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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(g(x)) -> G(h(g(x)))
G(g(x)) -> H(g(x))
H(h(x)) -> H(f(h(x), x))

Furthermore, R contains one SCC. 


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


Dependency Pair:

G(g(x)) -> G(h(g(x)))


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(g(x)) -> G(h(g(x)))


The following usable rule for innermost w.r.t. to the AFS can be oriented:

h(h(x)) -> h(f(h(x), x))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
g > h

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


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


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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