Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(x, y), 0)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

four new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, f), y'''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 29

                 ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:
innermost
Dependency Pairs:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'')), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 10

                 ↳Rewriting Transformation

Dependency Pairs:

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z''')), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), f)), ap(s, z'')), ap(ap(ap(ap(g, f), f), 0), 0))

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z''')), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 14

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 15

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 16

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 17

                 ↳Rewriting Transformation

Dependency Pairs:

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 18

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

Dependency Pairs:

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'0)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 20

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z''')), ap(ap(ap(ap(g, f), f), 0), 0)), 0))

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 21

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), y''')), ap(s, z'0)), ap(ap(ap(ap(g, f), y'''), ap(y''', 0)), 0)), 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 22

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 23

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 24

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 25

                 ↳Rewriting Transformation

Dependency Pairs:

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 26

                 ↳Rewriting Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(ap(f, s), 0)), 0)), 0)), 0))

one new Dependency Pair is created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 27

                 ↳Narrowing Transformation

Dependency Pairs:

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, x'), y')), ap(s, z'1)), ap(ap(ap(ap(g, x'), y'), ap(ap(x', y'), 0)), 0)), 0)), 0)), 0))

two new Dependency Pairs are created:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 28

                 ↳Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), f)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), f)), ap(s, z'0)), ap(ap(ap(ap(g, f), f), 0), 0)), 0)), 0))

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

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 29

                 ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:
innermost
Dependency Pairs:
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, f), y'')), ap(s, z'1)), ap(ap(ap(ap(g, f), y''), ap(y'', 0)), 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z'0)), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, f), s)), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, f), s)), ap(s, z''')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0'))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, x'''), y''')), ap(s, z'''''))), ap(s, z''0')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), s)), ap(s, z'')), ap(ap(ap(ap(g, f), s), ap(s, 0)), 0))
AP(ap(ap(g, ap(ap(g, ap(ap(g, f), y''''')), ap(s, z''''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, f), y''''')), ap(s, z'''')), ap(s, z''))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, x''), y''), ap(s, z''))
AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z'''))), ap(s, z'')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0))), ap(s, z''')), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1))), ap(s, z'0)), ap(ap(ap(ap(g, ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2))), ap(s, z'1)), ap(ap(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'2)), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0)), 0)), 0)), 0)), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:06 minutes