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)

Innermost 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)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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(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)


Strategy:

innermost




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
                 ↳Polynomial Ordering


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)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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(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 5
                 ↳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)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   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)


Strategy:

innermost




The following dependency pair can be strictly oriented:

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

resulting in one new DP problem.


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