GI001B

gaussian_signed_norm_integer_transport

The actual Gaussian norm is invariant under equality of both represented integer coordinates.

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.

The natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ N. a + f = e + b ∧ c + h = g + d → GaussianSignedNorm(a,b,c,d,N) → GaussianSignedNorm(e,f,g,h,N)

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

Definition DAG

GaussianSignedNorm(ap,an,bp,bn,N) · 2

Actual proof prerequisites

gaussian_signed_square_integer_transport
Original expanded first-order statement 
forall a b c d e f g h N. (((((a) + (f)) = ((e) + (b))) /\ (((c) + (h)) = ((g) + (d))))) -> (exists ge_real_square_transport_first ge_imaginary_square_transport_first. ((((((a) * (a))) + (((b) * (b)))) = ((ge_real_square_transport_first) + (((((a) * (b))) + (((b) * (a))))))) /\ ((((((c) * (c))) + (((d) * (d)))) = ((ge_imaginary_square_transport_first) + (((((c) * (d))) + (((d) * (c))))))) /\ ((N) = ge_real_square_transport_first + ge_imaginary_square_transport_first)))) -> (exists ge_real_square_transport_second ge_imaginary_square_transport_second. ((((((e) * (e))) + (((f) * (f)))) = ((ge_real_square_transport_second) + (((((e) * (f))) + (((f) * (e))))))) /\ ((((((g) * (g))) + (((h) * (h)))) = ((ge_imaginary_square_transport_second) + (((((g) * (h))) + (((h) * (g))))))) /\ ((N) = ge_real_square_transport_second + ge_imaginary_square_transport_second))))

Complete tactic proof in conservative notation

All 37 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

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.

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 (1)
gaussian signed square integer transport
01Fix variables and assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
  9. L9
    intro N
  10. L10
    intro hequal
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hnorm
03Separate the logical casesL12–16

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

  1. L12
    cases hequal
  2. L13
    cases hnorm
  3. L14
    cases hnorm_witness
  4. L15
    cases hnorm_witness_witness
  5. 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 a
  2. L21
    specialize gaussian_signed_square_integer_transport b
  3. L22
    specialize gaussian_signed_square_integer_transport e
  4. L23
    specialize gaussian_signed_square_integer_transport f
  5. L24
    specialize gaussian_signed_square_integer_transport x
  6. L25
    apply gaussian_signed_square_integer_transport
  7. L26
    exact hequal_left
  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 c
  2. L30
    specialize gaussian_signed_square_integer_transport d
  3. L31
    specialize gaussian_signed_square_integer_transport g
  4. L32
    specialize gaussian_signed_square_integer_transport h
  5. L33
    specialize gaussian_signed_square_integer_transport x1
  6. L34
    apply gaussian_signed_square_integer_transport
  7. L35
    exact hequal_right
  8. L36
    exact hnorm_witness_witness_right_left
  9. L37
    exact hnorm_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 37 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009intro N
  10. 0010intro hequal
  11. 0011intro hnorm
  12. 0012cases hequal
  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 a
  21. 0021specialize gaussian_signed_square_integer_transport b
  22. 0022specialize gaussian_signed_square_integer_transport e
  23. 0023specialize gaussian_signed_square_integer_transport f
  24. 0024specialize gaussian_signed_square_integer_transport x
  25. 0025apply gaussian_signed_square_integer_transport
  26. 0026exact hequal_left
  27. 0027exact hnorm_witness_witness_left
  28. 0028split
  29. 0029specialize gaussian_signed_square_integer_transport c
  30. 0030specialize gaussian_signed_square_integer_transport d
  31. 0031specialize gaussian_signed_square_integer_transport g
  32. 0032specialize gaussian_signed_square_integer_transport h
  33. 0033specialize gaussian_signed_square_integer_transport x1
  34. 0034apply gaussian_signed_square_integer_transport
  35. 0035exact hequal_right
  36. 0036exact hnorm_witness_witness_right_left
  37. 0037exact hnorm_witness_witness_right_right