Global Constraint Catalog

4.4.2. Functional dependency invariants involving three constraints

Proposition 188 Given the constraints

$𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗$ $(𝙸, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚙𝚎𝚊𝚔$ $(𝙿, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚟𝚊𝚕𝚕𝚎𝚢$ $(𝚅, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ :

𝙸 = 𝙿 + 𝚅

Proof 186 By definition of the $𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗$ , $𝚙𝚎𝚊𝚔$ and $𝚟𝚊𝚕𝚕𝚎𝚢$ constraints.

Proposition 189 Given the constraints

$𝚕𝚎𝚗𝚐𝚝𝚑_𝚏𝚒𝚛𝚜𝚝_𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎$ $(𝙻𝙴𝙽_𝙵𝙸𝚁𝚂𝚃, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚕𝚎𝚗𝚐𝚝𝚑_𝚕𝚊𝚜𝚝_𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎$ $(𝙻𝙴𝙽_𝙻𝙰𝚂𝚃, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚗𝚟𝚊𝚕𝚞𝚎$ $(𝙽𝚅𝙰𝙻, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ :

𝙽𝚅𝙰𝙻 > 1 \Rightarrow 𝙻𝙴𝙽_𝙵𝙸𝚁𝚂𝚃 + 𝙻𝙴𝙽_𝙻𝙰𝚂𝚃 + 𝙽𝚅𝙰𝙻 - 2 \leq | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |

Proof 187 Since we have at least two distinct values the first and last sequences do not overlap. Since we have at least $𝙽𝚅𝙰𝙻$ distinct values we have at least $𝙽𝚅𝙰𝙻 - 2$ additional values excluding the first and last sequences, which use at most two distinct values.

Proposition 190 Given the constraints

$𝚕𝚘𝚗𝚐𝚎𝚜𝚝_𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐_𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎$ $(𝙻, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚊𝚡𝚒𝚖𝚞𝚖$ $(𝙼𝙰𝚇, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ $(𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ :

𝙼𝙰𝚇 - 𝙼𝙸𝙽 \geq 𝙻

Proof 188 By definition of the $𝚕𝚘𝚗𝚐𝚎𝚜𝚝_𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐_𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎$ constraint.

Proposition 191 Given the constraints

$𝚕𝚘𝚗𝚐𝚎𝚜𝚝_𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐_𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎$ $(𝙻, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚊𝚡𝚒𝚖𝚞𝚖$ $(𝙼𝙰𝚇, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ $(𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ :

𝙼𝙰𝚇 - 𝙼𝙸𝙽 \geq 𝙻

Proof 189 By definition of the $𝚕𝚘𝚗𝚐𝚎𝚜𝚝_𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐_𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎$ constraint.

Proposition 192 Given the constraints

$𝚖𝚊𝚡𝚒𝚖𝚞𝚖$ $(𝙼𝙰𝚇, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ $(𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚗𝚟𝚊𝚕𝚞𝚎$ $(𝙽𝚅𝙰𝙻, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ :

𝙽𝚅𝙰𝙻 \leq 𝙼𝙰𝚇 - 𝙼𝙸𝙽 + 1

Proof 190 Since taking all values between $𝙼𝙸𝙽$ and $𝙼𝙰𝚇$ gives the maximum number of distinct values.

Proposition 193 Given the constraints

$𝚖𝚊𝚡𝚒𝚖𝚞𝚖$ $(𝙼𝙰𝚇, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ $(𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚗𝚟𝚊𝚕𝚞𝚎$ $(𝙽𝚅𝙰𝙻, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ :

𝙼𝙰𝚇 > 𝙼𝙸𝙽 \Rightarrow 𝙽𝚅𝙰𝙻 > 1

Proof 191 Since at least two distinct values if $𝙼𝙸𝙽$ and $𝙼𝙰𝚇$ are distinct.

Proposition 194 Given the constraints

$𝚖𝚊𝚡𝚒𝚖𝚞𝚖$ $(𝙼𝙰𝚇, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ $(𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚜𝚞𝚖_𝚌𝚝𝚛$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, =, 𝚂𝚄𝙼)$ :

𝚂𝚄𝙼 \geq (| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | - 1) \cdot 𝙼𝙸𝙽 + 𝙼𝙰𝚇

Proof 192 Since has also to use the maximum value at least once in the sum.

Proposition 195 Given the constraints

$𝚖𝚊𝚡𝚒𝚖𝚞𝚖$ $(𝙼𝙰𝚇, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ $(𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ ,
$𝚜𝚞𝚖_𝚌𝚝𝚛$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, =, 𝚂𝚄𝙼)$ :

𝚂𝚄𝙼 \leq (| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | - 1) \cdot 𝙼𝙰𝚇 + 𝙼𝙸𝙽

Proof 193 Since has also to use the minimum value at least once in the sum.

Global Constraint Catalog

4. Further Topics

4.4.2. Functional dependency invariants involving three constraints