Term Rewriting System R:
[x, y, u, z]
app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(perfectp, app(s, x)) -> APP(app(app(f, x), app(s, 0)), app(s, x))
APP(perfectp, app(s, x)) -> APP(app(f, x), app(s, 0))
APP(perfectp, app(s, x)) -> APP(f, x)
APP(perfectp, app(s, x)) -> APP(s, 0)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(f, x)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(minus, z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(if, app(app(le, x), y))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(le, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, app(s, x)), app(app(minus, y), x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(minus, y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(f, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(f, x), app(s, 0))
APP(perfectp, app(s, x)) -> APP(app(app(f, x), app(s, 0)), app(s, x))
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(perfectp, app(s, x)) -> APP(app(app(f, x), app(s, 0)), app(s, x))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(f, x), app(s, 0))
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(perfectp, app(s, x)) -> APP(app(f, x), app(s, 0))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(f, x), u)
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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(minus, z), app(s, x))
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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
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 7
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

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



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, app(s, x)), app(app(minus, y), x)), 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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(minus, y), x)
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 11
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), z)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(f, x), u), 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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(f, x), u)
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(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
four new Dependency Pairs are created:

APP(app(app(app(f, app(s, 0)), app(s, y)), 0), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), 0), u'')), true)
APP(app(app(app(f, app(s, 0)), app(s, y)), app(s, z'')), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), app(s, z'')), u'')), false)
APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), 0) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), 0)), app(app(app(app(f, x''), 0), app(app(minus, z''), app(s, x''))), 0))
APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), app(s, y'')) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), app(s, y''))), app(app(app(if, app(app(le, x''), y'')), app(app(app(app(f, app(s, x'')), app(app(minus, y''), x'')), z''), app(s, y''))), app(app(app(app(f, x''), app(s, y'')), z''), app(s, y''))))

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(app(app(f, app(s, app(s, x''))), app(s, y)), z''), app(s, y'')) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), app(s, y''))), app(app(app(if, app(app(le, x''), y'')), app(app(app(app(f, app(s, x'')), app(app(minus, y''), x'')), z''), app(s, y''))), app(app(app(app(f, x''), app(s, y'')), z''), app(s, y''))))
APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), 0) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), 0)), app(app(app(app(f, x''), 0), app(app(minus, z''), app(s, x''))), 0))
APP(app(app(app(f, app(s, 0)), app(s, y)), app(s, z'')), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), app(s, z'')), u'')), false)
APP(app(app(app(f, app(s, 0)), app(s, y)), 0), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), 0), u'')), true)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, 0)), app(s, y)), 0), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), 0), u'')), true)
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 15
                 ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), 0) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), 0)), app(app(app(app(f, x''), 0), app(app(minus, z''), app(s, x''))), 0))
APP(app(app(app(f, app(s, 0)), app(s, y)), app(s, z'')), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), app(s, z'')), u'')), false)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), app(s, y'')) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), app(s, y''))), app(app(app(if, app(app(le, x''), y'')), app(app(app(app(f, app(s, x'')), app(app(minus, y''), x'')), z''), app(s, y''))), app(app(app(app(f, x''), app(s, y'')), z''), app(s, y''))))


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost




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

APP(app(app(app(f, app(s, 0)), app(s, y)), app(s, z'')), u'') -> APP(app(app(if, app(app(le, 0), y)), app(app(app(app(f, app(s, 0)), app(app(minus, y), 0)), app(s, z'')), u'')), false)
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
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), app(s, y'')) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), app(s, y''))), app(app(app(if, app(app(le, x''), y'')), app(app(app(app(f, app(s, x'')), app(app(minus, y''), x'')), z''), app(s, y''))), app(app(app(app(f, x''), app(s, y'')), z''), app(s, y''))))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) -> APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), 0), z), u) -> APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(perfectp, app(s, x)) -> APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(f, app(s, app(s, x''))), app(s, y)), z''), 0) -> APP(app(app(if, app(app(le, app(s, x'')), y)), app(app(app(app(f, app(s, app(s, x''))), app(app(minus, y), app(s, x''))), z''), 0)), app(app(app(app(f, x''), 0), app(app(minus, z''), app(s, x''))), 0))


Rules:


app(perfectp, 0) -> false
app(perfectp, app(s, x)) -> app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) -> true
app(app(app(app(f, 0), y), app(s, z)), u) -> false
app(app(app(app(f, app(s, x)), 0), z), u) -> app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) -> app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))


Strategy:

innermost



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