+(0,

+(s(

-(0,

-(

-(s(

R

↳Dependency Pair Analysis

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

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

+(0,y) ->y

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

-(0,y) -> 0

-(x, 0) ->x

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

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(s(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}

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳AFS

+(0,y) ->y

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

-(0,y) -> 0

-(x, 0) ->x

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

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

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

+(0,y) ->y

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

-(0,y) -> 0

-(x, 0) ->x

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

The following dependency pair can be strictly oriented:

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

There are no usable rules 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(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 4

↳Dependency Graph

+(0,y) ->y

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

-(0,y) -> 0

-(x, 0) ->x

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

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes