Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(x, y) -> G(x, y)
G(h(x), y) -> F(x, y)
G(h(x), y) -> G(x, y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

G(h(x), y) -> G(x, y)
G(h(x), y) -> F(x, y)
F(x, y) -> G(x, y)

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

F(x, y) -> G(x, y)

one new Dependency Pair is created:

F(h(x''), y'') -> G(h(x''), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

F(h(x''), y'') -> G(h(x''), y'')
G(h(x), y) -> F(x, y)
G(h(x), y) -> G(x, y)

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

G(h(x), y) -> F(x, y)

one new Dependency Pair is created:

G(h(h(x'''')), y') -> F(h(x''''), y')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(h(h(x'''')), y') -> F(h(x''''), y')
G(h(x), y) -> G(x, y)
F(h(x''), y'') -> G(h(x''), y'')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

G(h(x), y) -> G(x, y)

two new Dependency Pairs are created:

G(h(h(x'')), y'') -> G(h(x''), y'')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
G(h(h(x'')), y'') -> G(h(x''), y'')
F(h(x''), y'') -> G(h(x''), y'')
G(h(h(x'''')), y') -> F(h(x''''), y')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

F(h(x''), y'') -> G(h(x''), y'')

three new Dependency Pairs are created:

F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(x'')), y'') -> G(h(x''), y'')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(x'''')), y') -> F(h(x''''), y')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

G(h(h(x'''')), y') -> F(h(x''''), y')

three new Dependency Pairs are created:

G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
G(h(h(x'')), y'') -> G(h(x''), y'')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

G(h(h(x'')), y'') -> G(h(x''), y'')

four new Dependency Pairs are created:

G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

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

G(h(h(h(x''''''))), y') -> G(h(h(x'''''')), y')

seven new Dependency Pairs are created:

G(h(h(h(h(x'''''''')))), y''') -> G(h(h(h(x''''''''))), y''')
G(h(h(h(h(x'''''''''')))), y'') -> G(h(h(h(x''''''''''))), y'')
G(h(h(h(h(x'''''''')))), y'') -> G(h(h(h(x''''''''))), y'')
G(h(h(h(h(h(x''''''''''''))))), y'') -> G(h(h(h(h(x'''''''''''')))), y'')
G(h(h(h(h(x''''''')))), y'') -> G(h(h(h(x'''''''))), y'')
G(h(h(h(h(h(x''''''''''))))), y'') -> G(h(h(h(h(x'''''''''')))), y'')
G(h(h(h(h(h(h(x'''''''''''''')))))), y'') -> G(h(h(h(h(h(x''''''''''''''))))), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Argument Filtering and Ordering

Dependency Pairs:

G(h(h(h(h(h(h(x'''''''''''''')))))), y'') -> G(h(h(h(h(h(x''''''''''''''))))), y'')
G(h(h(h(h(h(x''''''''''))))), y'') -> G(h(h(h(h(x'''''''''')))), y'')
G(h(h(h(h(x''''''')))), y'') -> G(h(h(h(x'''''''))), y'')
G(h(h(h(h(h(x''''''''''''))))), y'') -> G(h(h(h(h(x'''''''''''')))), y'')
G(h(h(h(h(x'''''''')))), y'') -> G(h(h(h(x''''''''))), y'')
G(h(h(h(h(x'''''''''')))), y'') -> G(h(h(h(x''''''''''))), y'')
G(h(h(h(h(x'''''''')))), y''') -> G(h(h(h(x''''''''))), y''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')
G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

G(h(h(h(h(h(h(x'''''''''''''')))))), y'') -> G(h(h(h(h(h(x''''''''''''''))))), y'')
G(h(h(h(h(h(x''''''''''))))), y'') -> G(h(h(h(h(x'''''''''')))), y'')
G(h(h(h(h(x''''''')))), y'') -> G(h(h(h(x'''''''))), y'')
G(h(h(h(h(h(x''''''''''''))))), y'') -> G(h(h(h(h(x'''''''''''')))), y'')
G(h(h(h(h(x'''''''')))), y'') -> G(h(h(h(x''''''''))), y'')
G(h(h(h(h(x'''''''''')))), y'') -> G(h(h(h(x''''''''''))), y'')
G(h(h(h(h(x'''''''')))), y''') -> G(h(h(h(x''''''''))), y''')
G(h(h(h(h(x'''''''''')))), y'''') -> G(h(h(h(x''''''''''))), y'''')
G(h(h(h(h(x'''''''')))), y'''') -> G(h(h(h(x''''''''))), y'''')
G(h(h(h(x''''))), y'''') -> G(h(h(x'''')), y'''')
F(h(h(h(x''''''''))), y'''') -> G(h(h(h(x''''''''))), y'''')
F(h(h(x'''')), y'''') -> G(h(h(x'''')), y'''')
F(h(h(x'''''')), y'''') -> G(h(h(x'''''')), y'''')
G(h(h(h(h(h(x''''''''''''))))), y'''') -> G(h(h(h(h(x'''''''''''')))), y'''')

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(G(x₁, x₂)) = x₁ + x₂

POL(h(x₁)) = 1 + x₁

POL(F(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

F(x₁, x₂) -> F(x₁, x₂)
G(x₁, x₂) -> G(x₁, x₂)
h(x₁) -> h(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pairs:

G(h(h(h(h(x'''''''''')))), y'') -> F(h(h(h(x''''''''''))), y'')
G(h(h(h(x''''''))), y'') -> F(h(h(x'''''')), y'')
G(h(h(h(x''''''''))), y'') -> F(h(h(x'''''''')), y'')

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes

POL(G(x₁, x₂))	= x₁ + x₂
POL(h(x₁))	= 1 + x₁
POL(F(x₁, x₂))	= 1 + x₁ + x₂