Term Rewriting System R:
[x, y]
f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x)) -> G(x, s(x))
G(s(x), y) -> G(x, +(y, s(x)))
G(s(x), y) -> +'(y, s(x))
G(s(x), y) -> G(x, s(+(y, x)))
G(s(x), y) -> +'(y, x)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs.


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


Dependency Pair:

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


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




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

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

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


Dependency Pair:

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


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




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

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

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


Dependency Pair:

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


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

+'(x'''', s(s(s(y'''')))) -> +'(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(s(x1))=  1 + x1  
  POL(+'(x1, x2))=  1 + x2  

resulting in one new DP problem.



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


Dependency Pair:


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

G(s(x), y) -> G(x, s(+(y, x)))
G(s(x), y) -> G(x, +(y, s(x)))


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




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

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

G(s(x), y) -> G(x, s(+(y, x)))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

G(s(x), y) -> G(x, s(+(y, x)))
G(s(x), y) -> G(x, s(+(y, x)))


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




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

G(s(x), y) -> G(x, s(+(y, x)))
two new Dependency Pairs are created:

G(s(0), y') -> G(0, s(y'))
G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))
G(s(x), y) -> G(x, s(+(y, x)))


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




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

G(s(x), y) -> G(x, s(+(y, x)))
two new Dependency Pairs are created:

G(s(0), y') -> G(0, s(y'))
G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))
G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




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

G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))
two new Dependency Pairs are created:

G(s(s(0)), y0') -> G(s(0), s(s(y0')))
G(s(s(s(y'))), y0') -> G(s(s(y')), s(s(s(+(y0', y')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Rw
           →DP Problem 6
Nar
             ...
               →DP Problem 9
Polynomial Ordering


Dependency Pairs:

G(s(s(s(y'))), y0') -> G(s(s(y')), s(s(s(+(y0', y')))))
G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, y''))))


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

G(s(s(s(y'))), y0') -> G(s(s(y')), s(s(s(+(y0', y')))))
G(s(s(y'')), y0) -> G(s(y''), s(s(+(y0, 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(0)=  0  
  POL(G(x1, x2))=  1 + x1  
  POL(s(x1))=  1 + x1  
  POL(+(x1, x2))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Rw
           →DP Problem 6
Nar
             ...
               →DP Problem 10
Dependency Graph


Dependency Pair:


Rules:


f(0) -> 1
f(s(x)) -> g(x, s(x))
g(0, y) -> y
g(s(x), y) -> g(x, +(y, s(x)))
g(s(x), y) -> g(x, s(+(y, x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes