EI0006

eisenstein_parallelogram_norm_strict

Every lattice residue pair 0≤a,b<m has norm strictly below m², including zero and a=b=m−1.

Alpha v34 checked-use · first admitted v28 · 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.

A floor quotient in the fundamental parallelogram already gives the required strict norm decrease; global nearest-point optimality is not asserted. The shared carrier is identical to the Gaussian carrier, but the multiplication law and norm are different. Eisenstein gcd, factorization, and prime classification remain separate targets.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ m. ∀ n. Lt(a,m) → Lt(b,m) → a · a + b · b = a · b + n → Lt(n,m · m)

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

Definition DAG

Lt(a,b) · 3

Actual proof prerequisites

le_total · checked external prerequisiteeisenstein_natural_norm_symmetriceisenstein_natural_norm_le_larger_squaresquare_lt_strict · checked external prerequisitelt_of_le_of_lt · checked external prerequisite
Original expanded first-order statement 
forall a b m n. (exists ee_gap_lt. ee_gap_lt + S (a) = (m)) -> (exists ee_gap_lt. ee_gap_lt + S (b) = (m)) -> ((((a) * (a)) + ((b) * (b))) = (((a) * (b)) + (n))) -> (exists ee_gap_lt. ee_gap_lt + S (n) = (m * m))

Complete tactic proof in conservative notation

All 42 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

42 script commands · 7 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 (2)
eisenstein natural norm le larger squareeisenstein natural norm symmetric
01Fix variables and assumptionsL1–7

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro m
  4. L4
    intro n
  5. L5
    intro ha
  6. L6
    intro hb
  7. L7
    intro hnorm
02Use earlier factsL8–9

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

  1. L8
    specialize le_total a
  2. L9
    specialize le_total b
03Separate the logical casesL10–10

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

  1. L10
    cases le_total
04Use earlier factsL11–20

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

  1. L11
    specialize lt_of_le_of_lt n
  2. L12
    specialize lt_of_le_of_lt (b * b)
  3. L13
    specialize lt_of_le_of_lt (m * m)
  4. L14
    apply lt_of_le_of_lt
  5. L15
    specialize eisenstein_natural_norm_le_larger_square a
  6. L16
    specialize eisenstein_natural_norm_le_larger_square b
  7. L17
    specialize eisenstein_natural_norm_le_larger_square n
  8. L18
    apply eisenstein_natural_norm_le_larger_square
  9. L19
    exact le_total_left
  10. L20
    exact hnorm
05Use earlier factsL21–30

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

  1. L21
    specialize square_lt_strict b
  2. L22
    specialize square_lt_strict m
  3. L23
    apply square_lt_strict
  4. L24
    exact hb
  5. L25
    specialize lt_of_le_of_lt n
  6. L26
    specialize lt_of_le_of_lt (a * a)
  7. L27
    specialize lt_of_le_of_lt (m * m)
  8. L28
    apply lt_of_le_of_lt
  9. L29
    specialize eisenstein_natural_norm_le_larger_square b
  10. L30
    specialize eisenstein_natural_norm_le_larger_square a
06Use earlier factsL31–40

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

  1. L31
    specialize eisenstein_natural_norm_le_larger_square n
  2. L32
    apply eisenstein_natural_norm_le_larger_square
  3. L33
    exact le_total_right
  4. L34
    specialize eisenstein_natural_norm_symmetric a
  5. L35
    specialize eisenstein_natural_norm_symmetric b
  6. L36
    specialize eisenstein_natural_norm_symmetric n
  7. L37
    apply eisenstein_natural_norm_symmetric
  8. L38
    exact hnorm
  9. L39
    specialize square_lt_strict a
  10. L40
    specialize square_lt_strict m
07Use earlier factsL41–42

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

  1. L41
    apply square_lt_strict
  2. L42
    exact ha

Library-wide reading audit

Original defined command ledger · 42 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro m
  4. 0004intro n
  5. 0005intro ha
  6. 0006intro hb
  7. 0007intro hnorm
  8. 0008specialize le_total a
  9. 0009specialize le_total b
  10. 0010cases le_total
  11. 0011specialize lt_of_le_of_lt n
  12. 0012specialize lt_of_le_of_lt (b * b)
  13. 0013specialize lt_of_le_of_lt (m * m)
  14. 0014apply lt_of_le_of_lt
  15. 0015specialize eisenstein_natural_norm_le_larger_square a
  16. 0016specialize eisenstein_natural_norm_le_larger_square b
  17. 0017specialize eisenstein_natural_norm_le_larger_square n
  18. 0018apply eisenstein_natural_norm_le_larger_square
  19. 0019exact le_total_left
  20. 0020exact hnorm
  21. 0021specialize square_lt_strict b
  22. 0022specialize square_lt_strict m
  23. 0023apply square_lt_strict
  24. 0024exact hb
  25. 0025specialize lt_of_le_of_lt n
  26. 0026specialize lt_of_le_of_lt (a * a)
  27. 0027specialize lt_of_le_of_lt (m * m)
  28. 0028apply lt_of_le_of_lt
  29. 0029specialize eisenstein_natural_norm_le_larger_square b
  30. 0030specialize eisenstein_natural_norm_le_larger_square a
  31. 0031specialize eisenstein_natural_norm_le_larger_square n
  32. 0032apply eisenstein_natural_norm_le_larger_square
  33. 0033exact le_total_right
  34. 0034specialize eisenstein_natural_norm_symmetric a
  35. 0035specialize eisenstein_natural_norm_symmetric b
  36. 0036specialize eisenstein_natural_norm_symmetric n
  37. 0037apply eisenstein_natural_norm_symmetric
  38. 0038exact hnorm
  39. 0039specialize square_lt_strict a
  40. 0040specialize square_lt_strict m
  41. 0041apply square_lt_strict
  42. 0042exact ha