EI0016

eisenstein_weighted_lagrange_compensation

A weight-three multiple of the summed-square equation cancels the exact difference-square cross terms.

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

∀ su. ∀ sv. ∀ sw. ∀ sx. ∀ R. ∀ I. ∀ P. ∀ Q. ∀ p. ∀ q. P = 3 · p → Q = 3 · q → R + (P + P) = su + sv + (Q + Q) → I + (q + q) = sw + sx + (p + p) → R + 3 · I = su + sv + 3 · (sw + sx)

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

Definition DAG

none

Actual proof prerequisites

add_cross_sum_chain · checked external prerequisitemul_add · 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 su sv sw sx R I P Q p q. P = 3 * p -> Q = 3 * q -> R + (P + P) = (su + sv) + (Q + Q) -> I + (q + q) = (sw + sx) + (p + p) -> R + 3 * I = (su + sv) + 3 * (sw + sx)

Complete tactic proof in conservative notation

All 39 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

39 script commands · 10 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 assumptionsL1–10

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

  1. L1
    intro su
  2. L2
    intro sv
  3. L3
    intro sw
  4. L4
    intro sx
  5. L5
    intro R
  6. L6
    intro I
  7. L7
    intro P
  8. L8
    intro Q
  9. L9
    intro p
  10. L10
    intro q
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hp
  2. L12
    intro hq
  3. L13
    intro hdifference
  4. L14
    intro hsum
03Establish hscaledL15–24

Establish this local claim before using it. It is not an additional assumption.

  1. L15
    have hscaled : 3 * I + (Q + Q) = 3 * (sw + sx) + (P + P)
  2. L16
    rewrite hp
  3. L17
    rewrite hp
  4. L18
    rewrite hq
  5. L19
    rewrite hq
  6. L20
    trans 3 * (I + (q + q))
  7. L21
    simp [mul_add, add_assoc, add_comm, four_square_add_swap_right_tail]
  8. L22
    trans 3 * ((sw + sx) + (p + p))
  9. L23
    congr
  10. L24
    refl
04Use earlier factsL25–25

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

  1. L25
    exact hsum
05Calculate and transport equalitiesL26–26

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

  1. L26
    simp [mul_add, add_assoc, add_comm, four_square_add_swap_right_tail]
06Use earlier factsL27–34

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

  1. L27
    specialize add_cross_sum_chain R
  2. L28
    specialize add_cross_sum_chain (su + sv)
  3. L29
    specialize add_cross_sum_chain (P + P)
  4. L30
    specialize add_cross_sum_chain (Q + Q)
  5. L31
    specialize add_cross_sum_chain (3 * I)
  6. L32
    specialize add_cross_sum_chain (3 * (sw + sx))
  7. L33
    apply add_cross_sum_chain
  8. L34
    exact hdifference
07Calculate and transport equalitiesL35–35

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

  1. L35
    trans 3 * I + (Q + Q)
08Use earlier factsL36–36

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

  1. L36
    apply add_comm
09Calculate and transport equalitiesL37–37

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

  1. L37
    trans 3 * (sw + sx) + (P + P)
10Use earlier factsL38–39

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

  1. L38
    exact hscaled
  2. L39
    apply add_comm

Library-wide reading audit

Original defined command ledger · 39 lines
  1. 0001intro su
  2. 0002intro sv
  3. 0003intro sw
  4. 0004intro sx
  5. 0005intro R
  6. 0006intro I
  7. 0007intro P
  8. 0008intro Q
  9. 0009intro p
  10. 0010intro q
  11. 0011intro hp
  12. 0012intro hq
  13. 0013intro hdifference
  14. 0014intro hsum
  15. 0015have hscaled : 3 * I + (Q + Q) = 3 * (sw + sx) + (P + P)
  16. 0016rewrite hp
  17. 0017rewrite hp
  18. 0018rewrite hq
  19. 0019rewrite hq
  20. 0020trans 3 * (I + (q + q))
  21. 0021simp [mul_add, add_assoc, add_comm, four_square_add_swap_right_tail]
  22. 0022trans 3 * ((sw + sx) + (p + p))
  23. 0023congr
  24. 0024refl
  25. 0025exact hsum
  26. 0026simp [mul_add, add_assoc, add_comm, four_square_add_swap_right_tail]
  27. 0027specialize add_cross_sum_chain R
  28. 0028specialize add_cross_sum_chain (su + sv)
  29. 0029specialize add_cross_sum_chain (P + P)
  30. 0030specialize add_cross_sum_chain (Q + Q)
  31. 0031specialize add_cross_sum_chain (3 * I)
  32. 0032specialize add_cross_sum_chain (3 * (sw + sx))
  33. 0033apply add_cross_sum_chain
  34. 0034exact hdifference
  35. 0035trans 3 * I + (Q + Q)
  36. 0036apply add_comm
  37. 0037trans 3 * (sw + sx) + (P + P)
  38. 0038exact hscaled
  39. 0039apply add_comm