f(c(s(

f(c(s(

g(c(

g(c(s(

R

↳Dependency Pair Analysis

F(c(s(x),y)) -> F(c(x, s(y)))

F(c(s(x), s(y))) -> G(c(x,y))

G(c(x, s(y))) -> G(c(s(x),y))

G(c(s(x), s(y))) -> F(c(x,y))

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

**G(c(s( x), s(y))) -> F(c(x, y))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

The following dependency pairs can be strictly oriented:

G(c(s(x), s(y))) -> F(c(x,y))

F(c(s(x), s(y))) -> G(c(x,y))

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Dependency Graph

**G(c( x, s(y))) -> G(c(s(x), y))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

Using the Dependency Graph the DP problem was split into 2 DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Instantiation Transformation

**G(c( x, s(y))) -> G(c(s(x), y))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

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

As a result of transforming the rule

one new Dependency Pair is created:

G(c(x, s(y))) -> G(c(s(x),y))

G(c(s(x''), s(y''))) -> G(c(s(s(x'')),y''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 5

↳Instantiation Transformation

**G(c(s( x''), s(y''))) -> G(c(s(s(x'')), y''))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

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

As a result of transforming the rule

one new Dependency Pair is created:

G(c(s(x''), s(y''))) -> G(c(s(s(x'')),y''))

G(c(s(s(x'''')), s(y''''))) -> G(c(s(s(s(x''''))),y''''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 7

↳Polynomial Ordering

**G(c(s(s( x'''')), s(y''''))) -> G(c(s(s(s(x''''))), y''''))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

The following dependency pair can be strictly oriented:

G(c(s(s(x'''')), s(y''''))) -> G(c(s(s(s(x''''))),y''''))

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 9

↳Dependency Graph

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Instantiation Transformation

**F(c(s( x), y)) -> F(c(x, s(y)))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

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

As a result of transforming the rule

one new Dependency Pair is created:

F(c(s(x),y)) -> F(c(x, s(y)))

F(c(s(x''), s(y''))) -> F(c(x'', s(s(y''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 6

↳Instantiation Transformation

**F(c(s( x''), s(y''))) -> F(c(x'', s(s(y''))))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

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

As a result of transforming the rule

one new Dependency Pair is created:

F(c(s(x''), s(y''))) -> F(c(x'', s(s(y''))))

F(c(s(x''''), s(s(y'''')))) -> F(c(x'''', s(s(s(y'''')))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 8

↳Polynomial Ordering

**F(c(s( x''''), s(s(y'''')))) -> F(c(x'''', s(s(s(y'''')))))**

f(c(s(x),y)) -> f(c(x, s(y)))

f(c(s(x), s(y))) -> g(c(x,y))

g(c(x, s(y))) -> g(c(s(x),y))

g(c(s(x), s(y))) -> f(c(x,y))

The following dependency pair can be strictly oriented:

F(c(s(x''''), s(s(y'''')))) -> F(c(x'''', s(s(s(y'''')))))

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Duration:

0:00 minutes