+(

+(

+(+(

*(

*(+(

R

↳Dependency Pair Analysis

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

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

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

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

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

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Polo

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

+(x, 0) ->x

+(x, i(x)) -> 0

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

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

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

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳Polo

+(x, 0) ->x

+(x, i(x)) -> 0

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

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polynomial Ordering

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

+(x, 0) ->x

+(x, i(x)) -> 0

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

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

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

innermost

The following dependency pairs can be strictly oriented:

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

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

There are no usable rules for innermost 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.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 4

↳Dependency Graph

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

+(x, 0) ->x

+(x, i(x)) -> 0

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

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

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

innermost

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

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 4

↳DGraph

...

→DP Problem 5

↳Polynomial Ordering

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

+(x, 0) ->x

+(x, i(x)) -> 0

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

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

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

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost 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(+(x)_{1}, x_{2})= 1 + x _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 4

↳DGraph

...

→DP Problem 6

↳Dependency Graph

+(x, 0) ->x

+(x, i(x)) -> 0

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

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes