not(true) -> false

not(false) -> true

odd(0) -> false

odd(s(

+(

+(

+(s(

R

↳Dependency Pair Analysis

ODD(s(x)) -> NOT(odd(x))

ODD(s(x)) -> ODD(x)

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Polo

**ODD(s( x)) -> ODD(x)**

not(true) -> false

not(false) -> true

odd(0) -> false

odd(s(x)) -> not(odd(x))

+(x, 0) ->x

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

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

innermost

The following dependency pair can be strictly oriented:

ODD(s(x)) -> ODD(x)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(ODD(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳Polo

not(true) -> false

not(false) -> true

odd(0) -> false

odd(s(x)) -> not(odd(x))

+(x, 0) ->x

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polynomial Ordering

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

not(true) -> false

not(false) -> true

odd(0) -> false

odd(s(x)) -> not(odd(x))

+(x, 0) ->x

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

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

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost 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}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 4

↳Polynomial Ordering

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

not(true) -> false

not(false) -> true

odd(0) -> false

odd(s(x)) -> not(odd(x))

+(x, 0) ->x

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

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

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost 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 _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 4

↳Polo

...

→DP Problem 5

↳Dependency Graph

not(true) -> false

not(false) -> true

odd(0) -> false

odd(s(x)) -> not(odd(x))

+(x, 0) ->x

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes