*(

*(1,

*(+(

*(

R

↳Dependency Pair Analysis

*'(x, *(y,z)) -> *'(otimes(x,y),z)

*'(+(x,y),z) -> *'(x,z)

*'(+(x,y),z) -> *'(y,z)

*'(x, oplus(y,z)) -> *'(x,y)

*'(x, oplus(y,z)) -> *'(x,z)

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

***'( x, oplus(y, z)) -> *'(x, z)**

*(x, *(y,z)) -> *(otimes(x,y),z)

*(1,y) ->y

*(+(x,y),z) -> oplus(*(x,z), *(y,z))

*(x, oplus(y,z)) -> oplus(*(x,y), *(x,z))

The following dependency pairs can be strictly oriented:

*'(x, oplus(y,z)) -> *'(x,z)

*'(x, oplus(y,z)) -> *'(x,y)

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(*'(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(otimes(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(*(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(oplus(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 0 _{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Dependency Graph

***'(+( x, y), z) -> *'(y, z)**

*(x, *(y,z)) -> *(otimes(x,y),z)

*(1,y) ->y

*(+(x,y),z) -> oplus(*(x,z), *(y,z))

*(x, oplus(y,z)) -> oplus(*(x,y), *(x,z))

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

↳Polynomial Ordering

***'( x, *(y, z)) -> *'(otimes(x, y), z)**

*(x, *(y,z)) -> *(otimes(x,y),z)

*(1,y) ->y

*(+(x,y),z) -> oplus(*(x,z), *(y,z))

*(x, oplus(y,z)) -> oplus(*(x,y), *(x,z))

The following dependency pair can be strictly oriented:

*'(x, *(y,z)) -> *'(otimes(x,y),z)

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(*'(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(otimes(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(*(x)_{1}, x_{2})= 1 + x _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 5

↳Dependency Graph

*(x, *(y,z)) -> *(otimes(x,y),z)

*(1,y) ->y

*(+(x,y),z) -> oplus(*(x,z), *(y,z))

*(x, oplus(y,z)) -> oplus(*(x,y), *(x,z))

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Polynomial Ordering

***'(+( x, y), z) -> *'(x, z)**

*(x, *(y,z)) -> *(otimes(x,y),z)

*(1,y) ->y

*(+(x,y),z) -> oplus(*(x,z), *(y,z))

*(x, oplus(y,z)) -> oplus(*(x,y), *(x,z))

The following dependency pairs can be strictly oriented:

*'(+(x,y),z) -> *'(x,z)

*'(+(x,y),z) -> *'(y,z)

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(*'(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

Duration:

0:00 minutes