fac(0) -> 1

fac(s(

fac(0) -> s(0)

floop(0,

floop(s(

*(

*(

+(

+(

1 -> s(0)

R

↳Dependency Pair Analysis

FAC(0) -> 1'

FAC(s(x)) -> *'(s(x), fac(x))

FAC(s(x)) -> FAC(x)

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

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

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

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

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

fac > * > + > s

fac > 1 > 0

fac > 1 > s

floop > * > + > s

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

1 -> 1

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 5

↳Dependency Graph

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

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

→DP Problem 4

↳AFS

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

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{fac, 1} > 0

{fac, 1} > * > + > s

floop > * > + > s

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

1 -> 1

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 6

↳Dependency Graph

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

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

→DP Problem 4

↳AFS

**FAC(s( x)) -> FAC(x)**

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

The following dependency pair can be strictly oriented:

FAC(s(x)) -> FAC(x)

The following rules can be oriented:

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{fac, 1} > 0

{fac, 1} > * > + > s

floop > * > + > s

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

1 -> 1

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 7

↳Dependency Graph

→DP Problem 4

↳AFS

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳Argument Filtering and Ordering

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

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

*(x, 0) -> 0

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

+(x, 0) ->x

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

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

1 -> s(0)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{fac, 1} > 0

{fac, 1} > * > + > s

FLOOP > * > + > s

floop > * > + > s

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

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

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

1 -> 1

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

→DP Problem 8

↳Dependency Graph

fac(0) -> 1

fac(s(x)) -> *(s(x), fac(x))

fac(0) -> s(0)

floop(0,y) ->y

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

*(x, 0) -> 0

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

+(x, 0) ->x

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

1 -> s(0)

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes