Termination Proof using AProVE

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.

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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:

     ↳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(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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳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(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)), 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(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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳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, 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(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)), 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, 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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳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, 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(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)), 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

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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳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, 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(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)), 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, 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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳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, 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(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)), 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

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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳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, 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(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)), 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, 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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳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, 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(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)), 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

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:

     ↳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''), 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(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)), 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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

Dependency Pairs:

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:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

Dependency Pairs:

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, 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))

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

Dependency Pairs:

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

three new Dependency Pairs are created:

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

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

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

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

Dependency Pairs:

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

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 20

                 ↳Polynomial Ordering

Dependency Pairs:

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

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(if) = 0

POL(0) = 0

POL(false) = 0

POL(perfectp) = 1

POL(minus) = 0

POL(true) = 0

POL(s) = 0

POL(le) = 0

POL(app(x₁, x₂)) = 0

POL(f) = 0

POL(APP(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 21

                 ↳Polynomial Ordering

Dependency Pairs:

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

The following dependency pairs can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(if) = 0

POL(0) = 0

POL(false) = 0

POL(perfectp) = 1

POL(minus) = 1

POL(true) = 0

POL(s) = 0

POL(le) = 0

POL(app(x₁, x₂)) = x₁

POL(f) = 1

POL(APP(x₁, x₂)) = 1 + x₁

resulting in one new DP problem.

     ↳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(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)

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

POL(if)	= 0
POL(0)	= 0
POL(false)	= 0
POL(perfectp)	= 1
POL(minus)	= 0
POL(true)	= 0
POL(s)	= 0
POL(le)	= 0
POL(app(x₁, x₂))	= 0
POL(f)	= 0
POL(APP(x₁, x₂))	= x₁