GI0012

gaussian_signed_norm_exists

Every signed Gaussian coordinate pair has a constructed actual nonnegative squared norm.

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

∀ p. ∀ n. ∀ q. ∀ m. ∃ N. GaussianSignedNorm(p,n,q,m,N)

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

Definition DAG

SignedDifferenceSquare(p,n,s) · 2GaussianSignedNorm(ap,an,bp,bn,N) · 1

Actual proof prerequisites

gaussian_signed_square_exists
Original expanded first-order statement 
forall p n q m. exists N. (exists ge_real_square_exists_total ge_imaginary_square_exists_total. ((((((p) * (p))) + (((n) * (n)))) = ((ge_real_square_exists_total) + (((((p) * (n))) + (((n) * (p))))))) /\ ((((((q) * (q))) + (((m) * (m)))) = ((ge_imaginary_square_exists_total) + (((((q) * (m))) + (((m) * (q))))))) /\ ((N) = ge_real_square_exists_total + ge_imaginary_square_exists_total))))

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 · 11 reading checkpoints · 2 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 exists
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro q
  4. L4
    intro m
02Establish hrealL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gaussian signed square exists.

  1. L5
    have hreal : ∃ u. SignedDifferenceSquare(p,n,u)Definitions: SignedDifferenceSquare(p,n,u)Original native command in the exact edition
  2. L6
    specialize gaussian_signed_square_exists p
  3. L7
    specialize gaussian_signed_square_exists n
  4. L8
    apply gaussian_signed_square_exists
03Separate the logical casesL9–9

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

  1. L9
    cases hreal
04Establish himaginaryL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gaussian signed square exists.

  1. L10
    have himaginary : ∃ v. SignedDifferenceSquare(q,m,v)Definitions: SignedDifferenceSquare(q,m,v)Original native command in the exact edition
  2. L11
    specialize gaussian_signed_square_exists q
  3. L12
    specialize gaussian_signed_square_exists m
  4. L13
    apply gaussian_signed_square_exists
05Separate the logical casesL14–14

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

  1. L14
    cases himaginary
06Construct an explicit witnessL15–17

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

  1. L15
    exists x + x1
  2. L16
    exists x
  3. L17
    exists x1
07Separate the logical casesL18–18

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

  1. L18
    split
08Use earlier factsL19–19

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

  1. L19
    exact hreal_witness
09Separate the logical casesL20–20

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

  1. L20
    split
10Use earlier factsL21–21

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

  1. L21
    exact himaginary_witness
11Calculate and transport equalitiesL22–22

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

  1. L22
    refl

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro m
  5. 0005have hreal : ∃ u. SignedDifferenceSquare(p,n,u)
  6. 0006specialize gaussian_signed_square_exists p
  7. 0007specialize gaussian_signed_square_exists n
  8. 0008apply gaussian_signed_square_exists
  9. 0009cases hreal
  10. 0010have himaginary : ∃ v. SignedDifferenceSquare(q,m,v)
  11. 0011specialize gaussian_signed_square_exists q
  12. 0012specialize gaussian_signed_square_exists m
  13. 0013apply gaussian_signed_square_exists
  14. 0014cases himaginary
  15. 0015exists x + x1
  16. 0016exists x
  17. 0017exists x1
  18. 0018split
  19. 0019exact hreal_witness
  20. 0020split
  21. 0021exact himaginary_witness
  22. 0022refl