f(

f(a(

f(b(

R

↳Dependency Pair Analysis

F(x, a(a(b(b(y))))) -> F(a(a(a(b(b(b(x)))))),y)

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Instantiation Transformation

**F(b( x), y) -> F(x, b(y))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, an Instantiation SCC transformation can be performed.

As a result of transforming the rule

three new Dependency Pairs are created:

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

F(a(x''), a(y'')) -> F(x'', a(a(y'')))

F(a(a(a(b(b(b(x'')))))),y'') -> F(a(a(b(b(b(x''))))), a(y''))

F(a(x''), b(y'')) -> F(x'', a(b(y'')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Instantiation Transformation

**F(a( x''), b(y'')) -> F(x'', a(b(y'')))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, an Instantiation SCC transformation can be performed.

As a result of transforming the rule

three new Dependency Pairs are created:

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

F(b(x''), b(y'')) -> F(x'', b(b(y'')))

F(b(x'), a(a(y''''))) -> F(x', b(a(a(y''''))))

F(b(x'), a(b(y''''))) -> F(x', b(a(b(y''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 3

↳Forward Instantiation Transformation

**F(b( x'), a(b(y''''))) -> F(x', b(a(b(y''''))))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

four new Dependency Pairs are created:

F(a(x''), a(y'')) -> F(x'', a(a(y'')))

F(a(x'''), a(b(b(y''')))) -> F(x''', a(a(b(b(y''')))))

F(a(a(x'''')), a(y'''')) -> F(a(x''''), a(a(y'''')))

F(a(a(a(a(b(b(b(x''''))))))), a(y'''')) -> F(a(a(a(b(b(b(x'''')))))), a(a(y'''')))

F(a(b(x'''')), a(y''')) -> F(b(x''''), a(a(y''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 4

↳Forward Instantiation Transformation

**F(a(b( x'''')), a(y''')) -> F(b(x''''), a(a(y''')))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

six new Dependency Pairs are created:

F(a(x''), b(y'')) -> F(x'', a(b(y'')))

F(a(a(a(a(b(b(b(x''''))))))), b(y'''')) -> F(a(a(a(b(b(b(x'''')))))), a(b(y'''')))

F(a(b(x'''')), b(y''')) -> F(b(x''''), a(b(y''')))

F(a(a(x''''')), b(b(y'''''))) -> F(a(x'''''), a(b(b(y'''''))))

F(a(a(a(x''''''))), b(y''')) -> F(a(a(x'''''')), a(b(y''')))

F(a(a(a(a(a(b(b(b(x'''''')))))))), b(y''')) -> F(a(a(a(a(b(b(b(x''''''))))))), a(b(y''')))

F(a(a(b(x''''''))), b(y''')) -> F(a(b(x'''''')), a(b(y''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 5

↳Forward Instantiation Transformation

**F(a(a( x''''')), b(b(y'''''))) -> F(a(x'''''), a(b(b(y'''''))))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

eight new Dependency Pairs are created:

F(b(x''), b(y'')) -> F(x'', b(b(y'')))

F(b(a(a(a(b(b(b(x''''))))))), b(y'''')) -> F(a(a(a(b(b(b(x'''')))))), b(b(y'''')))

F(b(b(x'''')), b(y'''')) -> F(b(x''''), b(b(y'''')))

F(b(a(a(a(a(b(b(b(x'''''')))))))), b(y''')) -> F(a(a(a(a(b(b(b(x''''''))))))), b(b(y''')))

F(b(a(b(x''''''))), b(y''')) -> F(a(b(x'''''')), b(b(y''')))

F(b(a(a(x'''''''))), b(y''')) -> F(a(a(x''''''')), b(b(y''')))

F(b(a(a(a(x'''''''')))), b(y''')) -> F(a(a(a(x''''''''))), b(b(y''')))

F(b(a(a(a(a(a(b(b(b(x''''''''))))))))), b(y''')) -> F(a(a(a(a(a(b(b(b(x'''''''')))))))), b(b(y''')))

F(b(a(a(b(x'''''''')))), b(y''')) -> F(a(a(b(x''''''''))), b(b(y''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 6

↳Forward Instantiation Transformation

**F(b(a(a(b( x'''''''')))), b(y''')) -> F(a(a(b(x''''''''))), b(b(y''')))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

14 new Dependency Pairs are created:

F(b(x'), a(a(y''''))) -> F(x', b(a(a(y''''))))

F(b(a(a(a(b(b(b(x''''))))))), a(a(y'''''))) -> F(a(a(a(b(b(b(x'''')))))), b(a(a(y'''''))))

F(b(a(a(a(a(b(b(b(x'''''')))))))), a(a(y''''''))) -> F(a(a(a(a(b(b(b(x''''''))))))), b(a(a(y''''''))))

F(b(a(b(x''''''))), a(a(y''''''))) -> F(a(b(x'''''')), b(a(a(y''''''))))

F(b(a(a(a(x'''''''')))), a(a(y''''''))) -> F(a(a(a(x''''''''))), b(a(a(y''''''))))

