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))

Innermost 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
Narrowing Transformation


Dependency Pairs:

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


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(:, app(app(g, z), y)), app(app(+, x), a))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Narrowing Transformation


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


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))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(:, z), app(app(+, x), app(f, y))) -> APP(app(g, z), y)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Argument Filtering and Ordering


Dependency Pairs:

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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

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))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
: > +

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> x1


   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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(:, 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))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:04 minutes