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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳A-Transformation


Dependency Pairs:

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


Rules:


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


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
     ↳DPs
       →DP Problem 1
         ↳ATrans
           →DP Problem 2
             ↳Negative Polynomial Order


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following Dependency Pairs can be strictly oriented using the given order.

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


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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


Used ordering:
Polynomial Order with Interpretation:

POL( F(x1) ) = x1

POL( g(x1) ) = x1 + 1

POL( f(x1) ) = x1

POL( h(x1) ) = 0


This results in one new DP problem. 


   R
     ↳DPs
       →DP Problem 1
         ↳ATrans
           →DP Problem 2
             ↳Neg POLO
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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