F(b(a(a(a(a(a(b(b(b(x''''''''))))))))), a(a(y''''''))) -> F(a(a(a(a(a(b(b(b(x'''''''')))))))), b(a(a(y''''''))))

F(b(a(a(b(x'''''''')))), a(a(y''''''))) -> F(a(a(b(x''''''''))), b(a(a(y''''''))))

F(b(b(a(a(a(b(b(b(x'''''')))))))), a(a(y''''''))) -> F(b(a(a(a(b(b(b(x''''''))))))), b(a(a(y''''''))))

F(b(b(b(x''''''))), a(a(y''''''))) -> F(b(b(x'''''')), b(a(a(y''''''))))

F(b(b(a(a(a(a(b(b(b(x''''''''))))))))), a(a(y''''''))) -> F(b(a(a(a(a(b(b(b(x'''''''')))))))), b(a(a(y''''''))))

F(b(b(a(b(x'''''''')))), a(a(y''''''))) -> F(b(a(b(x''''''''))), b(a(a(y''''''))))

F(b(b(a(a(x''''''''')))), a(a(y''''''))) -> F(b(a(a(x'''''''''))), b(a(a(y''''''))))

F(b(b(a(a(a(x''''''''''))))), a(a(y''''''))) -> F(b(a(a(a(x'''''''''')))), b(a(a(y''''''))))

F(b(b(a(a(a(a(a(b(b(b(x'''''''''')))))))))), a(a(y''''''))) -> F(b(a(a(a(a(a(b(b(b(x''''''''''))))))))), b(a(a(y''''''))))

F(b(b(a(a(b(x''''''''''))))), a(a(y''''''))) -> F(b(a(a(b(x'''''''''')))), b(a(a(y''''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 7

↳Forward Instantiation Transformation

**F(b(b(a(a(b( x''''''''''))))), a(a(y''''''))) -> F(b(a(a(b(x'''''''''')))), b(a(a(y''''''))))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

14 new Dependency Pairs are created:

F(b(x'), a(b(y''''))) -> F(x', b(a(b(y''''))))

F(b(a(a(a(b(b(b(x''''))))))), a(b(y'''''))) -> F(a(a(a(b(b(b(x'''')))))), b(a(b(y'''''))))

F(b(a(a(a(a(b(b(b(x'''''')))))))), a(b(y''''''))) -> F(a(a(a(a(b(b(b(x''''''))))))), b(a(b(y''''''))))

F(b(a(b(x''''''))), a(b(y''''''))) -> F(a(b(x'''''')), b(a(b(y''''''))))

F(b(a(a(a(x'''''''')))), a(b(y''''''))) -> F(a(a(a(x''''''''))), b(a(b(y''''''))))

F(b(a(a(a(a(a(b(b(b(x''''''''))))))))), a(b(y''''''))) -> F(a(a(a(a(a(b(b(b(x'''''''')))))))), b(a(b(y''''''))))

F(b(a(a(b(x'''''''')))), a(b(y''''''))) -> F(a(a(b(x''''''''))), b(a(b(y''''''))))

F(b(b(a(a(a(b(b(b(x'''''')))))))), a(b(y''''''))) -> F(b(a(a(a(b(b(b(x''''''))))))), b(a(b(y''''''))))

F(b(b(b(x''''''))), a(b(y''''''))) -> F(b(b(x'''''')), b(a(b(y''''''))))

F(b(b(a(a(a(a(b(b(b(x''''''''))))))))), a(b(y''''''))) -> F(b(a(a(a(a(b(b(b(x'''''''')))))))), b(a(b(y''''''))))

F(b(b(a(b(x'''''''')))), a(b(y''''''))) -> F(b(a(b(x''''''''))), b(a(b(y''''''))))

F(b(b(a(a(x''''''''')))), a(b(y''''''))) -> F(b(a(a(x'''''''''))), b(a(b(y''''''))))

F(b(b(a(a(a(x''''''''''))))), a(b(y''''''))) -> F(b(a(a(a(x'''''''''')))), b(a(b(y''''''))))

F(b(b(a(a(a(a(a(b(b(b(x'''''''''')))))))))), a(b(y''''''))) -> F(b(a(a(a(a(a(b(b(b(x''''''''''))))))))), b(a(b(y''''''))))

F(b(b(a(a(b(x''''''''''))))), a(b(y''''''))) -> F(b(a(a(b(x'''''''''')))), b(a(b(y''''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 8

↳Remaining Obligation(s)

The following remains to be proven:

**F(b(b(a(a(a(a(a(b(b(b( x'''''''''')))))))))), a(a(y''''''))) -> F(b(a(a(a(a(a(b(b(b(x''''''''''))))))))), b(a(a(y''''''))))**

f(x, a(a(b(b(y))))) -> f(a(a(a(b(b(b(x)))))),y)

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

f(b(x),y) -> f(x, b(y))

innermost

Duration:

0:13 minutes