+(

+(minus(

minus(0) -> 0

minus(minus(

minus(+(

*(

*(

*(

*(

R

↳Dependency Pair Analysis

MINUS(+(x,y)) -> +'(minus(y), minus(x))

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

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

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

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

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

*'(x, minus(y)) -> MINUS(*(x,y))

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

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

+(x, 0) ->x

+(minus(x),x) -> 0

minus(0) -> 0

minus(minus(x)) ->x

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

*(x, 1) ->x

*(x, 0) -> 0

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

*(x, minus(y)) -> minus(*(x,y))

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

trivial

resulting in one new DP problem.

Used Argument Filtering System:

MINUS(x) -> MINUS(_{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

+(minus(x),x) -> 0

minus(0) -> 0

minus(minus(x)) ->x

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

*(x, 1) ->x

*(x, 0) -> 0

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

*(x, minus(y)) -> minus(*(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

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

+(x, 0) ->x

+(minus(x),x) -> 0

minus(0) -> 0

minus(minus(x)) ->x

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

*(x, 1) ->x

*(x, 0) -> 0

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

*(x, minus(y)) -> minus(*(x,y))

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

trivial

resulting in one new DP problem.

Used Argument Filtering System:

*'(x,_{1}x) -> *'(_{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 4

↳Dependency Graph

+(x, 0) ->x

+(minus(x),x) -> 0

minus(0) -> 0

minus(minus(x)) ->x

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

*(x, 1) ->x

*(x, 0) -> 0

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

*(x, minus(y)) -> minus(*(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes