Term Rewriting System R:
[x, y, z]
f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, g(y)) -> F(h(x), i(x, y))
F(x, g(y)) -> I(x, y)
I(x, j(y, z)) -> J(g(y), i(x, z))
I(x, j(y, z)) -> I(x, z)
I(h(x), j(j(y, z), 0)) -> J(i(h(x), j(y, z)), i(x, j(y, z)))
I(h(x), j(j(y, z), 0)) -> I(h(x), j(y, z))
I(h(x), j(j(y, z), 0)) -> I(x, j(y, z))
J(g(x), g(y)) -> J(x, y)

Furthermore, R contains three SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Polo
       →DP Problem 3
Inst


Dependency Pair:

J(g(x), g(y)) -> J(x, y)


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





The following dependency pair can be strictly oriented:

J(g(x), g(y)) -> J(x, y)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(J(x1, x2))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 4
Dependency Graph
       →DP Problem 2
Polo
       →DP Problem 3
Inst


Dependency Pair:


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polynomial Ordering
       →DP Problem 3
Inst


Dependency Pairs:

I(h(x), j(j(y, z), 0)) -> I(x, j(y, z))
I(h(x), j(j(y, z), 0)) -> I(h(x), j(y, z))
I(x, j(y, z)) -> I(x, z)


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





The following dependency pair can be strictly oriented:

I(h(x), j(j(y, z), 0)) -> I(x, j(y, z))


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(I(x1, x2))=  x1  
  POL(0)=  0  
  POL(g(x1))=  0  
  POL(h(x1))=  1 + x1  
  POL(j(x1, x2))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 5
Polynomial Ordering
       →DP Problem 3
Inst


Dependency Pairs:

I(h(x), j(j(y, z), 0)) -> I(h(x), j(y, z))
I(x, j(y, z)) -> I(x, z)


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





The following dependency pair can be strictly oriented:

I(h(x), j(j(y, z), 0)) -> I(h(x), j(y, z))


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

j(g(x), g(y)) -> g(j(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(I(x1, x2))=  1 + x1 + x2  
  POL(0)=  1  
  POL(g(x1))=  0  
  POL(h(x1))=  0  
  POL(j(x1, x2))=  x1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 5
Polo
             ...
               →DP Problem 6
Polynomial Ordering
       →DP Problem 3
Inst


Dependency Pair:

I(x, j(y, z)) -> I(x, z)


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





The following dependency pair can be strictly oriented:

I(x, j(y, z)) -> I(x, z)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(I(x1, x2))=  x2  
  POL(j(x1, x2))=  1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 5
Polo
             ...
               →DP Problem 7
Dependency Graph
       →DP Problem 3
Inst


Dependency Pair:


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Instantiation Transformation


Dependency Pair:

F(x, g(y)) -> F(h(x), i(x, y))


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





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

F(x, g(y)) -> F(h(x), i(x, y))
one new Dependency Pair is created:

F(h(x''''), g(y')) -> F(h(h(x'''')), i(h(x''''), y'))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Inst
           →DP Problem 8
Instantiation Transformation


Dependency Pair:

F(h(x''''), g(y')) -> F(h(h(x'''')), i(h(x''''), y'))


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





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

F(h(x''''), g(y')) -> F(h(h(x'''')), i(h(x''''), y'))
one new Dependency Pair is created:

F(h(h(x'''''')), g(y'')) -> F(h(h(h(x''''''))), i(h(h(x'''''')), y''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Inst
           →DP Problem 8
Inst
             ...
               →DP Problem 9
Instantiation Transformation


Dependency Pair:

F(h(h(x'''''')), g(y'')) -> F(h(h(h(x''''''))), i(h(h(x'''''')), y''))


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





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

F(h(h(x'''''')), g(y'')) -> F(h(h(h(x''''''))), i(h(h(x'''''')), y''))
one new Dependency Pair is created:

F(h(h(h(x''''''''))), g(y''')) -> F(h(h(h(h(x'''''''')))), i(h(h(h(x''''''''))), y'''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Inst
           →DP Problem 8
Inst
             ...
               →DP Problem 10
Instantiation Transformation


Dependency Pair:

F(h(h(h(x''''''''))), g(y''')) -> F(h(h(h(h(x'''''''')))), i(h(h(h(x''''''''))), y'''))


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





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

F(h(h(h(x''''''''))), g(y''')) -> F(h(h(h(h(x'''''''')))), i(h(h(h(x''''''''))), y'''))
one new Dependency Pair is created:

F(h(h(h(h(x'''''''''')))), g(y'''')) -> F(h(h(h(h(h(x''''''''''))))), i(h(h(h(h(x'''''''''')))), y''''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Inst
           →DP Problem 8
Inst
             ...
               →DP Problem 11
Instantiation Transformation


Dependency Pair:

F(h(h(h(h(x'''''''''')))), g(y'''')) -> F(h(h(h(h(h(x''''''''''))))), i(h(h(h(h(x'''''''''')))), y''''))


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))





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

F(h(h(h(h(x'''''''''')))), g(y'''')) -> F(h(h(h(h(h(x''''''''''))))), i(h(h(h(h(x'''''''''')))), y''''))
one new Dependency Pair is created:

F(h(h(h(h(h(x''''''''''''))))), g(y''''')) -> F(h(h(h(h(h(h(x'''''''''''')))))), i(h(h(h(h(h(x''''''''''''))))), y'''''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Inst
           →DP Problem 8
Inst
             ...
               →DP Problem 12
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(h(h(h(h(h(x''''''''''''))))), g(y''''')) -> F(h(h(h(h(h(h(x'''''''''''')))))), i(h(h(h(h(h(x''''''''''''))))), y'''''))


Rules:


f(x, g(y)) -> f(h(x), i(x, y))
i(x, j(0, 0)) -> g(0)
i(x, j(y, z)) -> j(g(y), i(x, z))
i(h(x), j(j(y, z), 0)) -> j(i(h(x), j(y, z)), i(x, j(y, z)))
j(g(x), g(y)) -> g(j(x, y))




Termination of R could not be shown.
Duration:
0:00 minutes