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

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

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

where the Polynomial interpretation:
  POL(g)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(f)=  1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳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
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →DP Problem 1
                 ↳Usable Rules (Innermost)


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →DP Problem 2
                 ↳A-Transformation


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →DP Problem 3
                 ↳Size-Change Principle


Dependency Pair:

F(g(x)) -> F(x)


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1} , {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:
g(x1) -> g(x1)

We obtain no new DP problems.

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