EC0003

euclidean_next_division_step_exists

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

Every nonterminal Euclidean remainder constructively admits the next genuine bounded division.

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 q r. ((a = b * q + r /\ (exists ff_lt_ec_first_division. ff_lt_ec_first_division + S r = b))) -> ~(r = 0) -> exists Q t. ((b = r * Q + t /\ (exists ff_lt_ec_second_division. ff_lt_ec_second_division + S t = r)))

Constructive proof overview

Generated structural guide

Every nonterminal Euclidean remainder constructively admits the next genuine bounded division.

The unchanged tactic script uses 1 declared prerequisite and contains 10 exact native proof lines.

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

Proof neighborhood

Direct dependencies

EC0001 euclidean_division_step_exists

Direct dependents

none

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

10 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.

Named ingredients (1)
euclidean division step exists
01Fix variables and assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro q
  4. L4
    intro r
  5. L5
    intro hfirst
  6. L6
    intro hr
02Use earlier factsL7–10

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

  1. L7
    specialize euclidean_division_step_exists b
  2. L8
    specialize euclidean_division_step_exists r
  3. L9
    apply euclidean_division_step_exists
  4. L10
    exact hr

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro q
  4. 0004intro r
  5. 0005intro hfirst
  6. 0006intro hr
  7. 0007specialize euclidean_division_step_exists b
  8. 0008specialize euclidean_division_step_exists r
  9. 0009apply euclidean_division_step_exists
  10. 0010exact hr