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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(f(x, y), z) -> G(y, z)
G(h(x, y), z) -> G(x, f(y, z))
G(x, h(y, z)) -> G(x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

G(x, h(y, z)) -> G(x, y)
G(h(x, y), z) -> G(x, f(y, z))
G(f(x, y), z) -> G(y, z)


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





The following dependency pair can be strictly oriented:

G(f(x, y), z) -> G(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(G(x1, x2))=  x1  
  POL(h(x1, x2))=  x1  
  POL(f(x1, x2))=  1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Dependency Graph


Dependency Pairs:

G(x, h(y, z)) -> G(x, y)
G(h(x, y), z) -> G(x, f(y, z))


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





Using the Dependency Graph the DP problem was split into 2 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pair:

G(h(x, y), z) -> G(x, f(y, z))


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





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

G(h(x, y), z) -> G(x, f(y, z))
one new Dependency Pair is created:

G(h(x'', y0), f(y'', z'')) -> G(x'', f(y0, f(y'', z'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 5
                 ↳Instantiation Transformation


Dependency Pair:

G(h(x'', y0), f(y'', z'')) -> G(x'', f(y0, f(y'', z'')))


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





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

G(h(x'', y0), f(y'', z'')) -> G(x'', f(y0, f(y'', z'')))
one new Dependency Pair is created:

G(h(x'''', y0''), f(y''0, f(y'''', z''''))) -> G(x'''', f(y0'', f(y''0, f(y'''', z''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 6
                 ↳Polynomial Ordering


Dependency Pair:

G(h(x'''', y0''), f(y''0, f(y'''', z''''))) -> G(x'''', f(y0'', f(y''0, f(y'''', z''''))))


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





The following dependency pair can be strictly oriented:

G(h(x'''', y0''), f(y''0, f(y'''', z''''))) -> G(x'''', f(y0'', f(y''0, f(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(G(x1, x2))=  x1  
  POL(h(x1, x2))=  1 + x1  
  POL(f(x1, x2))=  0  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pair:

G(x, h(y, z)) -> G(x, y)


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





The following dependency pair can be strictly oriented:

G(x, h(y, z)) -> G(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, x2))=  x2  
  POL(h(x1, x2))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 7
                 ↳Dependency Graph


Dependency Pair:


Rules:


g(f(x, y), z) -> f(x, g(y, z))
g(h(x, y), z) -> g(x, f(y, z))
g(x, h(y, z)) -> h(g(x, y), z)





Using the Dependency Graph resulted in no new DP problems.

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