Term Rewriting System R:
[x, y]
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x, y)) -> G(f(x), f(y))
F(g(x, y)) -> F(x)
F(g(x, y)) -> F(y)
F(h(x, y)) -> G(h(y, f(x)), h(x, f(y)))
F(h(x, y)) -> F(x)
F(h(x, y)) -> F(y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1, x2))=  x1 + x2  
  POL(h(x1, x2))=  1 + x1 + x2  
  POL(F(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
h(x1, x2) -> h(x1, x2)
g(x1, x2) -> g(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(F(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
g(x1, x2) -> g(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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