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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. H(x, y) -> G(x, f(y))
  2. G(f(x), y) -> H(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
{2} , {2}
1>1
2=2

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
f(x1) -> f(x1)

We obtain no new DP problems.

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