Termination Proof using AProVE

Term Rewriting System R:
[y, x]
app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) -> APP(le, x)
APP(app(minus, x), app(s, y)) -> APP(pred, app(app(minus, x), y))
APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(if_gcd, app(app(le, y), x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(le, y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(gcd, app(app(minus, x), y))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(minus, x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(gcd, app(app(minus, y), x))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(minus, y)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(minus, x), app(s, y)) -> APP(app(minus, x), y)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(if_gcd) = 0

POL(0) = 0

POL(false) = 0

POL(pred) = 0

POL(minus) = 0

POL(true) = 0

POL(s) = 0

POL(le) = 0

POL(app(x₁, x₂)) = 1 + x₂

POL(APP(x₁, x₂)) = x₂

POL(gcd) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))

three new Dependency Pairs are created:

APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(if_gcd, app(app(le, y), x)), app(s, x))

three new Dependency Pairs are created:

APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, x), y)), app(s, y))

two new Dependency Pairs are created:

APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x)), app(s, y)) -> APP(app(minus, x), y)

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(gcd, app(app(minus, y), x)), app(s, x))

two new Dependency Pairs are created:

APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, x)), app(s, y)) -> APP(app(minus, y), x)

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y''))), app(s, app(s, x'')))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(if_gcd, true), app(s, x'))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(if_gcd, false), app(s, 0))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))
APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, y''))), app(s, app(s, x''))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, y'')))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(if_gcd, true), app(s, app(s, y''')))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, false), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, false), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(if_gcd, true), app(s, app(s, y''')))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x'')), app(s, app(s, y''))) -> APP(app(gcd, app(pred, app(app(minus, x''), y''))), app(s, app(s, y'')))

two new Dependency Pairs are created:

APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, false), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(if_gcd, true), app(s, app(s, y''')))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, y''))), app(s, y0)) -> APP(app(gcd, app(pred, app(app(minus, y0), y''))), app(s, app(s, y'')))

two new Dependency Pairs are created:

APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, false), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(if_gcd, true), app(s, app(s, y''')))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(if_gcd, true), app(s, app(s, y''')))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, false), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, false), app(s, app(s, 0)))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y')))), app(s, app(s, app(s, x'))))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, y')))), app(s, app(s, app(s, x')))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, y'))))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, 0))) -> APP(app(gcd, app(pred, x''')), app(s, app(s, 0)))

one new Dependency Pair is created:

APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x''')), app(s, app(s, app(s, y')))) -> APP(app(gcd, app(pred, app(pred, app(app(minus, x'''), y')))), app(s, app(s, app(s, y'))))

two new Dependency Pairs are created:

APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, y0')) -> APP(app(gcd, app(pred, y0')), app(s, app(s, 0)))

one new Dependency Pair is created:

APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 22

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, app(s, y')))), app(s, y0')) -> APP(app(gcd, app(pred, app(pred, app(app(minus, y0'), y')))), app(s, app(s, app(s, y'))))

two new Dependency Pairs are created:

APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 23

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x'')))))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 24

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(if_gcd, true), app(s, app(s, app(s, y''))))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 25

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, 0))))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 26

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, y''))))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, y'')))))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 27

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, app(pred, x''''))), app(s, app(s, app(s, 0))))

one new Dependency Pair is created:

APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 28

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x'''')), app(s, app(s, app(s, app(s, y''))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, x''''), y''))))), app(s, app(s, app(s, app(s, y'')))))

two new Dependency Pairs are created:

APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 29

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, y0''))), app(s, app(s, app(s, 0))))

one new Dependency Pair is created:

APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 30

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, y''))))), app(s, y0'')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(app(minus, y0''), y''))))), app(s, app(s, app(s, app(s, y'')))))

two new Dependency Pairs are created:

APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 31

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x'))))))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 32

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, y''')))))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 33

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, 0)))))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 34

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y')))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(if_gcd, app(app(le, x'), y')), app(s, app(s, app(s, app(s, app(s, y'))))))

three new Dependency Pairs are created:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 35

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))

one new Dependency Pair is created:

APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 36

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, app(pred, x''''')))), app(s, app(s, app(s, app(s, 0)))))

one new Dependency Pair is created:

APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 37

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, true), app(s, x''''')), app(s, app(s, app(s, app(s, app(s, y')))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, x'''''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))

two new Dependency Pairs are created:

APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 38

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, x')), app(s, app(s, app(s, 0))))

one new Dependency Pair is created:

APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, x'')))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 39

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, x'')))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, y0''')))), app(s, app(s, app(s, app(s, 0)))))

one new Dependency Pair is created:

APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 40

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, x'')))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, y')))))), app(s, y0''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0'''), y')))))), app(s, app(s, app(s, app(s, app(s, y'))))))

two new Dependency Pairs are created:

APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, y0''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 41

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, x'')))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, y0''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y''))))))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 42

                 ↳Narrowing Transformation

Dependency Pairs:

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, x'')))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, y0''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

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

APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0))))))

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

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 43

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, x''''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, app(s, app(s, app(s, app(s, app(s, x'')))))))
APP(app(app(if_gcd, true), app(s, x'''''')), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, x''''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0)))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, app(s, y'')))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(gcd, app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x')))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, app(s, app(s, app(s, x'))))))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0))))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, app(s, y'''))))), app(s, app(s, app(s, app(s, 0)))))
APP(app(gcd, app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x''))))) -> APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, app(s, x'')))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, app(pred, app(app(minus, y0''''), y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, 0)))), app(s, app(s, app(s, x'')))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(app(if_gcd, true), app(s, app(s, app(s, x'')))), app(s, app(s, app(s, 0)))) -> APP(app(gcd, x''), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0)))) -> APP(app(app(if_gcd, true), app(s, app(s, app(s, y'')))), app(s, app(s, app(s, 0))))
APP(app(gcd, app(s, app(s, 0))), app(s, app(s, app(s, x')))) -> APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, app(s, x'))))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, 0))))), app(s, app(s, x'))) -> APP(app(gcd, app(pred, app(pred, x'))), app(s, app(s, app(s, app(s, 0)))))
APP(app(app(if_gcd, false), app(s, app(s, 0))), app(s, app(s, x'))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(app(if_gcd, true), app(s, app(s, x'))), app(s, app(s, 0))) -> APP(app(gcd, x'), app(s, app(s, 0)))
APP(app(gcd, app(s, app(s, y'''))), app(s, app(s, 0))) -> APP(app(app(if_gcd, true), app(s, app(s, y'''))), app(s, app(s, 0)))
APP(app(gcd, app(s, 0)), app(s, app(s, x''))) -> APP(app(app(if_gcd, false), app(s, 0)), app(s, app(s, x'')))
APP(app(app(if_gcd, false), app(s, app(s, app(s, app(s, app(s, 0)))))), app(s, y0'''')) -> APP(app(gcd, app(pred, app(pred, app(pred, app(pred, y0''''))))), app(s, app(s, app(s, app(s, app(s, 0))))))
APP(app(app(if_gcd, false), app(s, 0)), app(s, y')) -> APP(app(gcd, y'), app(s, 0))
APP(app(app(if_gcd, true), app(s, x'')), app(s, 0)) -> APP(app(gcd, x''), app(s, 0))
APP(app(gcd, app(s, x')), app(s, 0)) -> APP(app(app(if_gcd, true), app(s, x')), app(s, 0))
APP(app(gcd, app(s, x)), app(s, y)) -> APP(app(le, y), x)
APP(app(le, app(s, x)), app(s, y)) -> APP(app(le, x), y)
APP(app(gcd, app(s, app(s, app(s, app(s, app(s, app(s, y''))))))), app(s, app(s, app(s, app(s, app(s, app(s, x''))))))) -> APP(app(if_gcd, app(app(le, x''), y'')), app(s, app(s, app(s, app(s, app(s, app(s, y'')))))))

Rules:

app(app(le, 0), y) -> true
app(app(le, app(s, x)), 0) -> false
app(app(le, app(s, x)), app(s, y)) -> app(app(le, x), y)
app(pred, app(s, x)) -> x
app(app(minus, x), 0) -> x
app(app(minus, x), app(s, y)) -> app(pred, app(app(minus, x), y))
app(app(gcd, 0), y) -> y
app(app(gcd, app(s, x)), 0) -> app(s, x)
app(app(gcd, app(s, x)), app(s, y)) -> app(app(app(if_gcd, app(app(le, y), x)), app(s, x)), app(s, y))
app(app(app(if_gcd, true), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, x), y)), app(s, y))
app(app(app(if_gcd, false), app(s, x)), app(s, y)) -> app(app(gcd, app(app(minus, y), x)), app(s, x))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:33 minutes

POL(if_gcd)	= 0
POL(0)	= 0
POL(false)	= 0
POL(pred)	= 0
POL(minus)	= 0
POL(true)	= 0
POL(s)	= 0
POL(le)	= 0
POL(app(x₁, x₂))	= 1 + x₂
POL(APP(x₁, x₂))	= x₂
POL(gcd)	= 0