a(f, a(f,

a(

R

↳Dependency Pair Analysis

A(f, a(f,x)) -> A(x, g)

A(x, g) -> A(f, a(g, a(f,x)))

A(x, g) -> A(g, a(f,x))

A(x, g) -> A(f,x)

Furthermore,

R

↳DPs

→DP Problem 1

↳Narrowing Transformation

**A( x, g) -> A(f, x)**

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

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

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

As a result of transforming the rule

two new Dependency Pairs are created:

A(x, g) -> A(f, a(g, a(f,x)))

A(a(f,x''), g) -> A(f, a(g, a(x'', g)))

A(g, g) -> A(f, a(g, a(f, a(g, a(f, f)))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Narrowing Transformation

**A(a(f, x''), g) -> A(f, a(g, a(x'', g)))**

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

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

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

As a result of transforming the rule

two new Dependency Pairs are created:

A(x, g) -> A(g, a(f,x))

A(a(f,x''), g) -> A(g, a(x'', g))

A(g, g) -> A(g, a(f, a(g, a(f, f))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 3

↳Narrowing Transformation

**A(g, g) -> A(g, a(f, a(g, a(f, f))))****A(a(f, x''), g) -> A(g, a(x'', g))**

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

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

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

As a result of transforming the rule

one new Dependency Pair is created:

A(a(f,x''), g) -> A(f, a(g, a(x'', g)))

A(a(f,x'''), g) -> A(f, a(g, a(f, a(g, a(f,x''')))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 4

↳Narrowing Transformation

**A(a(f, x'''), g) -> A(f, a(g, a(f, a(g, a(f, x''')))))**

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

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

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

As a result of transforming the rule

no new Dependency Pairs are created.

A(g, g) -> A(f, a(g, a(f, a(g, a(f, f)))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 5

↳Narrowing Transformation

**A(g, g) -> A(g, a(f, a(g, a(f, f))))****A(a(f, x''), g) -> A(g, a(x'', g))**

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

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

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

As a result of transforming the rule

one new Dependency Pair is created:

A(a(f,x''), g) -> A(g, a(x'', g))

A(a(f,x'''), g) -> A(g, a(f, a(g, a(f,x'''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 6

↳Narrowing Transformation

**A(a(f, x'''), g) -> A(g, a(f, a(g, a(f, x'''))))**

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

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

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

As a result of transforming the rule

no new Dependency Pairs are created.

A(g, g) -> A(g, a(f, a(g, a(f, f))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 7

↳Forward Instantiation Transformation

**A(a(f, x'''), g) -> A(f, a(g, a(f, a(g, a(f, x''')))))**

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

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

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

As a result of transforming the rule

no new Dependency Pairs are created.

A(a(f,x'''), g) -> A(g, a(f, a(g, a(f,x'''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 8

↳Narrowing Transformation

**A( x, g) -> A(f, x)**

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

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

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

As a result of transforming the rule

two new Dependency Pairs are created:

A(a(f,x'''), g) -> A(f, a(g, a(f, a(g, a(f,x''')))))

A(a(f, a(f,x')), g) -> A(f, a(g, a(f, a(g, a(x', g)))))

A(a(f, g), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 9

↳Narrowing Transformation

**A(a(f, g), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))****A(a(f, a(f, x')), g) -> A(f, a(g, a(f, a(g, a(x', g)))))**

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

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

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

As a result of transforming the rule

one new Dependency Pair is created:

A(a(f, a(f,x')), g) -> A(f, a(g, a(f, a(g, a(x', g)))))

A(a(f, a(f,x'')), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f,x'')))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 10

↳Narrowing Transformation

**A(a(f, a(f, x'')), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, x'')))))))**

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

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

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

As a result of transforming the rule

no new Dependency Pairs are created.

A(a(f, g), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 11

↳Narrowing Transformation

**A( x, g) -> A(f, x)**

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

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

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

As a result of transforming the rule

two new Dependency Pairs are created:

A(a(f, a(f,x'')), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f,x'')))))))

A(a(f, a(f, a(f,x'))), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(x', g)))))))

A(a(f, a(f, g)), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 12

↳Narrowing Transformation

**A(a(f, a(f, g)), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))))****A(a(f, a(f, a(f, x'))), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(x', g)))))))**

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

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

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

As a result of transforming the rule

no new Dependency Pairs are created.

A(a(f, a(f, g)), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 13

↳Remaining Obligation(s)

The following remains to be proven:

**A( x, g) -> A(f, x)**

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

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

Duration:

0:00 minutes