Term Rewriting System R:
[x, y]
app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(f, app(s, x)) -> APP(f, x)
APP(app(g, x), app(c, y)) -> APP(c, app(app(g, x), y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(if, app(f, x))
APP(app(g, x), app(c, y)) -> APP(f, x)
APP(app(g, x), app(c, y)) -> APP(c, app(app(g, app(s, x)), y))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(g, app(s, x))
APP(app(g, x), app(c, y)) -> APP(s, x)

Furthermore, R contains two SCCs.


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


Dependency Pair:

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


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> app(x1, x2)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 3
Dependency Graph
       →DP Problem 2
Nar


Dependency Pair:


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))


Strategy:

innermost




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

APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
five new Dependency Pairs are created:

APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Nar
           →DP Problem 4
Narrowing Transformation


Dependency Pairs:

APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))


Strategy:

innermost




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

APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
five new Dependency Pairs are created:

APP(app(g, 0), app(c, y)) -> APP(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y))
APP(app(g, 1), app(c, y)) -> APP(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y'')))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 5
Narrowing Transformation


Dependency Pairs:

APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y'')))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y'')))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y))
APP(app(g, 1), app(c, y)) -> APP(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y))
APP(app(g, 0), app(c, y)) -> APP(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))


Strategy:

innermost




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

APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
five new Dependency Pairs are created:

APP(app(g, 0), app(c, y)) -> APP(app(if, true), app(c, app(app(g, app(s, 0)), y)))
APP(app(g, 1), app(c, y)) -> APP(app(if, false), app(c, app(app(g, app(s, 1)), y)))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y''))))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y'')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 6
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y'')))))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y''))))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y)))
APP(app(g, 1), app(c, y)) -> APP(app(if, false), app(c, app(app(g, app(s, 1)), y)))
APP(app(g, 0), app(c, y)) -> APP(app(if, true), app(c, app(app(g, app(s, 0)), y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y'')))
APP(app(g, 1), app(c, y)) -> APP(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y))
APP(app(g, 0), app(c, y)) -> APP(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y))
APP(app(g, app(s, x'')), app(c, y)) -> APP(app(g, app(s, x'')), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, app(s, x''))), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y''))))
APP(app(g, 1), app(c, y)) -> APP(app(g, 1), app(app(app(if, false), app(c, app(app(g, app(s, 1)), y))), app(c, y)))
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(g, x''), app(app(app(if, app(f, x'')), app(c, app(c, app(app(g, app(s, x'')), y'')))), app(c, app(c, y''))))
APP(app(g, 0), app(c, y)) -> APP(app(g, 0), app(app(app(if, true), app(c, app(app(g, app(s, 0)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x''), app(c, app(c, y''))) -> APP(app(app(if, app(f, x'')), app(c, app(app(g, app(s, x'')), app(app(app(if, app(f, app(s, x''))), app(c, app(app(g, app(s, app(s, x''))), y''))), app(c, y''))))), app(c, app(c, y'')))


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:18 minutes