+(

+(

+(0, s(

s(+(0,

R

↳Dependency Pair Analysis

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

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

S(+(0,y)) -> S(y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

**S(+(0, y)) -> S(y)**

+(x, 0) ->x

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

+(0, s(y)) -> s(y)

s(+(0,y)) -> s(y)

The following dependency pair can be strictly oriented:

S(+(0,y)) -> S(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(0)= 0 _{ }^{ }_{ }^{ }POL(S(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳AFS

+(x, 0) ->x

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

+(0, s(y)) -> s(y)

s(+(0,y)) -> s(y)

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

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

+(x, 0) ->x

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

+(0, s(y)) -> s(y)

s(+(0,y)) -> s(y)

The following dependency pair can be strictly oriented:

+'(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(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 2

↳AFS

→DP Problem 4

↳Dependency Graph

+(x, 0) ->x

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

+(0, s(y)) -> s(y)

s(+(0,y)) -> s(y)

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes