EC0001

euclidean_division_step_exists

A nonzero Euclidean divisor constructively yields an exact quotient and strictly smaller remainder.

Alpha v34 checked-use · first admitted v21 · independently kernel and Lean verified; not Stable

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.

G101 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: the actual anchored Euclidean history, terminal gcd, and exact bound steps≤2*BitLen(b)+1 are proved.

Exact theorem in conservative defined notation

∀ a. ∀ b. ¬b = 0 → ∃ x. ∃ y. EuclideanDivision(a,b,x,y)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

EuclideanDivision(a,b,q,r) · 1

Actual proof prerequisites

division_remainder_exists · checked external prerequisite
Original expanded first-order statement 
forall a b. ~(b = 0) -> exists q r. ((a = b * q + r /\ (exists ff_lt_ec_first_division. ff_lt_ec_first_division + S r = b)))

Complete unchanged native tactic proof

All 7 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

7 script commands · 2 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro hb
02Use earlier factsL4–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L4
    specialize division_remainder_exists b
  2. L5
    specialize division_remainder_exists a
  3. L6
    apply division_remainder_exists
  4. L7
    exact hb

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hb
  4. 0004specialize division_remainder_exists b
  5. 0005specialize division_remainder_exists a
  6. 0006apply division_remainder_exists
  7. 0007exact hb