Term Rewriting System R:
[x, y, z]
app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) -> APP(f, app(s, x))
APP(app(app(f, 0), 1), x) -> APP(s, x)
APP(app(app(f, x), y), app(s, z)) -> APP(s, app(app(app(f, 0), 1), z))
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, x), y), app(s, z)) -> APP(f, 0)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)


Rules:


app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)


The following usable rules for innermost can be oriented:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


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

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


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 2
Argument Filtering and Ordering


Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)


Rules:


app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)


The following usable rules for innermost can be oriented:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


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

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


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 2
AFS
             ...
               →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)


Rules:


app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))


Strategy:

innermost



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