Term Rewriting System R:
[f, x]
app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(F, app(app(F, f), x)), x) -> APP(app(F, app(G, app(app(F, f), x))), app(f, x))
APP(app(F, app(app(F, f), x)), x) -> APP(F, app(G, app(app(F, f), x)))
APP(app(F, app(app(F, f), x)), x) -> APP(G, app(app(F, f), x))
APP(app(F, app(app(F, f), x)), x) -> APP(f, x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)


Rule:


app(app(F, app(app(F, f), x)), x) -> app(app(F, app(G, app(app(F, f), x))), app(f, x))


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


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


Dependency Pair:

APP(app(F, app(app(F, f), x)), x) -> APP(f, x)


Rule:

none


Strategy:

innermost




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

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

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems.

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