p(s(

fact(0) -> s(0)

fact(s(

*(0,

*(s(

+(

+(

R

↳Dependency Pair Analysis

FACT(s(x)) -> *'(s(x), fact(p(s(x))))

FACT(s(x)) -> FACT(p(s(x)))

FACT(s(x)) -> P(s(x))

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Polo

→DP Problem 3

↳Rw

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

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

+(x, s(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 w.r.t. to the implicit 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 _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 4

↳Dependency Graph

→DP Problem 2

↳Polo

→DP Problem 3

↳Rw

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

+(x, s(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

→DP Problem 3

↳Rw

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

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

+(x, s(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 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(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 5

↳Dependency Graph

→DP Problem 3

↳Rw

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 3

↳Rewriting Transformation

**FACT(s( x)) -> FACT(p(s(x)))**

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

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

innermost

On this DP problem, a Rewriting SCC transformation can be performed.

As a result of transforming the rule

one new Dependency Pair is created:

FACT(s(x)) -> FACT(p(s(x)))

FACT(s(x)) -> FACT(x)

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 3

↳Rw

→DP Problem 6

↳Polynomial Ordering

**FACT(s( x)) -> FACT(x)**

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

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

innermost

The following dependency pair can be strictly oriented:

FACT(s(x)) -> FACT(x)

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(FACT(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 2

↳Polo

→DP Problem 3

↳Rw

→DP Problem 6

↳Polo

...

→DP Problem 7

↳Dependency Graph

p(s(x)) ->x

fact(0) -> s(0)

fact(s(x)) -> *(s(x), fact(p(s(x))))

*(0,y) -> 0

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

+(x, 0) ->x

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes