Term Rewriting System R:

   [x, y, z]
   app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
   APP(app(*, x), app(app(+, y), z)) -> APP(+, app(app(*, x), y))
   APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
   APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)

Furthermore, R contains one SCC.


SCC1:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rule can be oriented: 

   app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(*) = 1
POL(APP(x_1, x_2)) = x_1
POL(+) = 0
POL(app(x_1, x_2)) = x_1

 resulting in one subcycle.


SCC1.Polo1:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(*) = 1
POL(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(+) = 1
POL(app(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
   APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.741 seconds.