EC0003

euclidean_next_division_step_exists

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

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

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

Definition DAG

EuclideanDivision(a,b,q,r) · 2

Actual proof prerequisites

euclidean_division_step_exists
Original expanded first-order 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)))

Complete unchanged native tactic proof

All 10 lines are the exact independently kernel-checked original script.

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.

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

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 defined 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