O(0) -> 0

+(0,

+(

+(O(

+(O(

+(I(

+(I(

*(0,

*(

*(O(

*(I(

-(

-(0,

-(O(

-(O(

-(I(

-(I(

R

↳Dependency Pair Analysis

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

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

+'(O(x), I(y)) -> +'(x,y)

+'(I(x), O(y)) -> +'(x,y)

+'(I(x), I(y)) -> O'(+(+(x,y), I(0)))

+'(I(x), I(y)) -> +'(+(x,y), I(0))

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

*'(O(x),y) -> O'(*(x,y))

*'(O(x),y) -> *'(x,y)

*'(I(x),y) -> +'(O(*(x,y)),y)

*'(I(x),y) -> O'(*(x,y))

*'(I(x),y) -> *'(x,y)

-'(O(x), O(y)) -> O'(-(x,y))

-'(O(x), O(y)) -> -'(x,y)

-'(O(x), I(y)) -> -'(-(x,y), I(1))

-'(O(x), I(y)) -> -'(x,y)

-'(I(x), O(y)) -> -'(x,y)

-'(I(x), I(y)) -> O'(-(x,y))

-'(I(x), I(y)) -> -'(x,y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

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

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

The following dependency pairs can be strictly oriented:

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

+'(I(x), I(y)) -> +'(+(x,y), I(0))

+'(I(x), O(y)) -> +'(x,y)

+'(O(x), I(y)) -> +'(x,y)

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

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

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

O(0) -> 0

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

+'(x,_{1}x) -> +'(_{2}x,_{1}x)_{2}

O(x) -> O(_{1}x)_{1}

I(x) -> I(_{1}x)_{1}

+(x,_{1}x) -> +(_{2}x,_{1}x)_{2}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 4

↳Dependency Graph

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

→DP Problem 3

↳AFS

**-'(I( x), I(y)) -> -'(x, y)**

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

The following dependency pairs can be strictly oriented:

-'(I(x), I(y)) -> -'(x,y)

-'(I(x), O(y)) -> -'(x,y)

-'(O(x), I(y)) -> -'(x,y)

-'(O(x), I(y)) -> -'(-(x,y), I(1))

-'(O(x), O(y)) -> -'(x,y)

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

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

O(0) -> 0

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(I(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(-'(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(0)= 0 _{ }^{ }_{ }^{ }POL(1)= 0 _{ }^{ }_{ }^{ }POL(O(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(-(x)_{1}, x_{2})= x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

-'(x,_{1}x) -> -'(_{2}x,_{1}x)_{2}

I(x) -> I(_{1}x)_{1}

O(x) -> O(_{1}x)_{1}

-(x,_{1}x) -> -(_{2}x,_{1}x)_{2}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 5

↳Dependency Graph

→DP Problem 3

↳AFS

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳Argument Filtering and Ordering

***'(I( x), y) -> *'(x, y)**

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

The following dependency pair can be strictly oriented:

*'(I(x),y) -> *'(x,y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

*'(x,_{1}x) -> *'(_{2}x,_{1}x)_{2}

I(x) -> I(_{1}x)_{1}

O(x) -> O(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 6

↳Argument Filtering and Ordering

***'(O( x), y) -> *'(x, y)**

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

The following dependency pair can be strictly oriented:

*'(O(x),y) -> *'(x,y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

*'(x,_{1}x) -> *'(_{2}x,_{1}x)_{2}

O(x) -> O(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 6

↳AFS

...

→DP Problem 7

↳Dependency Graph

O(0) -> 0

+(0,x) ->x

+(x, 0) ->x

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

+(O(x), I(y)) -> I(+(x,y))

+(I(x), O(y)) -> I(+(x,y))

+(I(x), I(y)) -> O(+(+(x,y), I(0)))

*(0,x) -> 0

*(x, 0) -> 0

*(O(x),y) -> O(*(x,y))

*(I(x),y) -> +(O(*(x,y)),y)

-(x, 0) ->x

-(0,x) -> 0

-(O(x), O(y)) -> O(-(x,y))

-(O(x), I(y)) -> I(-(-(x,y), I(1)))

-(I(x), O(y)) -> I(-(x,y))

-(I(x), I(y)) -> O(-(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes