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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(:, app(app(:, x), y)), z) -> APP(app(:, x), app(app(:, y), z))
APP(app(:, app(app(:, x), y)), z) -> APP(app(:, y), z)
APP(app(:, app(app(:, x), y)), z) -> APP(:, y)
APP(app(:, app(app(+, x), y)), z) -> APP(app(+, app(app(:, x), z)), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) -> APP(+, app(app(:, x), z))
APP(app(:, app(app(+, x), y)), z) -> APP(app(:, x), z)
APP(app(:, app(app(+, x), y)), z) -> APP(:, x)
APP(app(:, app(app(+, x), y)), z) -> APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) -> APP(:, y)
APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(:, app(app(g, z), y)), app(app(+, x), a))
APP(app(:, z), app(app(+, x), app(f, y))) -> APP(:, app(app(g, z), y))
APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(g, z), y)
APP(app(:, z), app(app(+, x), app(f, y))) -> APP(g, z)
APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(+, x), a)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a))





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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


app(app(:, app(app(:, x), y)), z) -> app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) -> app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) -> app(app(:, app(app(g, z), y)), app(app(+, x), a))


It is proper and hence, it can be A-transformed which results in the DP problem

:'(+(x, y), z) -> :'(y, z)
:'(+(x, y), z) -> :'(x, z)
:'(:(x, y), z) -> :'(y, z)
:'(:(x, y), z) -> :'(x, :(y, z))


:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))


We number the DPs as follows:
  1. :'(+(x, y), z) -> :'(y, z)
  2. :'(+(x, y), z) -> :'(x, z)
  3. :'(:(x, y), z) -> :'(y, z)
  4. :'(:(x, y), z) -> :'(x, :(y, z))
and get the following Size-Change Graph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2=2
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2=2
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems.

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