Global Constraint Catalog

4.4.3. Functional dependency invariants involving four constraints

Proposition 196 Given the constraints

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

𝙽𝚅𝙰𝙻 > 1 \Rightarrow 2 \cdot 𝙿 \leq | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | - max (1, 𝙻𝙴𝙽_𝙵𝙸𝚁𝚂𝚃) - max (1, 𝙻𝙴𝙽_𝙻𝙰𝚂𝚃) + 1

Proof 194 Beside the first and the last sequence with a small value, we alternate between large and small values.

Proposition 197 Given the constraints

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

𝙽𝚅𝙰𝙻 > 1 \Rightarrow 2 \cdot 𝚅 \leq | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | - max (1, 𝙻𝙴𝙽_𝙵𝙸𝚁𝚂𝚃) - max (1, 𝙻𝙴𝙽_𝙻𝙰𝚂𝚃) + 1

Proof 195 Beside the first and the last sequence with a large value, we alternate between small and large values.