EI0005

eisenstein_natural_norm_le_larger_square

If a≤b then a²−ab+b²≤b², with an actual natural gap witness.

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. ∀ n. a ≤ b → a · a + b · b = a · b + n → n ≤ b · b

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

Definition DAG

none

Actual proof prerequisites

mul_le_mul_left · checked external prerequisiteadd_left_cancel · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall a b n. (exists ee_bound_le. ee_bound_le + (a) = (b)) -> ((((a) * (a)) + ((b) * (b))) = (((a) * (b)) + (n))) -> (exists ee_bound_le. ee_bound_le + (n) = (b * b))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 7 reading checkpoints · 1 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro n
  4. L4
    intro hle
  5. L5
    intro hnorm
02Establish hproductL6–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L6
    have hproduct : exists k. k + a * a = a * b
  2. L7
    specialize mul_le_mul_left a
  3. L8
    specialize mul_le_mul_left b
  4. L9
    specialize mul_le_mul_left a
  5. L10
    apply mul_le_mul_left
  6. L11
    exact hle
03Separate the logical casesL12–12

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

  1. L12
    cases hproduct
04Construct an explicit witnessL13–13

Supply the displayed value, then prove that it has the required property.

  1. L13
    exists x
05Use earlier factsL14–17

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

  1. L14
    specialize add_left_cancel (a * a)
  2. L15
    specialize add_left_cancel (x + n)
  3. L16
    specialize add_left_cancel (b * b)
  4. L17
    apply add_left_cancel
06Calculate and transport equalitiesL18–21

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L18
    trans (x + a * a) + n
  2. L19
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L20
    rewrite hproduct_witness
  4. L21
    symm
07Use earlier factsL22–22

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

  1. L22
    exact hnorm

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro n
  4. 0004intro hle
  5. 0005intro hnorm
  6. 0006have hproduct : exists k. k + a * a = a * b
  7. 0007specialize mul_le_mul_left a
  8. 0008specialize mul_le_mul_left b
  9. 0009specialize mul_le_mul_left a
  10. 0010apply mul_le_mul_left
  11. 0011exact hle
  12. 0012cases hproduct
  13. 0013exists x
  14. 0014specialize add_left_cancel (a * a)
  15. 0015specialize add_left_cancel (x + n)
  16. 0016specialize add_left_cancel (b * b)
  17. 0017apply add_left_cancel
  18. 0018trans (x + a * a) + n
  19. 0019simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  20. 0020rewrite hproduct_witness
  21. 0021symm
  22. 0022exact hnorm