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)))

innermost

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

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

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

innermost

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 two new DP problems:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 3

↳Polynomial Ordering

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

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

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

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(g)= 0 _{ }^{ }_{ }^{ }POL(a(x)_{1}, x_{2})= 1 + x _{2}_{ }^{ }_{ }^{ }POL(A(x)_{1}, x_{2})= x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(f)= 0 _{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 5

↳Instantiation 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)))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in no new DP problems.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 4

↳Narrowing Transformation

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

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

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

innermost

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 no new DP problems.

Duration:

0:00 minutes