a(b(

a(u(

b(c(

b(v(

c(a(

c(w(

u(a(

v(b(

w(c(

R

↳Dependency Pair Analysis

A(b(x)) -> B(a(a(x)))

A(b(x)) -> A(a(x))

A(b(x)) -> A(x)

B(c(x)) -> C(b(b(x)))

B(c(x)) -> B(b(x))

B(c(x)) -> B(x)

C(a(x)) -> A(c(c(x)))

C(a(x)) -> C(c(x))

C(a(x)) -> C(x)

Furthermore,

R

↳DPs

→DP Problem 1

↳Narrowing Transformation

**B(c( x)) -> B(x)**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->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(b(x)) -> B(a(a(x)))

A(b(b(x''))) -> B(a(b(a(a(x'')))))

A(b(u(x''))) -> B(a(x''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Narrowing Transformation

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->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(b(x)) -> A(a(x))

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

A(b(u(x''))) -> A(x'')

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(b(u( x''))) -> A(x'')**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->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:

B(c(x)) -> C(b(b(x)))

B(c(c(x''))) -> C(b(c(b(b(x'')))))

B(c(v(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 4

↳Narrowing Transformation

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->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:

B(c(x)) -> B(b(x))

B(c(c(x''))) -> B(c(b(b(x''))))

B(c(v(x''))) -> B(x'')

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(b(u( x''))) -> A(x'')**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->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:

C(a(x)) -> A(c(c(x)))

C(a(a(x''))) -> A(c(a(c(c(x'')))))

C(a(w(x''))) -> A(c(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

**B(c(v( x''))) -> B(x'')**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->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:

C(a(x)) -> C(c(x))

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

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

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

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

As a result of transforming the rule

three new Dependency Pairs are created:

A(b(x)) -> A(x)

A(b(b(x''))) -> A(b(x''))

A(b(b(b(x'''')))) -> A(b(b(x'''')))

A(b(b(u(x'''')))) -> A(b(u(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

**A(b(b(u( x'''')))) -> A(b(u(x'''')))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

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

As a result of transforming the rule

three new Dependency Pairs are created:

B(c(x)) -> B(x)

B(c(c(x''))) -> B(c(x''))

B(c(c(c(x'''')))) -> B(c(c(x'''')))

B(c(c(v(x'''')))) -> B(c(v(x'''')))

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

**B(c(c(v( x'''')))) -> B(c(v(x'''')))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

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

As a result of transforming the rule

three new Dependency Pairs are created:

C(a(x)) -> C(x)

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

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 10

↳Forward Instantiation Transformation

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

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:

A(b(u(x''))) -> A(x'')

A(b(u(b(b(x''''))))) -> A(b(b(x'''')))

A(b(u(b(u(x''''))))) -> A(b(u(x'''')))

A(b(u(b(b(b(x'''''')))))) -> A(b(b(b(x''''''))))

A(b(u(b(b(u(x'''''')))))) -> A(b(b(u(x''''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 11

↳Forward Instantiation Transformation

**A(b(u(b(b(u( x'''''')))))) -> A(b(b(u(x''''''))))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

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:

B(c(v(x''))) -> B(x'')

B(c(v(c(c(x''''))))) -> B(c(c(x'''')))

B(c(v(c(v(x''''))))) -> B(c(v(x'''')))

B(c(v(c(c(c(x'''''')))))) -> B(c(c(c(x''''''))))

B(c(v(c(c(v(x'''''')))))) -> B(c(c(v(x''''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 12

↳Forward Instantiation Transformation

**B(c(v(c(c(v( x'''''')))))) -> B(c(c(v(x''''''))))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

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:

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

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

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

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 13

↳Polynomial Ordering

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

The following dependency pairs can be strictly oriented:

A(b(u(b(b(u(x'''''')))))) -> A(b(b(u(x''''''))))

A(b(u(b(b(b(x'''''')))))) -> A(b(b(b(x''''''))))

A(b(u(b(u(x''''))))) -> A(b(u(x'''')))

A(b(u(b(b(x''''))))) -> A(b(b(x'''')))

A(b(u(x''))) -> B(a(x''))

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

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

w(c(x)) ->x

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(C(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(v(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(B(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(b(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(a(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(w(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(u(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(A(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 14

↳Dependency Graph

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 15

↳Polynomial Ordering

**B(c(v(c(c(v( x'''''')))))) -> B(c(c(v(x''''''))))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

The following dependency pairs can be strictly oriented:

B(c(v(c(c(v(x'''''')))))) -> B(c(c(v(x''''''))))

B(c(v(c(c(c(x'''''')))))) -> B(c(c(c(x''''''))))

B(c(v(c(v(x''''))))) -> B(c(v(x'''')))

B(c(v(c(c(x''''))))) -> B(c(c(x'''')))

B(c(v(x''))) -> C(b(x''))

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

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

v(b(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

w(c(x)) ->x

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(C(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(v(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(B(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(b(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(a(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(w(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(u(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(A(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 16

↳Dependency Graph

**B(c(c(v( x'''')))) -> B(c(v(x'''')))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 17

↳Polynomial Ordering

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

C(a(w(x''))) -> A(c(x''))

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

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

w(c(x)) ->x

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(C(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(v(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(B(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(b(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(a(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(w(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(u(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(A(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 18

↳Dependency Graph

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

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 19

↳Remaining Obligation(s)

The following remains to be proven:

**A(b(b(b( x'''')))) -> A(b(b(x'''')))**

a(b(x)) -> b(a(a(x)))

a(u(x)) ->x

b(c(x)) -> c(b(b(x)))

b(v(x)) ->x

c(a(x)) -> a(c(c(x)))

c(w(x)) ->x

u(a(x)) ->x

v(b(x)) ->x

w(c(x)) ->x

innermost

Duration:

0:08 minutes