Term Rewriting System R:
[z, x, y]
h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

H(z, e(x)) -> H(c(z), d(z, x))
H(z, e(x)) -> D(z, x)
D(z, g(x, y)) -> G(e(x), d(z, y))
D(z, g(x, y)) -> D(z, y)
D(c(z), g(g(x, y), 0)) -> G(d(c(z), g(x, y)), d(z, g(x, y)))
D(c(z), g(g(x, y), 0)) -> D(c(z), g(x, y))
D(c(z), g(g(x, y), 0)) -> D(z, g(x, y))
G(e(x), e(y)) -> G(x, y)

Furthermore, R contains three SCCs.


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


Dependency Pair:

G(e(x), e(y)) -> G(x, y)


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(e(x), e(y)) -> G(x, y)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1, x2))=  x1  
  POL(e(x1))=  1 + 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
Nar


Dependency Pair:


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

D(c(z), g(g(x, y), 0)) -> D(z, g(x, y))
D(c(z), g(g(x, y), 0)) -> D(c(z), g(x, y))
D(z, g(x, y)) -> D(z, y)


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

D(c(z), g(g(x, y), 0)) -> D(z, g(x, y))


Additionally, the following usable rule for innermost can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c(x1))=  1 + x1  
  POL(0)=  0  
  POL(g(x1, x2))=  0  
  POL(e(x1))=  0  
  POL(D(x1, x2))=  x1  

resulting in one new DP problem.



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


Dependency Pairs:

D(c(z), g(g(x, y), 0)) -> D(c(z), g(x, y))
D(z, g(x, y)) -> D(z, y)


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

D(c(z), g(g(x, y), 0)) -> D(c(z), g(x, y))


Additionally, the following usable rule for innermost can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c(x1))=  0  
  POL(0)=  1  
  POL(g(x1, x2))=  x1 + x2  
  POL(e(x1))=  0  
  POL(D(x1, x2))=  1 + 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
Nar


Dependency Pair:

D(z, g(x, y)) -> D(z, y)


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

D(z, g(x, y)) -> D(z, y)


There are no usable rules for innermost that need to be oriented.

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


Dependency Pair:


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

H(z, e(x)) -> H(c(z), d(z, x))


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




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

H(z, e(x)) -> H(c(z), d(z, x))
three new Dependency Pairs are created:

H(z'', e(g(0, 0))) -> H(c(z''), e(0))
H(z'', e(g(x'', y'))) -> H(c(z''), g(e(x''), d(z'', y')))
H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))
H(z'', e(g(x'', y'))) -> H(c(z''), g(e(x''), d(z'', y')))


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




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

H(z'', e(g(x'', y'))) -> H(c(z''), g(e(x''), d(z'', y')))
two new Dependency Pairs are created:

H(c(z''''), e(g(x''', y''))) -> H(c(c(z'''')), g(e(x'''), d(c(z''''), y'')))
H(c(c(z'''')), e(g(x''', y''))) -> H(c(c(c(z''''))), g(e(x'''), d(c(c(z'''')), y'')))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

H(c(c(z'''')), e(g(x''', y''))) -> H(c(c(c(z''''))), g(e(x'''), d(c(c(z'''')), y'')))
H(c(z''''), e(g(x''', y''))) -> H(c(c(z'''')), g(e(x'''), d(c(z''''), y'')))
H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost




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

H(c(z''), e(g(g(x'', y'), 0))) -> H(c(c(z'')), g(d(c(z''), g(x'', y')), d(z'', g(x'', y'))))
three new Dependency Pairs are created:

H(c(c(z'''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''))), g(d(c(c(z'''')), g(x''', y'')), d(c(z''''), g(x''', y''))))
H(c(c(z'''''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''''))), g(d(c(c(z'''''')), g(x''', y'')), d(c(z''''''), g(x''', y''))))
H(c(c(c(z''''''))), e(g(g(x''', y''), 0))) -> H(c(c(c(c(z'''''')))), g(d(c(c(c(z''''''))), g(x''', y'')), d(c(c(z'''''')), g(x''', y''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Nar
           →DP Problem 8
Inst
             ...
               →DP Problem 10
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

H(c(c(c(z''''''))), e(g(g(x''', y''), 0))) -> H(c(c(c(c(z'''''')))), g(d(c(c(c(z''''''))), g(x''', y'')), d(c(c(z'''''')), g(x''', y''))))
H(c(c(z'''''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''''))), g(d(c(c(z'''''')), g(x''', y'')), d(c(z''''''), g(x''', y''))))
H(c(c(z'''')), e(g(g(x''', y''), 0))) -> H(c(c(c(z''''))), g(d(c(c(z'''')), g(x''', y'')), d(c(z''''), g(x''', y''))))
H(c(z''''), e(g(x''', y''))) -> H(c(c(z'''')), g(e(x'''), d(c(z''''), y'')))
H(c(c(z'''')), e(g(x''', y''))) -> H(c(c(c(z''''))), g(e(x'''), d(c(c(z'''')), y'')))


Rules:


h(z, e(x)) -> h(c(z), d(z, x))
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) -> e(g(x, y))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:03 minutes