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
         ↳Usable Rules (Innermost)


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




As we are in the innermost case, we can delete all 5 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Size-Change Principle


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)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(h(x, y)) -> F(y)
  2. F(h(x, y)) -> F(x)
  3. F(g(x, y)) -> F(y)
  4. F(g(x, y)) -> F(x)
and get the following Size-Change Graph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
g(x1, x2) -> g(x1, x2)
h(x1, x2) -> h(x1, x2)

We obtain no new DP problems.

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