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

f(a, b) -> f(s(a), c)

f(s(

f(c, c) -> f(a, a)

R

↳Dependency Pair Analysis

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

F(a, b) -> F(s(a), c)

F(s(X), c) -> F(X, c)

F(c, c) -> F(a, a)

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

**F(c, c) -> F(a, a)****F(s( X), c) -> F(X, c)**

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

f(a, b) -> f(s(a), c)

f(s(X), c) -> f(X, c)

f(c, c) -> f(a, a)

The following dependency pair can be strictly oriented:

F(c, c) -> F(a, a)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c)= 1 _{ }^{ }_{ }^{ }POL(b)= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(a)= 0 _{ }^{ }_{ }^{ }POL(F(x)_{1}, x_{2})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Dependency Graph

**F(s( X), c) -> F(X, c)**

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

f(a, b) -> f(s(a), c)

f(s(X), c) -> f(X, c)

f(c, c) -> f(a, a)

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

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Forward Instantiation Transformation

**F(s( X), c) -> F(X, c)**

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

f(a, b) -> f(s(a), c)

f(s(X), c) -> f(X, c)

f(c, c) -> f(a, a)

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:

F(s(X), c) -> F(X, c)

F(s(s(X'')), c) -> F(s(X''), c)

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Forward Instantiation Transformation

**F(s(s( X'')), c) -> F(s(X''), c)**

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

f(a, b) -> f(s(a), c)

f(s(X), c) -> f(X, c)

f(c, c) -> f(a, a)

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:

F(s(s(X'')), c) -> F(s(X''), c)

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 5

↳Polynomial Ordering

**F(s(s(s( X''''))), c) -> F(s(s(X'''')), c)**

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

f(a, b) -> f(s(a), c)

f(s(X), c) -> f(X, c)

f(c, c) -> f(a, a)

The following dependency pair can be strictly oriented:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c)= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(F(x)_{1}, x_{2})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 6

↳Dependency Graph

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

f(a, b) -> f(s(a), c)

f(s(X), c) -> f(X, c)

f(c, c) -> f(a, a)

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes