Term Rewriting System R:
[y, x, z]
app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(le, x)
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(eq, app(s, x)), app(s, y)) -> APP(eq, x)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(minsort, app(app(cons, x), y)) -> APP(cons, app(app(min, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(min, x)
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(del, app(app(min, x), y))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(if, app(app(le, x), y))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(le, x)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(min, y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(if, app(app(eq, x), y))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(eq, x)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, x), app(app(cons, y), z)) -> APP(app(if, app(app(eq, x), y)), z)
four new Dependency Pairs are created:

APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, x), app(app(cons, y), z)) -> APP(app(cons, y), app(app(del, x), z))
two new Dependency Pairs are created:

APP(app(del, x''), app(app(cons, y), nil)) -> APP(app(cons, y), nil)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, 0), app(app(cons, 0), z)) -> APP(app(if, true), z)
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 4
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, 0), app(app(cons, app(s, y'')), z)) -> APP(app(if, false), z)
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 5
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, x'')), app(app(cons, 0), z)) -> APP(app(if, false), z)
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 6
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, x'')), app(app(cons, app(s, y'')), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
four new Dependency Pairs are created:

APP(app(del, app(s, 0)), app(app(cons, app(s, 0)), z)) -> APP(app(if, true), z)
APP(app(del, app(s, 0)), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, 0)), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, app(app(eq, x'), y')), z)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, app(app(eq, x'), y')), z)
APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, 0)), z)) -> APP(app(if, false), z)
APP(app(del, app(s, 0)), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, 0)), app(app(cons, app(s, 0)), z)) -> APP(app(if, true), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, 0)), app(app(cons, app(s, 0)), z)) -> APP(app(if, true), z)
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 8
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, 0)), z)) -> APP(app(if, false), z)
APP(app(del, app(s, 0)), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, false), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, app(app(eq, x'), y')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, 0)), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, false), z)
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 9
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, app(app(eq, x'), y')), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, 0)), z)) -> APP(app(if, false), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, 0)), z)) -> APP(app(if, false), z)
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 10
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, app(app(eq, x'), y')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, x'))), app(app(cons, app(s, app(s, y'))), z)) -> APP(app(if, app(app(eq, x'), y')), z)
four new Dependency Pairs are created:

APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, true), z)
APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 11
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, true), z)
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 12
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, false), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, 0))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, false), z)
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 13
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, false), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, 0))), z)) -> APP(app(if, false), z)
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 14
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, x'')))), app(app(cons, app(s, app(s, app(s, y'')))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
four new Dependency Pairs are created:

APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, true), z)
APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, app(app(eq, x'), y')), z)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 15
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, app(app(eq, x'), y')), z)
APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, true), z)
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 16
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, false), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, app(app(eq, x'), y')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, 0)))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, false), z)
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 17
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, app(app(eq, x'), y')), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, false), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, 0)))), z)) -> APP(app(if, false), z)
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 18
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, app(app(eq, x'), y')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, app(s, x'))))), app(app(cons, app(s, app(s, app(s, app(s, y'))))), z)) -> APP(app(if, app(app(eq, x'), y')), z)
four new Dependency Pairs are created:

APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, true), z)
APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 19
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, true), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, true), z)
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 20
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, false), z)
APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, false), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, app(s, 0))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, false), z)
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 21
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)
APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, false), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))





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

APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, 0))))), z)) -> APP(app(if, false), z)
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 22
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(del, x''), app(app(cons, y), app(app(cons, y''), z''))) -> APP(app(cons, y), app(app(app(if, app(app(eq, x''), y'')), z''), app(app(cons, y''), app(app(del, x''), z''))))
APP(app(del, x), app(app(cons, y), z)) -> APP(app(del, x), z)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) -> APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) -> APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(minsort, app(app(cons, x), y)) -> APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) -> APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) -> APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) -> APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(app(eq, app(s, x)), app(s, y)) -> APP(app(eq, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(del, app(s, app(s, app(s, app(s, app(s, x'')))))), app(app(cons, app(s, app(s, app(s, app(s, app(s, y'')))))), z)) -> APP(app(if, app(app(eq, x''), y'')), z)


Rules:


app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(app(eq, 0), 0) -> true
app(app(eq, 0), app(s, y)) -> false
app(app(eq, app(s, x)), 0) -> false
app(app(eq, app(s, x)), app(s, y)) -> app(app(eq, x), y)
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(minsort, nil) -> nil
app(minsort, app(app(cons, x), y)) -> app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) -> x
app(app(min, x), app(app(cons, y), z)) -> app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) -> nil
app(app(del, x), app(app(cons, y), z)) -> app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))




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