GI0013

gaussian_signed_norm_functional

The squared Gaussian norm is functional across every possible square-witness choice.

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. ∀ M. GaussianSignedNorm(p,n,q,m,N) → GaussianSignedNorm(p,n,q,m,M) → N = M

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_functional
Original expanded first-order statement 
forall p n q m N M. (exists ge_real_square_first ge_imaginary_square_first. ((((((p) * (p))) + (((n) * (n)))) = ((ge_real_square_first) + (((((p) * (n))) + (((n) * (p))))))) /\ ((((((q) * (q))) + (((m) * (m)))) = ((ge_imaginary_square_first) + (((((q) * (m))) + (((m) * (q))))))) /\ ((N) = ge_real_square_first + ge_imaginary_square_first)))) -> (exists ge_real_square_second ge_imaginary_square_second. ((((((p) * (p))) + (((n) * (n)))) = ((ge_real_square_second) + (((((p) * (n))) + (((n) * (p))))))) /\ ((((((q) * (q))) + (((m) * (m)))) = ((ge_imaginary_square_second) + (((((q) * (m))) + (((m) * (q))))))) /\ ((M) = ge_real_square_second + ge_imaginary_square_second)))) -> N = M

Complete tactic proof in conservative notation

All 28 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

28 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 functional
01Fix variables and assumptionsL1–8

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
  5. L5
    intro N
  6. L6
    intro M
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Separate the logical casesL9–16

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

  1. L9
    cases hfirst
  2. L10
    cases hfirst_witness
  3. L11
    cases hfirst_witness_witness
  4. L12
    cases hfirst_witness_witness_right
  5. L13
    cases hsecond
  6. L14
    cases hsecond_witness
  7. L15
    cases hsecond_witness_witness
  8. L16
    cases hsecond_witness_witness_right
03Calculate and transport equalitiesL17–17

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

  1. L17
    trans x + x1
04Use earlier factsL18–18

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

  1. L18
    exact hfirst_witness_witness_right_right
05Calculate and transport equalitiesL19–20

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

  1. L19
    trans x2 + x3
  2. L20
    congr
06Use earlier factsL21–26

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

  1. L21
    apply gaussian_signed_square_functional
  2. L22
    exact hfirst_witness_witness_left
  3. L23
    exact hsecond_witness_witness_left
  4. L24
    apply gaussian_signed_square_functional
  5. L25
    exact hfirst_witness_witness_right_left
  6. L26
    exact hsecond_witness_witness_right_left
07Calculate and transport equalitiesL27–27

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

  1. L27
    symm
08Use earlier factsL28–28

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

  1. L28
    exact hsecond_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro m
  5. 0005intro N
  6. 0006intro M
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009cases hfirst
  10. 0010cases hfirst_witness
  11. 0011cases hfirst_witness_witness
  12. 0012cases hfirst_witness_witness_right
  13. 0013cases hsecond
  14. 0014cases hsecond_witness
  15. 0015cases hsecond_witness_witness
  16. 0016cases hsecond_witness_witness_right
  17. 0017trans x + x1
  18. 0018exact hfirst_witness_witness_right_right
  19. 0019trans x2 + x3
  20. 0020congr
  21. 0021apply gaussian_signed_square_functional
  22. 0022exact hfirst_witness_witness_left
  23. 0023exact hsecond_witness_witness_left
  24. 0024apply gaussian_signed_square_functional
  25. 0025exact hfirst_witness_witness_right_left
  26. 0026exact hsecond_witness_witness_right_left
  27. 0027symm
  28. 0028exact hsecond_witness_witness_right_right