### 3.7.61. Contractible

$•$   $\mathrm{𝚊𝚕𝚕}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚎𝚡𝚊𝚌𝚝𝚕𝚢}_𝚔_\mathrm{𝚙𝚘𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕}_\mathrm{𝚒𝚗𝚌𝚘𝚖𝚙𝚊𝚛𝚊𝚋𝚕𝚎}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚜𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚘𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚘𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚕𝚕𝚙𝚎𝚛𝚖}$ (suffix-contractible wrt $\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.\mathrm{𝚟𝚎𝚌}$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚍𝚒𝚏𝚏}_\mathtt{0}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚍𝚒𝚏𝚏}_\mathtt{0}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚄𝙿}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚄𝙿}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚄𝙿}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚂𝙴𝚀}=1$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚟𝚊𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚟𝚊𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚊𝚛𝚒𝚝𝚑}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚘𝚛}$ (contractible wrt $\left[\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}\right]$),

$•$   $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$ and

$\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$),

$•$   $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚜𝚜𝚒𝚐𝚗}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝𝚜}$ (contractible wrt $\mathrm{𝙸𝚃𝙴𝙼𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚊𝚜𝚜𝚒𝚐𝚗}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝙸𝚃𝙴𝙼𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚋𝚒𝚗}_\mathrm{𝚙𝚊𝚌𝚔𝚒𝚗𝚐}$ (contractible wrt $\mathrm{𝙸𝚃𝙴𝙼𝚂}$),

$•$   $\mathrm{𝚋𝚒𝚗}_\mathrm{𝚙𝚊𝚌𝚔𝚒𝚗𝚐}_\mathrm{𝚌𝚊𝚙𝚊}$ (contractible wrt $\mathrm{𝙸𝚃𝙴𝙼𝚂}$),

$•$   $\mathrm{𝚌𝚊𝚕𝚎𝚗𝚍𝚊𝚛}$ (contractible wrt $\mathrm{𝙸𝙽𝚂𝚃𝙰𝙽𝚃𝚂}$),

$•$   $\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[\ne ,<,\ge ,>,\le \right]$ and $\mathrm{𝙽𝙲𝙷𝙰𝙽𝙶𝙴}=0$),

$•$   $\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[=,<,\ge ,>,\le \right]$ and

$\mathrm{𝙽𝙲𝙷𝙰𝙽𝙶𝙴}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}-1|$),

$•$   $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚌𝚘𝚖𝚙𝚊𝚛𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}$ (contractible wrt $\left[\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}\right]$ when $\mathrm{𝙲𝙾𝚄𝙽𝚃}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚌𝚘𝚗𝚝𝚊𝚒𝚗𝚜}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚌𝚘𝚞𝚗𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚌𝚘𝚞𝚗𝚝𝚜}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚌𝚘𝚟𝚎𝚛𝚜}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚙𝚛𝚘𝚍𝚞𝚌𝚝}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍$ (contractible wrt $\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚠𝚒𝚝𝚑}_\mathrm{𝚕𝚎𝚟𝚎𝚕}_\mathrm{𝚘𝚏}_\mathrm{𝚙𝚛𝚒𝚘𝚛𝚒𝚝𝚢}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[\le \right]$ and $\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)\ge 0$),

$•$   $\mathrm{𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚏𝚏𝚗}$ (contractible wrt $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚏𝚏𝚗}_\mathrm{𝚌𝚘𝚕𝚞𝚖𝚗}$ (contractible wrt $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚏𝚏𝚗}_\mathrm{𝚒𝚗𝚌𝚕𝚞𝚍𝚎}$ (contractible wrt $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚝𝚊𝚜𝚔𝚜}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}\mathtt{1}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚝𝚊𝚜𝚔𝚜}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚎𝚗𝚍}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚝𝚊𝚛𝚝}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}$ (contractible wrt $\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚜𝚞𝚖}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $𝚔_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝚂}$),

$•$   $𝚔_\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚜𝚊𝚖𝚎}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚂𝙴𝚃𝚂}$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}$ (suffix-contractible wrt $\left[\mathrm{𝙻𝙾𝚆𝙴𝚁}_\mathrm{𝙱𝙾𝚄𝙽𝙳},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚄𝙿𝙿𝙴𝚁}_\mathrm{𝙱𝙾𝚄𝙽𝙳}_\mathrm{𝙱𝙾𝚄𝙽𝙳}\right]$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$ (suffix-contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}.\mathrm{𝚟𝚎𝚌}$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚎𝚚𝚞𝚊𝚕}$ (contractible wrt $\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right]$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$ (suffix-contractible wrt $\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right]$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$ (suffix-contractible wrt $\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right]$),

$•$   $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}_\mathrm{𝚊𝚕𝚕𝚙𝚎𝚛𝚖}$ (suffix-contractible wrt $\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right]$),

$•$   $\mathrm{𝚖𝚊𝚡}_\mathrm{𝚘𝚌𝚌}_\mathrm{𝚘𝚏}_\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙼𝙰𝚇}=1$),

$•$   $\mathrm{𝚖𝚊𝚡}_\mathrm{𝚘𝚌𝚌}_\mathrm{𝚘𝚏}_\mathrm{𝚜𝚘𝚛𝚝𝚎𝚍}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙼𝙰𝚇}=1$),

$•$   $\mathrm{𝚖𝚊𝚡}_\mathrm{𝚘𝚌𝚌}_\mathrm{𝚘𝚏}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙼𝙰𝚇}=1$),

$•$   $\mathrm{𝚖𝚎𝚎𝚝}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚖𝚞𝚕𝚝𝚒}_\mathrm{𝚒𝚗𝚝𝚎𝚛}_\mathrm{𝚍𝚒𝚜𝚝𝚊𝚗𝚌𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚖𝚞𝚕𝚝𝚒}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚗𝚊𝚗𝚍}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝙴𝚀𝚄𝙸𝚅}=1$ and $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0$),

$•$   $\mathrm{𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝙴𝚀𝚄𝙸𝚅}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝙻}=1$ and $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0$),

$•$   $\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝙻}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚗𝚘}_\mathrm{𝚙𝚎𝚊𝚔}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚗𝚘}_\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚗𝚘𝚗}_\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚗𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚅𝙰𝚁}=1$),

$•$   $\mathrm{𝚗𝚘𝚝}_\mathrm{𝚒𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚗𝚙𝚊𝚒𝚛}$ (contractible wrt $\mathrm{𝙿𝙰𝙸𝚁𝚂}$ when $\mathrm{𝙽𝙿𝙰𝙸𝚁𝚂}=1$ and $|\mathrm{𝙿𝙰𝙸𝚁𝚂}|>0$),

$•$   $\mathrm{𝚗𝚙𝚊𝚒𝚛}$ (contractible wrt $\mathrm{𝙿𝙰𝙸𝚁𝚂}$ when $\mathrm{𝙽𝙿𝙰𝙸𝚁𝚂}=|\mathrm{𝙿𝙰𝙸𝚁𝚂}|$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝙻}=1$ and $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝙻}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚘𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}$ when $\mathrm{𝙽𝚅𝙰𝙻}=0$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚘𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$ when $\mathrm{𝙽𝚅𝙰𝙻}=0$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[=\right]$ and $\mathrm{𝙻𝙸𝙼𝙸𝚃}=1$ and $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[=\right]$ and $\mathrm{𝙻𝙸𝙼𝙸𝚃}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$),

$•$   $\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙽𝚅𝙴𝙲}=1$ and $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|>0$),

$•$   $\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙽𝚅𝙴𝙲}=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|$,

$•$   $\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚖𝚘𝚗𝚐}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚅𝙰𝚁}=0$),

$•$   $\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$),

$•$   $\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙽𝚅𝙴𝙲}=1$ and $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|>0$),

$•$   $\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}$ (contractible wrt $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙽𝚅𝙴𝙲}=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|$),

$•$   $\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚘𝚛𝚒}_\mathrm{𝚜𝚒𝚣}_\mathrm{𝚎𝚗𝚍}$ (contractible wrt $\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}$),

$•$   $\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ (suffix-contractible wrt $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$),

$•$   $\mathrm{𝚙𝚊𝚝𝚝𝚎𝚛𝚗}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚊𝚝𝚝𝚎𝚛𝚗}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚎𝚊𝚔}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $𝙽=0$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[=\right]$ and $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}=1$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[=\right]$ and $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}=1$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚙𝚛𝚘𝚍𝚞𝚌𝚝}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$ and

$\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>0$),

$•$   $\mathrm{𝚛𝚊𝚗𝚐𝚎}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$),

$•$   $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚌𝚊𝚕𝚊𝚛}_\mathrm{𝚙𝚛𝚘𝚍𝚞𝚌𝚝}$ (contractible wrt $\mathrm{𝙻𝙸𝙽𝙴𝙰𝚁𝚃𝙴𝚁𝙼}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$,

$\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝙻𝙸𝙽𝙴𝙰𝚁𝚃𝙴𝚁𝙼}.\mathrm{𝚌𝚘𝚎𝚏𝚏}\right)\ge 0$ and $\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝙻𝙸𝙽𝙴𝙰𝚁𝚃𝙴𝚁𝙼}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$),

$•$   $\mathrm{𝚜𝚎𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚂𝙴𝚀}=1$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝚂𝙴𝚀}=1$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚝𝚒𝚖𝚎}_\mathrm{𝚠𝚒𝚗𝚍𝚘𝚠}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚝𝚒𝚖𝚎}_\mathrm{𝚠𝚒𝚗𝚍𝚘𝚠}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚜𝚝𝚊𝚛𝚝}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚝𝚒𝚖𝚎}_\mathrm{𝚠𝚒𝚗𝚍𝚘𝚠}_\mathrm{𝚜𝚞𝚖}$ (contractible wrt $\mathrm{𝚃𝙰𝚂𝙺𝚂}$),

$•$   $\mathrm{𝚜𝚖𝚘𝚘𝚝𝚑}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝙲𝙷𝙰𝙽𝙶𝙴}=0$),

$•$   $\mathrm{𝚜𝚖𝚘𝚘𝚝𝚑}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙽𝙲𝙷𝙰𝙽𝙶𝙴}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-1$),

$•$   $\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$ and

$\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[\ge ,>\right]$ and

$\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\le 0$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚞𝚋𝚎𝚜}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$ and

$\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚞𝚋𝚎𝚜}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[\ge ,>\right]$ and

$\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\le 0$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚙𝚘𝚠𝚎𝚛𝚜}\mathtt{4}_\mathrm{𝚌𝚝𝚛}$ ($\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$) when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$,

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚙𝚘𝚠𝚎𝚛𝚜}\mathtt{5}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$ and

$\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚙𝚘𝚠𝚎𝚛𝚜}\mathtt{5}_\mathrm{𝚌𝚝𝚛}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[\ge ,>\right]$ and

$\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\le 0$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚙𝚘𝚠𝚎𝚛𝚜}\mathtt{6}_\mathrm{𝚌𝚝𝚛}$ ($\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$) when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$,

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚖𝚎𝚗𝚝𝚜}$ (prefix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚖𝚎𝚗𝚝𝚜}$ (suffix-contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$),

$•$   $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚜𝚚𝚞𝚊𝚛𝚎𝚜}_\mathrm{𝚌𝚝𝚛}$ ($\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$) when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$,

$•$   $\mathrm{𝚝𝚠𝚒𝚗}$ (contractible wrt $\mathrm{𝙿𝙰𝙸𝚁𝚂}$),

$•$   $\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}$ ($\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚞𝚜𝚎𝚜}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}$),

$•$   $\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ (contractible wrt $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $𝙽=0$),

$•$   $\mathrm{𝚟𝚎𝚌}_\mathrm{𝚎𝚚}_\mathrm{𝚝𝚞𝚙𝚕𝚎}$ (contractible wrt $\left[\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚃𝚄𝙿𝙻𝙴}\right]$).

A contractible constraint is a constraint for which, given any satisfied ground instance, one can remove any item from one of its collection arguments, without affecting that the resulting constraint still holds, assuming all its restrictions hold. A typical example of a contractible constraint is the $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$ constraint: given any ground satisfied instance, e.g., $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$$\left(〈3,8,1〉\right)$, we can remove any value from its unique argument without affecting that the resulting constraint still holds. We generalise slightly the original definition of contractibility introduced by  [Maher09c] in the following ways:

• The sequence of variables is replaced by a collection. Consequently, variables are replaced by items. For instance, in the context of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$$\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)$ constraint, we can remove any task from $\mathrm{𝚃𝙰𝚂𝙺𝚂}$ from any satisfied instance without affecting that the resulting constraint still holds (e.g., if the resource limit $\mathrm{𝙻𝙸𝙼𝙸𝚃}$ is not exceeded at any point in time, this still is the case if we remove any task, i.e., since task heights are restricted to be non negative).

• Since the constraint may have more than one argument, one has to explicitly specify the argument from which one may remove items.

• Items cannot only be removed from the end of a collection like in  [Maher09c], but also from the beginning or from any part. Allowing to remove items from the beginning is called prefix-contractibility, while permitting to remove items from the end is called suffix-contractibility. Removing items from any part is just called contractibility. As an example, consider the $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$$\left(\mathrm{𝙻𝙾𝚆},\mathrm{𝚄𝙿},\mathrm{𝚂𝙴𝚀},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)$ constraint which forces all sequences of $\mathrm{𝚂𝙴𝚀}$ consecutive variables of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ to be assigned at least $\mathrm{𝙻𝙾𝚆}$ and at most $\mathrm{𝚄𝙿}$ values from $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$. The constraint $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$ is not contractible w.r.t. the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$, since removing an item in the middle of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ creates a new sequence for which the restriction with respect to $\mathrm{𝙻𝙾𝚆}$ and $\mathrm{𝚄𝙿}$ may not hold. However, if we restrict ourselves to removing just a prefix or suffix from $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$, then the corresponding $\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}$ constraint still holds, since no new sequence is created.

• A constraint may be contractible only if certain restrictions apply to some of its arguments. This is done by explicitly providing a list of restrictions, each restriction corresponding to one of the restrictions described in Section 2.2.3. We call this conditional contractibility. Given a source and a target constraint (i.e., the target constraint corresponds to the source constraint from which we remove some items in some arguments) all arguments of the target constraint should be identical to the arguments of the source constraint, except:

• Argument corresponding to a collection from which we remove items.

• Argument $\mathrm{𝚊𝚛𝚐}$ occurring in the list of conditional restrictions with of restriction of the form $\mathrm{𝚊𝚛𝚐}=f\left(|c|\right)$, where $c$ is an argument corresponding to a collection from which we remove items and $f$ a function.

In addition, all restrictions from the list of restrictions should apply both to the source and target constraints.

We now provide two examples of conditional contractibility with respect to the $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(\mathrm{𝙽𝚅𝙰𝚁},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)$ constraint, which forces $\mathrm{𝙽𝚅𝙰𝚁}$ to be the number of variables of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ that are assigned a value in $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$.

• In general $\mathrm{𝚊𝚖𝚘𝚗𝚐}$ is not contractible since removing an item from $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ may change the value of $\mathrm{𝙽𝚅𝙰𝚁}$. However, given a ground satisfied instance for which $\mathrm{𝙽𝚅𝙰𝚁}$ is set to 0, we can remove any item from $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ without affecting that the constraint still holds. In this context, the two arguments $\mathrm{𝙽𝚅𝙰𝚁}$ and $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ are left unchanged within the source and the target constraint.

As an illustration, consider the source constraint $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(0,〈2,4,2〉,〈1,5〉\right)$ and the target constraint $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(0,〈2,2〉,〈1,5〉\right)$. Since $\mathrm{𝙽𝚅𝙰𝚁}$ is set to 0 both in the source and the target constraint and since $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ is set to the same list of values both in the source and the target constraint, we have that $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(0,〈2,4,2〉,〈1,5〉\right)$ implies $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(0,〈2,2〉,〈1,5〉\right)$.

• Similarly, when $\mathrm{𝙽𝚅𝙰𝚁}$ is equal to $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$, all variables are assigned a value in $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$. In this context, we can remove any variable from $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ to get a new constraint that still holds, provided that the restriction $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$ still holds. In this example only the argument $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ is left unchanged between the source and the target constraint. $\mathrm{𝙽𝚅𝙰𝚁}$ changes since it occurs in a restriction of the form $\mathrm{𝙽𝚅𝙰𝚁}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$ in the list of conditional restrictions.

As an illustration, consider the source constraint $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(3,〈2,4,2〉,〈0,2,4,6,8〉\right)$ and the target constraint $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(2,〈4,2〉,〈0,2,4,6,8〉\right)$. Since $\mathrm{𝙽𝚅𝙰𝚁}$ is set to the number of items of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection both in the source and the target constraint, and since $\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}$ is set to the same list of values both in the source and the target constraint, we have that $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(3,〈2,4,2〉,〈0,2,4,6,8〉\right)$ implies $\mathrm{𝚊𝚖𝚘𝚗𝚐}$$\left(2,〈4,2〉,〈0,2,4,6,8〉\right)$.

• Finally, a last extension corresponds to the fact that the sequence of variables from which we remove elements may be replaced by several collections. In this context, items are removed simultaneously from all collections from exactly the same set of positions. A set of collections is either defined by a list of collections, or by a collection and one of its attributes, which is itself a collection.

As a first example, consider the $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)$ constraint, which given two vectors each defined by a collection of variables of the same length, forces that $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ is lexicographically greater than or equal to $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$. We have that $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$ is suffix-contractible with respect to $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ and $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$. This means that we can remove the $k$ $\left(1\le k\le |\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|\right)$ last items from collections $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ and $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$. Note that the $k$ items should be removed from both collections simultaneously. As an illustration, consider the source constraint $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$$\left(〈5,2,8,9〉,〈5,2,6,2〉\right)$ and the target constraint $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$$\left(〈5,2,8〉,〈5,2,6〉\right)$. Since $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$ is suffix-contractible with respect to the two collections $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}$ and $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}$, we have that $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$$\left(〈5,2,8,9〉,〈5,2,6,2〉\right)$ implies $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}$$\left(〈5,2,8〉,〈5,2,6〉\right)$.

As a second example, consider the $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}\right)$ constraint, which given a collection of vectors each of them defined by a collection of variables of the same length, forces the ${i}^{\mathrm{𝑡ℎ}}$ vector to be lexicographically less than or equal to the ${\left(i+1\right)}^{\mathrm{𝑡ℎ}}$ vector $\left(1\le i<|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|\right)$. We have that $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$ is suffix-contractible with respect to $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}.\mathrm{𝚟𝚎𝚌}$. This means that we can remove the $k$ last components of each vectors of the $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ collection. As in the previous example the $k$ items should be removed from all collections simultaneously. As an illustration, consider the source constraint $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$$\left(〈\mathrm{𝚟𝚎𝚌}-〈5,2,3,9〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6,2〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6,2〉〉\right)$ and the target constraint $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$$\left(〈\mathrm{𝚟𝚎𝚌}-〈5,2,3〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6〉〉\right)$. Since $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$ is suffix-contractible with respect to $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}.\mathrm{𝚟𝚎𝚌}$, we have that $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$$\left(〈\mathrm{𝚟𝚎𝚌}-〈5,2,3,9〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6,2〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6,2〉〉\right)$ implies $\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$$\left(〈\mathrm{𝚟𝚎𝚌}-〈5,2,3〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6〉,\mathrm{𝚟𝚎𝚌}-〈5,2,6〉〉\right)$.

The keyword extensible introduces a dual notion, where items can be added to a collection that is passed as an argument of a satisfied global constraint without affecting the fact that the resulting constraint is satisfied. Contractibility is a more common property than extensibility.