c(b(a(

c(

a(

b(

R

↳Dependency Pair Analysis

C(b(a(X))) -> A(a(b(b(c(c(X))))))

C(b(a(X))) -> A(b(b(c(c(X)))))

C(b(a(X))) -> B(b(c(c(X))))

C(b(a(X))) -> B(c(c(X)))

C(b(a(X))) -> C(c(X))

C(b(a(X))) -> C(X)

Furthermore,

R

↳DPs

→DP Problem 1

↳Narrowing Transformation

**C(b(a( X))) -> C(X)**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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

As a result of transforming the rule

two new Dependency Pairs are created:

C(b(a(X))) -> C(c(X))

C(b(a(b(a(X''))))) -> C(a(a(b(b(c(c(X'')))))))

C(b(a(X''))) -> C(e)

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Narrowing Transformation

**C(b(a(b(a( X''))))) -> C(a(a(b(b(c(c(X'')))))))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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

As a result of transforming the rule

seven new Dependency Pairs are created:

C(b(a(b(a(X''))))) -> C(a(a(b(b(c(c(X'')))))))

C(b(a(b(a(X'''))))) -> C(e)

C(b(a(b(a(X'''))))) -> C(a(e))

C(b(a(b(a(X'''))))) -> C(a(a(e)))

C(b(a(b(a(X'''))))) -> C(a(a(b(e))))

C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))

C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))

C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 3

↳Forward Instantiation Transformation

**C(b(a(b(a( X'''))))) -> C(a(a(b(b(c(e))))))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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.

C(b(a(b(a(X'''))))) -> C(a(e))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 4

↳Forward Instantiation Transformation

**C(b(a(b(a(b(a( X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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.

C(b(a(b(a(X'''))))) -> C(a(a(e)))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 5

↳Forward Instantiation Transformation

**C(b(a(b(a( X'''))))) -> C(a(a(b(b(c(e))))))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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.

C(b(a(b(a(X'''))))) -> C(a(a(b(e))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 6

↳Forward Instantiation Transformation

**C(b(a(b(a(b(a( X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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.

C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))

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

**C(b(a(b(a( X'''))))) -> C(a(a(b(b(c(e))))))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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.

C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 8

↳Forward Instantiation Transformation

**C(b(a( X))) -> C(X)**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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.

C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 9

↳Forward Instantiation Transformation

**C(b(a( X))) -> C(X)**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

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

As a result of transforming the rule

one new Dependency Pair is created:

C(b(a(X))) -> C(X)

C(b(a(b(x'')))) -> C(b(x''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 10

↳Polynomial Ordering

**C(b(a(b( x'')))) -> C(b(x''))**

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

The following dependency pair can be strictly oriented:

C(b(a(b(x'')))) -> C(b(x''))

Additionally, the following usable rule w.r.t. to the implicit AFS can be oriented:

b(X) -> e

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(C(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(e)= 0 _{ }^{ }_{ }^{ }POL(b(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(a(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 11

↳Dependency Graph

c(b(a(X))) -> a(a(b(b(c(c(X))))))

c(X) -> e

a(X) -> e

b(X) -> e

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:01 minutes