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

EL0006 · euclidean_log_budget_extendEL0007 · euclidean_log_budget_extend_twice

Separate complete second-wave branches: Full T13 proof · Alpha v27.