EC0001

euclidean_division_step_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 7 exact native proof lines.

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

Proof neighborhood

Direct dependencies

division_remainder_exists Stable theorem; checked-use authorized

Direct dependents

EC0003 euclidean_next_division_step_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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