EC0002

euclidean_division_step_functional

Two exact bounded Euclidean divisions of the same inputs have identical quotients and remainders.

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. ∀ Q. ∀ R. EuclideanDivision(a,b,q,r) → EuclideanDivision(a,b,Q,R) → q = Q ∧ r = R

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

Definition DAG

EuclideanDivision(a,b,q,r) · 2

Actual proof prerequisites

division_remainder_unique · checked external prerequisite
Original expanded first-order statement 
forall a b q r Q R. ((a = b * q + r /\ (exists ff_lt_ec_first_division. ff_lt_ec_first_division + S r = b))) -> ((a = b * Q + R /\ (exists ff_lt_ec_other_division. ff_lt_ec_other_division + S R = b))) -> q = Q /\ r = R

Complete unchanged native tactic proof

All 21 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

21 script commands · 4 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 assumptionsL1–8

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 Q
  6. L6
    intro R
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Separate the logical casesL9–10

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases hfirst
  2. L10
    cases hsecond
03Use earlier factsL11–20

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

  1. L11
    specialize division_remainder_unique b
  2. L12
    specialize division_remainder_unique a
  3. L13
    specialize division_remainder_unique q
  4. L14
    specialize division_remainder_unique r
  5. L15
    specialize division_remainder_unique Q
  6. L16
    specialize division_remainder_unique R
  7. L17
    apply division_remainder_unique
  8. L18
    exact hfirst_left
  9. L19
    exact hfirst_right
  10. L20
    exact hsecond_left
04Use earlier factsL21–21

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

  1. L21
    exact hsecond_right

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro q
  4. 0004intro r
  5. 0005intro Q
  6. 0006intro R
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009cases hfirst
  10. 0010cases hsecond
  11. 0011specialize division_remainder_unique b
  12. 0012specialize division_remainder_unique a
  13. 0013specialize division_remainder_unique q
  14. 0014specialize division_remainder_unique r
  15. 0015specialize division_remainder_unique Q
  16. 0016specialize division_remainder_unique R
  17. 0017apply division_remainder_unique
  18. 0018exact hfirst_left
  19. 0019exact hfirst_right
  20. 0020exact hsecond_left
  21. 0021exact hsecond_right