GF0008

gaussian_multiply_input_right_valid

Actual Gaussian multiply certifies the input right carrier, without treating every natural code as valid.

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

∀ a. ∀ b. ∀ c. GMul(a,b,c) → ZPairValid(b)

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

Definition DAG

ZPairValid(z) · 1GMul(a,b,c) · 1

Actual proof prerequisites

gaussian_representation_is_gaussian · checked external prerequisite
Original expanded first-order statement 
forall a b c. (exists ge_first_rp_ring_multiply_input_right ge_first_rn_ring_multiply_input_right ge_first_ip_ring_multiply_input_right ge_first_in_ring_multiply_input_right ge_second_rp_ring_multiply_input_right ge_second_rn_ring_multiply_input_right ge_second_ip_ring_multiply_input_right ge_second_in_ring_multiply_input_right. ((exists ge_representation_real_code_ring_multiply_input_rightfirst ge_representation_imaginary_code_ring_multiply_input_rightfirst. (((a) = ((ge_representation_real_code_ring_multiply_input_rightfirst) + (ge_representation_imaginary_code_ring_multiply_input_rightfirst)) * S ((ge_representation_real_code_ring_multiply_input_rightfirst) + (ge_representation_imaginary_code_ring_multiply_input_rightfirst)) + ((ge_representation_imaginary_code_ring_multiply_input_rightfirst) + (ge_representation_imaginary_code_ring_multiply_input_rightfirst))) /\ ((exists ge_balance_positive_ring_multiply_input_rightfirstreal ge_balance_negative_ring_multiply_input_rightfirstreal. (((((ge_representation_real_code_ring_multiply_input_rightfirst) = 2 * (ge_balance_positive_ring_multiply_input_rightfirstreal) /\ (ge_balance_negative_ring_multiply_input_rightfirstreal) = 0) \/ exists ge_signed_half_ring_multiply_input_rightfirstrealdecode. (((ge_representation_real_code_ring_multiply_input_rightfirst) = 2 * ge_signed_half_ring_multiply_input_rightfirstrealdecode + 1 /\ (ge_balance_positive_ring_multiply_input_rightfirstreal) = 0) /\ (ge_balance_negative_ring_multiply_input_rightfirstreal) = S ge_signed_half_ring_multiply_input_rightfirstrealdecode))) /\ ((ge_first_rp_ring_multiply_input_right) + ge_balance_negative_ring_multiply_input_rightfirstreal = (ge_first_rn_ring_multiply_input_right) + ge_balance_positive_ring_multiply_input_rightfirstreal))) /\ (exists ge_balance_positive_ring_multiply_input_rightfirstimaginary ge_balance_negative_ring_multiply_input_rightfirstimaginary. (((((ge_representation_imaginary_code_ring_multiply_input_rightfirst) = 2 * (ge_balance_positive_ring_multiply_input_rightfirstimaginary) /\ (ge_balance_negative_ring_multiply_input_rightfirstimaginary) = 0) \/ exists ge_signed_half_ring_multiply_input_rightfirstimaginarydecode. (((ge_representation_imaginary_code_ring_multiply_input_rightfirst) = 2 * ge_signed_half_ring_multiply_input_rightfirstimaginarydecode + 1 /\ (ge_balance_positive_ring_multiply_input_rightfirstimaginary) = 0) /\ (ge_balance_negative_ring_multiply_input_rightfirstimaginary) = S ge_signed_half_ring_multiply_input_rightfirstimaginarydecode))) /\ ((ge_first_ip_ring_multiply_input_right) + ge_balance_negative_ring_multiply_input_rightfirstimaginary = (ge_first_in_ring_multiply_input_right) + ge_balance_positive_ring_multiply_input_rightfirstimaginary)))))) /\ ((exists ge_representation_real_code_ring_multiply_input_rightsecond ge_representation_imaginary_code_ring_multiply_input_rightsecond. (((b) = ((ge_representation_real_code_ring_multiply_input_rightsecond) + (ge_representation_imaginary_code_ring_multiply_input_rightsecond)) * S ((ge_representation_real_code_ring_multiply_input_rightsecond) + (ge_representation_imaginary_code_ring_multiply_input_rightsecond)) + ((ge_representation_imaginary_code_ring_multiply_input_rightsecond) + (ge_representation_imaginary_code_ring_multiply_input_rightsecond))) /\ ((exists ge_balance_positive_ring_multiply_input_rightsecondreal ge_balance_negative_ring_multiply_input_rightsecondreal. (((((ge_representation_real_code_ring_multiply_input_rightsecond) = 2 * (ge_balance_positive_ring_multiply_input_rightsecondreal) /\ (ge_balance_negative_ring_multiply_input_rightsecondreal) = 0) \/ exists ge_signed_half_ring_multiply_input_rightsecondrealdecode. (((ge_representation_real_code_ring_multiply_input_rightsecond) = 2 * ge_signed_half_ring_multiply_input_rightsecondrealdecode + 1 /\ (ge_balance_positive_ring_multiply_input_rightsecondreal) = 0) /\ (ge_balance_negative_ring_multiply_input_rightsecondreal) = S ge_signed_half_ring_multiply_input_rightsecondrealdecode))) /\ ((ge_second_rp_ring_multiply_input_right) + ge_balance_negative_ring_multiply_input_rightsecondreal = (ge_second_rn_ring_multiply_input_right) + ge_balance_positive_ring_multiply_input_rightsecondreal))) /\ (exists ge_balance_positive_ring_multiply_input_rightsecondimaginary ge_balance_negative_ring_multiply_input_rightsecondimaginary. (((((ge_representation_imaginary_code_ring_multiply_input_rightsecond) = 2 * (ge_balance_positive_ring_multiply_input_rightsecondimaginary) /\ (ge_balance_negative_ring_multiply_input_rightsecondimaginary) = 0) \/ exists ge_signed_half_ring_multiply_input_rightsecondimaginarydecode. (((ge_representation_imaginary_code_ring_multiply_input_rightsecond) = 2 * ge_signed_half_ring_multiply_input_rightsecondimaginarydecode + 1 /\ (ge_balance_positive_ring_multiply_input_rightsecondimaginary) = 0) /\ (ge_balance_negative_ring_multiply_input_rightsecondimaginary) = S ge_signed_half_ring_multiply_input_rightsecondimaginarydecode))) /\ ((ge_second_ip_ring_multiply_input_right) + ge_balance_negative_ring_multiply_input_rightsecondimaginary = (ge_second_in_ring_multiply_input_right) + ge_balance_positive_ring_multiply_input_rightsecondimaginary)))))) /\ (exists ge_representation_real_code_ring_multiply_input_rightoutput ge_representation_imaginary_code_ring_multiply_input_rightoutput. (((c) = ((ge_representation_real_code_ring_multiply_input_rightoutput) + (ge_representation_imaginary_code_ring_multiply_input_rightoutput)) * S ((ge_representation_real_code_ring_multiply_input_rightoutput) + (ge_representation_imaginary_code_ring_multiply_input_rightoutput)) + ((ge_representation_imaginary_code_ring_multiply_input_rightoutput) + (ge_representation_imaginary_code_ring_multiply_input_rightoutput))) /\ ((exists ge_balance_positive_ring_multiply_input_rightoutputreal ge_balance_negative_ring_multiply_input_rightoutputreal. (((((ge_representation_real_code_ring_multiply_input_rightoutput) = 2 * (ge_balance_positive_ring_multiply_input_rightoutputreal) /\ (ge_balance_negative_ring_multiply_input_rightoutputreal) = 0) \/ exists ge_signed_half_ring_multiply_input_rightoutputrealdecode. (((ge_representation_real_code_ring_multiply_input_rightoutput) = 2 * ge_signed_half_ring_multiply_input_rightoutputrealdecode + 1 /\ (ge_balance_positive_ring_multiply_input_rightoutputreal) = 0) /\ (ge_balance_negative_ring_multiply_input_rightoutputreal) = S ge_signed_half_ring_multiply_input_rightoutputrealdecode))) /\ ((((((((ge_first_rp_ring_multiply_input_right) * (ge_second_rp_ring_multiply_input_right))) + (((ge_first_rn_ring_multiply_input_right) * (ge_second_rn_ring_multiply_input_right))))) + (((((ge_first_ip_ring_multiply_input_right) * (ge_second_in_ring_multiply_input_right))) + (((ge_first_in_ring_multiply_input_right) * (ge_second_ip_ring_multiply_input_right))))))) + ge_balance_negative_ring_multiply_input_rightoutputreal = (((((((ge_first_rp_ring_multiply_input_right) * (ge_second_rn_ring_multiply_input_right))) + (((ge_first_rn_ring_multiply_input_right) * (ge_second_rp_ring_multiply_input_right))))) + (((((ge_first_ip_ring_multiply_input_right) * (ge_second_ip_ring_multiply_input_right))) + (((ge_first_in_ring_multiply_input_right) * (ge_second_in_ring_multiply_input_right))))))) + ge_balance_positive_ring_multiply_input_rightoutputreal))) /\ (exists ge_balance_positive_ring_multiply_input_rightoutputimaginary ge_balance_negative_ring_multiply_input_rightoutputimaginary. (((((ge_representation_imaginary_code_ring_multiply_input_rightoutput) = 2 * (ge_balance_positive_ring_multiply_input_rightoutputimaginary) /\ (ge_balance_negative_ring_multiply_input_rightoutputimaginary) = 0) \/ exists ge_signed_half_ring_multiply_input_rightoutputimaginarydecode. (((ge_representation_imaginary_code_ring_multiply_input_rightoutput) = 2 * ge_signed_half_ring_multiply_input_rightoutputimaginarydecode + 1 /\ (ge_balance_positive_ring_multiply_input_rightoutputimaginary) = 0) /\ (ge_balance_negative_ring_multiply_input_rightoutputimaginary) = S ge_signed_half_ring_multiply_input_rightoutputimaginarydecode))) /\ ((((((((ge_first_rp_ring_multiply_input_right) * (ge_second_ip_ring_multiply_input_right))) + (((ge_first_rn_ring_multiply_input_right) * (ge_second_in_ring_multiply_input_right))))) + (((((ge_first_ip_ring_multiply_input_right) * (ge_second_rp_ring_multiply_input_right))) + (((ge_first_in_ring_multiply_input_right) * (ge_second_rn_ring_multiply_input_right))))))) + ge_balance_negative_ring_multiply_input_rightoutputimaginary = (((((((ge_first_rp_ring_multiply_input_right) * (ge_second_in_ring_multiply_input_right))) + (((ge_first_rn_ring_multiply_input_right) * (ge_second_ip_ring_multiply_input_right))))) + (((((ge_first_ip_ring_multiply_input_right) * (ge_second_rn_ring_multiply_input_right))) + (((ge_first_in_ring_multiply_input_right) * (ge_second_rp_ring_multiply_input_right))))))) + ge_balance_positive_ring_multiply_input_rightoutputimaginary))))))))) -> (exists ge_real_positive_ring_multiply_input_right_domain ge_real_negative_ring_multiply_input_right_domain ge_imaginary_positive_ring_multiply_input_right_domain ge_imaginary_negative_ring_multiply_input_right_domain. (exists ge_real_code_ring_multiply_input_right_domaindecode ge_imaginary_code_ring_multiply_input_right_domaindecode. (((b) = ((ge_real_code_ring_multiply_input_right_domaindecode) + (ge_imaginary_code_ring_multiply_input_right_domaindecode)) * S ((ge_real_code_ring_multiply_input_right_domaindecode) + (ge_imaginary_code_ring_multiply_input_right_domaindecode)) + ((ge_imaginary_code_ring_multiply_input_right_domaindecode) + (ge_imaginary_code_ring_multiply_input_right_domaindecode))) /\ (((((ge_real_code_ring_multiply_input_right_domaindecode) = 2 * (ge_real_positive_ring_multiply_input_right_domain) /\ (ge_real_negative_ring_multiply_input_right_domain) = 0) \/ exists ge_signed_half_ge_ring_multiply_input_right_domaindecode_real. (((ge_real_code_ring_multiply_input_right_domaindecode) = 2 * ge_signed_half_ge_ring_multiply_input_right_domaindecode_real + 1 /\ (ge_real_positive_ring_multiply_input_right_domain) = 0) /\ (ge_real_negative_ring_multiply_input_right_domain) = S ge_signed_half_ge_ring_multiply_input_right_domaindecode_real))) /\ ((((ge_imaginary_code_ring_multiply_input_right_domaindecode) = 2 * (ge_imaginary_positive_ring_multiply_input_right_domain) /\ (ge_imaginary_negative_ring_multiply_input_right_domain) = 0) \/ exists ge_signed_half_ge_ring_multiply_input_right_domaindecode_imaginary. (((ge_imaginary_code_ring_multiply_input_right_domaindecode) = 2 * ge_signed_half_ge_ring_multiply_input_right_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_ring_multiply_input_right_domain) = 0) /\ (ge_imaginary_negative_ring_multiply_input_right_domain) = S ge_signed_half_ge_ring_multiply_input_right_domaindecode_imaginary)))))))

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 · 3 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.

01Fix variables and assumptionsL1–4

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 h
02Separate the logical casesL5–14

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

  1. L5
    cases h
  2. L6
    cases h_witness
  3. L7
    cases h_witness_witness
  4. L8
    cases h_witness_witness_witness
  5. L9
    cases h_witness_witness_witness_witness
  6. L10
    cases h_witness_witness_witness_witness_witness
  7. L11
    cases h_witness_witness_witness_witness_witness_witness
  8. L12
    cases h_witness_witness_witness_witness_witness_witness_witness
  9. L13
    cases h_witness_witness_witness_witness_witness_witness_witness_witness
  10. L14
    cases h_witness_witness_witness_witness_witness_witness_witness_witness_right
03Use earlier factsL15–21

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

  1. L15
    specialize gaussian_representation_is_gaussian (b)
  2. L16
    specialize gaussian_representation_is_gaussian (x4)
  3. L17
    specialize gaussian_representation_is_gaussian (x5)
  4. L18
    specialize gaussian_representation_is_gaussian (x6)
  5. L19
    specialize gaussian_representation_is_gaussian (x7)
  6. L20
    apply gaussian_representation_is_gaussian
  7. L21
    exact h_witness_witness_witness_witness_witness_witness_witness_witness_right_left

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005cases h
  6. 0006cases h_witness
  7. 0007cases h_witness_witness
  8. 0008cases h_witness_witness_witness
  9. 0009cases h_witness_witness_witness_witness
  10. 0010cases h_witness_witness_witness_witness_witness
  11. 0011cases h_witness_witness_witness_witness_witness_witness
  12. 0012cases h_witness_witness_witness_witness_witness_witness_witness
  13. 0013cases h_witness_witness_witness_witness_witness_witness_witness_witness
  14. 0014cases h_witness_witness_witness_witness_witness_witness_witness_witness_right
  15. 0015specialize gaussian_representation_is_gaussian (b)
  16. 0016specialize gaussian_representation_is_gaussian (x4)
  17. 0017specialize gaussian_representation_is_gaussian (x5)
  18. 0018specialize gaussian_representation_is_gaussian (x6)
  19. 0019specialize gaussian_representation_is_gaussian (x7)
  20. 0020apply gaussian_representation_is_gaussian
  21. 0021exact h_witness_witness_witness_witness_witness_witness_witness_witness_right_left