Termination Proof using AProVE

Term Rewriting System R:
[x, y]
f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(x, x) -> F(i(x), g(g(x)))
F(x, x) -> G(g(x))
F(x, x) -> G(x)
F(x, i(x)) -> F(x, x)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Rewriting Transformation

Dependency Pairs:

F(x, i(x)) -> F(x, x)
F(x, x) -> F(i(x), g(g(x)))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(x, x) -> F(i(x), g(g(x)))

one new Dependency Pair is created:

F(x, x) -> F(i(x), i(g(x)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rewriting Transformation

Dependency Pairs:

F(x, x) -> F(i(x), i(g(x)))
F(x, i(x)) -> F(x, x)

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(x, x) -> F(i(x), i(g(x)))

one new Dependency Pair is created:

F(x, x) -> F(i(x), i(i(x)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 3

                 ↳Instantiation Transformation

Dependency Pairs:

F(x, x) -> F(i(x), i(i(x)))
F(x, i(x)) -> F(x, x)

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(x, i(x)) -> F(x, x)

one new Dependency Pair is created:

F(i(x''), i(i(x''))) -> F(i(x''), i(x''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 4

                 ↳Instantiation Transformation

Dependency Pairs:

F(i(x''), i(i(x''))) -> F(i(x''), i(x''))
F(x, x) -> F(i(x), i(i(x)))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(x, x) -> F(i(x), i(i(x)))

one new Dependency Pair is created:

F(i(x''''), i(x'''')) -> F(i(i(x'''')), i(i(i(x''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 5

                 ↳Instantiation Transformation

Dependency Pairs:

F(i(x''''), i(x'''')) -> F(i(i(x'''')), i(i(i(x''''))))
F(i(x''), i(i(x''))) -> F(i(x''), i(x''))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(i(x''), i(i(x''))) -> F(i(x''), i(x''))

one new Dependency Pair is created:

F(i(i(x'''''')), i(i(i(x'''''')))) -> F(i(i(x'''''')), i(i(x'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 6

                 ↳Instantiation Transformation

Dependency Pairs:

F(i(i(x'''''')), i(i(i(x'''''')))) -> F(i(i(x'''''')), i(i(x'''''')))
F(i(x''''), i(x'''')) -> F(i(i(x'''')), i(i(i(x''''))))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(i(x''''), i(x'''')) -> F(i(i(x'''')), i(i(i(x''''))))

one new Dependency Pair is created:

F(i(i(x'''''''')), i(i(x''''''''))) -> F(i(i(i(x''''''''))), i(i(i(i(x'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 7

                 ↳Instantiation Transformation

Dependency Pairs:

F(i(i(x'''''''')), i(i(x''''''''))) -> F(i(i(i(x''''''''))), i(i(i(i(x'''''''')))))
F(i(i(x'''''')), i(i(i(x'''''')))) -> F(i(i(x'''''')), i(i(x'''''')))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(i(i(x'''''')), i(i(i(x'''''')))) -> F(i(i(x'''''')), i(i(x'''''')))

one new Dependency Pair is created:

F(i(i(i(x''''''''''))), i(i(i(i(x''''''''''))))) -> F(i(i(i(x''''''''''))), i(i(i(x''''''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 8

                 ↳Instantiation Transformation

Dependency Pairs:

F(i(i(i(x''''''''''))), i(i(i(i(x''''''''''))))) -> F(i(i(i(x''''''''''))), i(i(i(x''''''''''))))
F(i(i(x'''''''')), i(i(x''''''''))) -> F(i(i(i(x''''''''))), i(i(i(i(x'''''''')))))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(i(i(x'''''''')), i(i(x''''''''))) -> F(i(i(i(x''''''''))), i(i(i(i(x'''''''')))))

one new Dependency Pair is created:

F(i(i(i(x''''''''''''))), i(i(i(x'''''''''''')))) -> F(i(i(i(i(x'''''''''''')))), i(i(i(i(i(x''''''''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 9

                 ↳Instantiation Transformation

Dependency Pairs:

F(i(i(i(x''''''''''''))), i(i(i(x'''''''''''')))) -> F(i(i(i(i(x'''''''''''')))), i(i(i(i(i(x''''''''''''))))))
F(i(i(i(x''''''''''))), i(i(i(i(x''''''''''))))) -> F(i(i(i(x''''''''''))), i(i(i(x''''''''''))))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

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

F(i(i(i(x''''''''''))), i(i(i(i(x''''''''''))))) -> F(i(i(i(x''''''''''))), i(i(i(x''''''''''))))

one new Dependency Pair is created:

F(i(i(i(i(x'''''''''''''')))), i(i(i(i(i(x'''''''''''''')))))) -> F(i(i(i(i(x'''''''''''''')))), i(i(i(i(x'''''''''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Rw

           →DP Problem 2

             ↳Rw

...

               →DP Problem 10

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

F(i(i(i(i(x'''''''''''''')))), i(i(i(i(i(x'''''''''''''')))))) -> F(i(i(i(i(x'''''''''''''')))), i(i(i(i(x'''''''''''''')))))
F(i(i(i(x''''''''''''))), i(i(i(x'''''''''''')))) -> F(i(i(i(i(x'''''''''''')))), i(i(i(i(i(x''''''''''''))))))

Rules:

f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
g(x) -> i(x)

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes