ND0018

EuclideanDivision(a,b,q,r)

A genuine Euclidean division a=b*q+r with strictly bounded natural remainder r<b.

Conservative notation; not a theorem, primitive, or axiom.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Hygienic expanded first-order definition

(a = b * q + r /\ (exists ff_lt_ec_advanced_division. ff_lt_ec_advanced_division + S r = b))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Lt(a,b)

Definitions depending on this notation

none

Checked theorems using this definition

EC0001 · euclidean_division_step_existsEC0002 · euclidean_division_step_functionalEC0003 · euclidean_next_division_step_existsEC0005 · euclidean_two_step_quotient_nonzeroEC0006 · euclidean_two_step_halving