Term Rewriting System R:
[xs, ys, p, x]
app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(if, app(p, x))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(cons, x), app(app(filter, p), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Rules:


app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Strategy:

innermost




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

APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
eight new Dependency Pairs are created:

APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs))), app(app(filter, app(app(if, true), xs'')), xs))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs))), app(app(filter, app(app(if, false), xs'')), xs))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(app(if, app(p'', x)), app(app(cons, x), nil)), app(app(filter, p''), nil))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(filter, p''), app(app(cons, x''), xs'')))
APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(filter, p''), nil))), nil)
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(filter, p''), app(app(cons, x''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(filter, p''), app(app(cons, x''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(filter, p''), app(app(cons, x''), xs'')))
APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(app(if, app(p'', x)), app(app(cons, x), nil)), app(app(filter, p''), nil))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs))), app(app(filter, app(app(if, false), xs'')), xs))
APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs))), app(app(filter, app(app(if, true), xs'')), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)


Rules:


app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Strategy:

innermost




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

APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(app(if, app(p'', x)), app(app(cons, x), nil)), app(app(filter, p''), nil))
one new Dependency Pair is created:

APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(app(if, app(p'', x)), app(app(cons, x), nil)), nil)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Rewriting Transformation


Dependency Pairs:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(filter, p''), app(app(cons, x''), xs'')))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs))), app(app(filter, app(app(if, false), xs'')), xs))
APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs))), app(app(filter, app(app(if, true), xs'')), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(filter, p''), app(app(cons, x''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))


Rules:


app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Strategy:

innermost




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

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(filter, p''), app(app(cons, x''), xs'')))
one new Dependency Pair is created:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Rewriting Transformation


Dependency Pairs:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(filter, p''), app(app(cons, x''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs))), app(app(filter, app(app(if, false), xs'')), xs))
APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs))), app(app(filter, app(app(if, true), xs'')), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))


Rules:


app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Strategy:

innermost




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

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(filter, p''), app(app(cons, x''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))
one new Dependency Pair is created:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 5
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs))), app(app(filter, app(app(if, false), xs'')), xs))
APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs))), app(app(filter, app(app(if, true), xs'')), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))


Rules:


app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Strategy:

innermost




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

APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
six new Dependency Pairs are created:

APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs)))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs)))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs)))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs)))
APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(if, app(p'', x)), app(app(cons, x), nil))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 6
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))))
APP(app(filter, p''), app(app(cons, x), nil)) -> APP(app(if, app(p'', x)), app(app(cons, x), nil))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs)))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs)))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs)))
APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs)))
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))
APP(app(filter, app(filter, p'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(app(if, app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs''))), app(app(cons, app(app(cons, x''), xs'')), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(filter, p'')), app(app(cons, nil), xs)) -> APP(app(app(if, nil), app(app(cons, nil), app(app(filter, app(filter, p'')), xs))), app(app(filter, app(filter, p'')), xs))
APP(app(filter, app(app(if, false), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, x'), app(app(cons, x'), app(app(filter, app(app(if, false), xs'')), xs))), app(app(filter, app(app(if, false), xs'')), xs))
APP(app(filter, app(app(if, true), xs'')), app(app(cons, x'), xs)) -> APP(app(app(if, xs''), app(app(cons, x'), app(app(filter, app(app(if, true), xs'')), xs))), app(app(filter, app(app(if, true), xs'')), xs))
APP(app(filter, p), app(app(cons, x), xs)) -> APP(app(filter, p), xs)
APP(app(filter, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(filter, p''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(app(if, app(p'', x)), app(app(cons, x), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))), app(app(app(if, app(p'', x'')), app(app(cons, x''), app(app(filter, p''), xs''))), app(app(filter, p''), xs'')))


Rules:


app(app(app(if, true), xs), ys) -> xs
app(app(app(if, false), xs), ys) -> ys
app(app(filter, p), nil) -> nil
app(app(filter, p), app(app(cons, x), xs)) -> app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:28 minutes