GF0036

gaussian_ring_raw_multiply_add_distributive

Actual Gaussian multiplication distributes over addition in both signed integer coordinates.

Alpha v34 checked-use · first admitted v30 · 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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

∀ ap. ∀ an. ∀ bp. ∀ bn. ∀ cp. ∀ cn. ∀ dp. ∀ dn. ∀ ep. ∀ en. ∀ fp. ∀ fn. ap · (cp + ep) + an · (cn + en) + (bp · (dn + fn) + bn · (dp + fp)) + (ap · cn + an · cp + (bp · dp + bn · dn) + (ap · en + an · ep + (bp · fp + bn · fn))) = ap · cp + an · cn + (bp · dn + bn · dp) + (ap · ep + an · en + (bp · fn + bn · fp)) + (ap · (cn + en) + an · (cp + ep) + (bp · (dp + fp) + bn · (dn + fn))) ∧ ap · (dp + fp) + an · (dn + fn) + (bp · (cp + ep) + bn · (cn + en)) + (ap · dn + an · dp + (bp · cn + bn · cp) + (ap · fn + an · fp + (bp · en + bn · ep))) = ap · dp + an · dn + (bp · cp + bn · cn) + (ap · fp + an · fn + (bp · ep + bn · en)) + (ap · (dn + fn) + an · (dp + fp) + (bp · (cn + en) + bn · (cp + ep)))

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

Definition DAG

none

Actual proof prerequisites

gaussian_ring_multiply_add_real_positivegaussian_ring_multiply_add_real_negativegaussian_ring_multiply_add_imaginary_positivegaussian_ring_multiply_add_imaginary_negative
Original expanded first-order statement 
forall ap an bp bn cp cn dp dn ep en fp fn. (((((((((((ap) * (((cp) + (ep))))) + (((an) * (((cn) + (en))))))) + (((((bp) * (((dn) + (fn))))) + (((bn) * (((dp) + (fp))))))))) + (((((((((ap) * (cn))) + (((an) * (cp))))) + (((((bp) * (dp))) + (((bn) * (dn))))))) + (((((((ap) * (en))) + (((an) * (ep))))) + (((((bp) * (fp))) + (((bn) * (fn)))))))))) = ((((((((((ap) * (cp))) + (((an) * (cn))))) + (((((bp) * (dn))) + (((bn) * (dp))))))) + (((((((ap) * (ep))) + (((an) * (en))))) + (((((bp) * (fn))) + (((bn) * (fp))))))))) + (((((((ap) * (((cn) + (en))))) + (((an) * (((cp) + (ep))))))) + (((((bp) * (((dp) + (fp))))) + (((bn) * (((dn) + (fn))))))))))) /\ (((((((((ap) * (((dp) + (fp))))) + (((an) * (((dn) + (fn))))))) + (((((bp) * (((cp) + (ep))))) + (((bn) * (((cn) + (en))))))))) + (((((((((ap) * (dn))) + (((an) * (dp))))) + (((((bp) * (cn))) + (((bn) * (cp))))))) + (((((((ap) * (fn))) + (((an) * (fp))))) + (((((bp) * (en))) + (((bn) * (ep)))))))))) = ((((((((((ap) * (dp))) + (((an) * (dn))))) + (((((bp) * (cp))) + (((bn) * (cn))))))) + (((((((ap) * (fp))) + (((an) * (fn))))) + (((((bp) * (ep))) + (((bn) * (en))))))))) + (((((((ap) * (((dn) + (fn))))) + (((an) * (((dp) + (fp))))))) + (((((bp) * (((cn) + (en))))) + (((bn) * (((cp) + (ep)))))))))))))

Complete tactic proof in conservative notation

All 21 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

21 script commands · 11 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 (4)
gaussian ring multiply add real positivegaussian ring multiply add real negativegaussian ring multiply add imaginary positivegaussian ring multiply add imaginary negative
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 ep
  10. L10
    intro en
02Fix variables and assumptionsL11–12

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

  1. L11
    intro fp
  2. L12
    intro fn
03Separate the logical casesL13–13

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

  1. L13
    split
04Calculate and transport equalitiesL14–14

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

  1. L14
    congr
05Use earlier factsL15–15

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

  1. L15
    apply gaussian_ring_multiply_add_real_positive
06Calculate and transport equalitiesL16–16

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

  1. L16
    symm
07Use earlier factsL17–17

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

  1. L17
    apply gaussian_ring_multiply_add_real_negative
08Calculate and transport equalitiesL18–18

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

  1. L18
    congr
09Use earlier factsL19–19

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

  1. L19
    apply gaussian_ring_multiply_add_imaginary_positive
10Calculate and transport equalitiesL20–20

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

  1. L20
    symm
11Use earlier factsL21–21

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

  1. L21
    apply gaussian_ring_multiply_add_imaginary_negative

Library-wide reading audit

Original defined command ledger · 21 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 ep
  10. 0010intro en
  11. 0011intro fp
  12. 0012intro fn
  13. 0013split
  14. 0014congr
  15. 0015apply gaussian_ring_multiply_add_real_positive
  16. 0016symm
  17. 0017apply gaussian_ring_multiply_add_real_negative
  18. 0018congr
  19. 0019apply gaussian_ring_multiply_add_imaginary_positive
  20. 0020symm
  21. 0021apply gaussian_ring_multiply_add_imaginary_negative