EI0013

eisenstein_weighted_norm_transport

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The positive-definite weighted norm respects actual equality of both integer coordinates.

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.

Exact expanded first-order arithmetic statement

forall ap an bp bn cp cn dp dn N. (((ap) + (cn)) = ((cp) + (an))) -> (((bp) + (dn)) = ((dp) + (bn))) -> (exists ee_real_square_source ee_imag_square_source. (((((ap) * (ap))) + (((an) * (an)))) = ((ee_real_square_source) + (((((ap) * (an))) + (((an) * (ap))))))) /\ ((((((bp) * (bp))) + (((bn) * (bn)))) = ((ee_imag_square_source) + (((((bp) * (bn))) + (((bn) * (bp))))))) /\ (N) = ee_real_square_source + 3 * ee_imag_square_source)) -> (exists ee_real_square_target ee_imag_square_target. (((((cp) * (cp))) + (((cn) * (cn)))) = ((ee_real_square_target) + (((((cp) * (cn))) + (((cn) * (cp))))))) /\ ((((((dp) * (dp))) + (((dn) * (dn)))) = ((ee_imag_square_target) + (((((dp) * (dn))) + (((dn) * (dp))))))) /\ (N) = ee_real_square_target + 3 * ee_imag_square_target))

Constructive proof overview

Generated structural guide

The positive-definite weighted norm respects actual equality of both integer coordinates.

The unchanged tactic script uses 1 declared prerequisite and contains 37 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

gaussian_signed_square_integer_transport Alpha theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

37 script commands · 8 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro cp
  6. L6
    intro cn
  7. L7
    intro dp
  8. L8
    intro dn
  9. L9
    intro N
  10. L10
    intro ha
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hb
  2. L12
    intro hnorm
03Separate the logical casesL13–16

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

  1. L13
    cases hnorm
  2. L14
    cases hnorm_witness
  3. L15
    cases hnorm_witness_witness
  4. L16
    cases hnorm_witness_witness_right
04Construct an explicit witnessL17–18

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

  1. L17
    exists x
  2. L18
    exists x1
05Separate the logical casesL19–19

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

  1. L19
    split
06Use earlier factsL20–27

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

  1. L20
    specialize gaussian_signed_square_integer_transport ap
  2. L21
    specialize gaussian_signed_square_integer_transport an
  3. L22
    specialize gaussian_signed_square_integer_transport cp
  4. L23
    specialize gaussian_signed_square_integer_transport cn
  5. L24
    specialize gaussian_signed_square_integer_transport x
  6. L25
    apply gaussian_signed_square_integer_transport
  7. L26
    exact ha
  8. L27
    exact hnorm_witness_witness_left
07Separate the logical casesL28–28

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

  1. L28
    split
08Use earlier factsL29–37

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

  1. L29
    specialize gaussian_signed_square_integer_transport bp
  2. L30
    specialize gaussian_signed_square_integer_transport bn
  3. L31
    specialize gaussian_signed_square_integer_transport dp
  4. L32
    specialize gaussian_signed_square_integer_transport dn
  5. L33
    specialize gaussian_signed_square_integer_transport x1
  6. L34
    apply gaussian_signed_square_integer_transport
  7. L35
    exact hb
  8. L36
    exact hnorm_witness_witness_right_left
  9. L37
    exact hnorm_witness_witness_right_right

Library-wide reading audit

Original exact command ledger · 37 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro cp
  6. 0006intro cn
  7. 0007intro dp
  8. 0008intro dn
  9. 0009intro N
  10. 0010intro ha
  11. 0011intro hb
  12. 0012intro hnorm
  13. 0013cases hnorm
  14. 0014cases hnorm_witness
  15. 0015cases hnorm_witness_witness
  16. 0016cases hnorm_witness_witness_right
  17. 0017exists x
  18. 0018exists x1
  19. 0019split
  20. 0020specialize gaussian_signed_square_integer_transport ap
  21. 0021specialize gaussian_signed_square_integer_transport an
  22. 0022specialize gaussian_signed_square_integer_transport cp
  23. 0023specialize gaussian_signed_square_integer_transport cn
  24. 0024specialize gaussian_signed_square_integer_transport x
  25. 0025apply gaussian_signed_square_integer_transport
  26. 0026exact ha
  27. 0027exact hnorm_witness_witness_left
  28. 0028split
  29. 0029specialize gaussian_signed_square_integer_transport bp
  30. 0030specialize gaussian_signed_square_integer_transport bn
  31. 0031specialize gaussian_signed_square_integer_transport dp
  32. 0032specialize gaussian_signed_square_integer_transport dn
  33. 0033specialize gaussian_signed_square_integer_transport x1
  34. 0034apply gaussian_signed_square_integer_transport
  35. 0035exact hb
  36. 0036exact hnorm_witness_witness_right_left
  37. 0037exact hnorm_witness_witness_right_right