0)) -> F(p(-(y'', y0)), p(-(y0, y'')))
F(s(x0), s(x'')) -> F(p(-(x0, x'')), p(-(x'', x0)))
F(s(x''), s(y'')) -> F(p(-(x'', y'')), p(-(y'', x'')))


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

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  1  
  POL(s(x1))=  1 + x1  
  POL(-(x1, x2))=  x1  
  POL(F(x1, x2))=  x1  
  POL(p(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Nar
           →DP Problem 6
Rw
             ...
               →DP Problem 16
Dependency Graph


Dependency Pair:


Rules:


-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes