Termination Proof using AProVE

Term Rewriting System R:
[x, y, z]
:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

:'(:(x, y), z) -> :'(x, :(z, i(y)))
:'(:(x, y), z) -> :'(z, i(y))
:'(:(x, y), z) -> I(y)
:'(e, x) -> I(x)
:'(x, :(y, i(x))) -> I(y)
:'(x, :(y, :(i(x), z))) -> :'(i(z), y)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(i(x), :(y, x)) -> I(y)
:'(i(x), :(y, :(x, z))) -> :'(i(z), y)
:'(i(x), :(y, :(x, z))) -> I(z)
I(:(x, y)) -> :'(y, x)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

:'(i(x), :(y, :(x, z))) -> I(z)
:'(i(x), :(y, :(x, z))) -> :'(i(z), y)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(e, x) -> I(x)
I(:(x, y)) -> :'(y, x)
:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
:'(x, :(y, :(i(x), z))) -> :'(i(z), y)
:'(:(x, y), z) -> :'(x, :(z, i(y)))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(:(x, y), z) -> :'(x, :(z, i(y)))

five new Dependency Pairs are created:

:'(:(x, y'), i(y')) -> :'(x, e)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(:(x, y'), e) -> :'(x, i(i(y')))
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

:'(:(x, y'), e) -> :'(x, i(i(y')))
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
:'(i(x), :(y, :(x, z))) -> :'(i(z), y)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(x, :(y, :(i(x), z))) -> :'(i(z), y)
:'(e, x) -> I(x)
:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
:'(i(x), :(y, :(x, z))) -> I(z)

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(x, :(y, :(i(x), z))) -> :'(i(z), y)

two new Dependency Pairs are created:

:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(x, :(y, :(i(x), i(x'')))) -> :'(x'', y)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(i(x), :(y, :(x, z))) -> I(z)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(i(x), :(y, :(x, z))) -> :'(i(z), y)
:'(e, x) -> I(x)
:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(:(x, y'), e) -> :'(x, i(i(y')))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(i(x), :(y, :(x, z))) -> :'(i(z), y)

two new Dependency Pairs are created:

:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(i(x), :(y, :(x, i(x'')))) -> :'(x'', y)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y'), e) -> :'(x, i(i(y')))
:'(i(x), :(y, :(x, z))) -> I(z)
:'(e, x) -> I(x)
:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(:(x, y), z) -> I(y)

one new Dependency Pair is created:

:'(:(x, :(x'', y'')), z) -> I(:(x'', y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

:'(:(x, y'), e) -> :'(x, i(i(y')))
:'(:(x, :(x'', y'')), z) -> I(:(x'', y''))
:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(i(x), :(y, :(x, z))) -> I(z)
:'(e, x) -> I(x)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(e, x) -> I(x)

one new Dependency Pair is created:

:'(e, :(x'', y'')) -> I(:(x'', y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

:'(e, :(x'', y'')) -> I(:(x'', y''))
:'(:(x, :(x'', y'')), z) -> I(:(x'', y''))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(i(x), :(y, :(x, z))) -> I(z)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
:'(x, :(y, :(i(x), z))) -> I(z)
:'(:(x, y'), e) -> :'(x, i(i(y')))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(x, :(y, :(i(x), z))) -> I(z)

one new Dependency Pair is created:

:'(x, :(y, :(i(x), :(x'', y'')))) -> I(:(x'', y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

:'(:(x, y'), e) -> :'(x, i(i(y')))
:'(:(x, :(x'', y'')), z) -> I(:(x'', y''))
:'(x, :(y, :(i(x), :(x'', y'')))) -> I(:(x'', y''))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(i(x), :(y, :(x, z))) -> I(z)
:'(:(x, y), z) -> :'(z, i(y))
I(:(x, y)) -> :'(y, x)
:'(e, :(x'', y'')) -> I(:(x'', y''))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

:'(i(x), :(y, :(x, z))) -> I(z)

one new Dependency Pair is created:

:'(i(x), :(y, :(x, :(x'', y'')))) -> I(:(x'', y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

:'(e, :(x'', y'')) -> I(:(x'', y''))
:'(:(x, :(x'', y'')), z) -> I(:(x'', y''))
:'(i(x), :(y, :(x, :(x'', y'')))) -> I(:(x'', y''))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
I(:(x, y)) -> :'(y, x)
:'(x, :(y, :(i(x), :(x'', y'')))) -> I(:(x'', y''))
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(:(x, y), z) -> :'(z, i(y))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y'), e) -> :'(x, i(i(y')))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

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

I(:(x, y)) -> :'(y, x)

three new Dependency Pairs are created:

I(:(x0, :(x'', y''))) -> :'(:(x'', y''), x0)
I(:(x0, :(x'', :(x'''', y'''')))) -> :'(:(x'', :(x'''', y'''')), x0)
I(:(x0, :(x'', i(x'''')))) -> :'(:(x'', i(x'''')), x0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

:'(:(x, y'), e) -> :'(x, i(i(y')))
:'(i(x), :(y, :(x, :(x'', y'')))) -> I(:(x'', y''))
I(:(x0, :(x'', i(x'''')))) -> :'(:(x'', i(x'''')), x0)
:'(:(x, :(x'', y'')), z) -> I(:(x'', y''))
I(:(x0, :(x'', :(x'''', y'''')))) -> :'(:(x'', :(x'''', y'''')), x0)
:'(x, :(y, :(i(x), :(x'', y'')))) -> I(:(x'', y''))
:'(:(x, i(x'')), z) -> :'(x, :(z, x''))
:'(i(x), :(y, :(x, :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, :(x'', y'')), z) -> :'(x, :(z, :(y'', x'')))
:'(x, :(y, :(i(x), :(x'', y'')))) -> :'(:(y'', x''), y)
:'(:(x, y0), :(x'', y'')) -> :'(x, :(x'', :(i(y0), i(y''))))
:'(:(x, y), z) -> :'(z, i(y))
I(:(x0, :(x'', y''))) -> :'(:(x'', y''), x0)
:'(e, :(x'', y'')) -> I(:(x'', y''))

Rules:

:(x, x) -> e
:(x, e) -> x
:(:(x, y), z) -> :(x, :(z, i(y)))
:(e, x) -> i(x)
:(x, :(y, i(x))) -> i(y)
:(x, :(y, :(i(x), z))) -> :(i(z), y)
:(i(x), :(y, x)) -> i(y)
:(i(x), :(y, :(x, z))) -> :(i(z), y)
i(:(x, y)) -> :(y, x)
i(i(x)) -> x
i(e) -> e

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:03 minutes