GF0043

gaussian_divides_value_valid

An actually divisible Gaussian value has a valid canonical carrier code.

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

∀ d. ∀ z. GDvd(d,z) → ZPairValid(z)

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

Definition DAG

ZPairValid(z) · 1GDvd(d,z) · 1

Actual proof prerequisites

gaussian_multiply_output_valid
Original expanded first-order statement 
forall d z. (exists gr_quotient_dividend_domain. (exists ge_first_rp_dividend_domainproduct ge_first_rn_dividend_domainproduct ge_first_ip_dividend_domainproduct ge_first_in_dividend_domainproduct ge_second_rp_dividend_domainproduct ge_second_rn_dividend_domainproduct ge_second_ip_dividend_domainproduct ge_second_in_dividend_domainproduct. ((exists ge_representation_real_code_dividend_domainproductfirst ge_representation_imaginary_code_dividend_domainproductfirst. (((d) = ((ge_representation_real_code_dividend_domainproductfirst) + (ge_representation_imaginary_code_dividend_domainproductfirst)) * S ((ge_representation_real_code_dividend_domainproductfirst) + (ge_representation_imaginary_code_dividend_domainproductfirst)) + ((ge_representation_imaginary_code_dividend_domainproductfirst) + (ge_representation_imaginary_code_dividend_domainproductfirst))) /\ ((exists ge_balance_positive_dividend_domainproductfirstreal ge_balance_negative_dividend_domainproductfirstreal. (((((ge_representation_real_code_dividend_domainproductfirst) = 2 * (ge_balance_positive_dividend_domainproductfirstreal) /\ (ge_balance_negative_dividend_domainproductfirstreal) = 0) \/ exists ge_signed_half_dividend_domainproductfirstrealdecode. (((ge_representation_real_code_dividend_domainproductfirst) = 2 * ge_signed_half_dividend_domainproductfirstrealdecode + 1 /\ (ge_balance_positive_dividend_domainproductfirstreal) = 0) /\ (ge_balance_negative_dividend_domainproductfirstreal) = S ge_signed_half_dividend_domainproductfirstrealdecode))) /\ ((ge_first_rp_dividend_domainproduct) + ge_balance_negative_dividend_domainproductfirstreal = (ge_first_rn_dividend_domainproduct) + ge_balance_positive_dividend_domainproductfirstreal))) /\ (exists ge_balance_positive_dividend_domainproductfirstimaginary ge_balance_negative_dividend_domainproductfirstimaginary. (((((ge_representation_imaginary_code_dividend_domainproductfirst) = 2 * (ge_balance_positive_dividend_domainproductfirstimaginary) /\ (ge_balance_negative_dividend_domainproductfirstimaginary) = 0) \/ exists ge_signed_half_dividend_domainproductfirstimaginarydecode. (((ge_representation_imaginary_code_dividend_domainproductfirst) = 2 * ge_signed_half_dividend_domainproductfirstimaginarydecode + 1 /\ (ge_balance_positive_dividend_domainproductfirstimaginary) = 0) /\ (ge_balance_negative_dividend_domainproductfirstimaginary) = S ge_signed_half_dividend_domainproductfirstimaginarydecode))) /\ ((ge_first_ip_dividend_domainproduct) + ge_balance_negative_dividend_domainproductfirstimaginary = (ge_first_in_dividend_domainproduct) + ge_balance_positive_dividend_domainproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_dividend_domainproductsecond ge_representation_imaginary_code_dividend_domainproductsecond. (((gr_quotient_dividend_domain) = ((ge_representation_real_code_dividend_domainproductsecond) + (ge_representation_imaginary_code_dividend_domainproductsecond)) * S ((ge_representation_real_code_dividend_domainproductsecond) + (ge_representation_imaginary_code_dividend_domainproductsecond)) + ((ge_representation_imaginary_code_dividend_domainproductsecond) + (ge_representation_imaginary_code_dividend_domainproductsecond))) /\ ((exists ge_balance_positive_dividend_domainproductsecondreal ge_balance_negative_dividend_domainproductsecondreal. (((((ge_representation_real_code_dividend_domainproductsecond) = 2 * (ge_balance_positive_dividend_domainproductsecondreal) /\ (ge_balance_negative_dividend_domainproductsecondreal) = 0) \/ exists ge_signed_half_dividend_domainproductsecondrealdecode. (((ge_representation_real_code_dividend_domainproductsecond) = 2 * ge_signed_half_dividend_domainproductsecondrealdecode + 1 /\ (ge_balance_positive_dividend_domainproductsecondreal) = 0) /\ (ge_balance_negative_dividend_domainproductsecondreal) = S ge_signed_half_dividend_domainproductsecondrealdecode))) /\ ((ge_second_rp_dividend_domainproduct) + ge_balance_negative_dividend_domainproductsecondreal = (ge_second_rn_dividend_domainproduct) + ge_balance_positive_dividend_domainproductsecondreal))) /\ (exists ge_balance_positive_dividend_domainproductsecondimaginary ge_balance_negative_dividend_domainproductsecondimaginary. (((((ge_representation_imaginary_code_dividend_domainproductsecond) = 2 * (ge_balance_positive_dividend_domainproductsecondimaginary) /\ (ge_balance_negative_dividend_domainproductsecondimaginary) = 0) \/ exists ge_signed_half_dividend_domainproductsecondimaginarydecode. (((ge_representation_imaginary_code_dividend_domainproductsecond) = 2 * ge_signed_half_dividend_domainproductsecondimaginarydecode + 1 /\ (ge_balance_positive_dividend_domainproductsecondimaginary) = 0) /\ (ge_balance_negative_dividend_domainproductsecondimaginary) = S ge_signed_half_dividend_domainproductsecondimaginarydecode))) /\ ((ge_second_ip_dividend_domainproduct) + ge_balance_negative_dividend_domainproductsecondimaginary = (ge_second_in_dividend_domainproduct) + ge_balance_positive_dividend_domainproductsecondimaginary)))))) /\ (exists ge_representation_real_code_dividend_domainproductoutput ge_representation_imaginary_code_dividend_domainproductoutput. (((z) = ((ge_representation_real_code_dividend_domainproductoutput) + (ge_representation_imaginary_code_dividend_domainproductoutput)) * S ((ge_representation_real_code_dividend_domainproductoutput) + (ge_representation_imaginary_code_dividend_domainproductoutput)) + ((ge_representation_imaginary_code_dividend_domainproductoutput) + (ge_representation_imaginary_code_dividend_domainproductoutput))) /\ ((exists ge_balance_positive_dividend_domainproductoutputreal ge_balance_negative_dividend_domainproductoutputreal. (((((ge_representation_real_code_dividend_domainproductoutput) = 2 * (ge_balance_positive_dividend_domainproductoutputreal) /\ (ge_balance_negative_dividend_domainproductoutputreal) = 0) \/ exists ge_signed_half_dividend_domainproductoutputrealdecode. (((ge_representation_real_code_dividend_domainproductoutput) = 2 * ge_signed_half_dividend_domainproductoutputrealdecode + 1 /\ (ge_balance_positive_dividend_domainproductoutputreal) = 0) /\ (ge_balance_negative_dividend_domainproductoutputreal) = S ge_signed_half_dividend_domainproductoutputrealdecode))) /\ ((((((((ge_first_rp_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))))))) + ge_balance_negative_dividend_domainproductoutputreal = (((((((ge_first_rp_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))))))) + ge_balance_positive_dividend_domainproductoutputreal))) /\ (exists ge_balance_positive_dividend_domainproductoutputimaginary ge_balance_negative_dividend_domainproductoutputimaginary. (((((ge_representation_imaginary_code_dividend_domainproductoutput) = 2 * (ge_balance_positive_dividend_domainproductoutputimaginary) /\ (ge_balance_negative_dividend_domainproductoutputimaginary) = 0) \/ exists ge_signed_half_dividend_domainproductoutputimaginarydecode. (((ge_representation_imaginary_code_dividend_domainproductoutput) = 2 * ge_signed_half_dividend_domainproductoutputimaginarydecode + 1 /\ (ge_balance_positive_dividend_domainproductoutputimaginary) = 0) /\ (ge_balance_negative_dividend_domainproductoutputimaginary) = S ge_signed_half_dividend_domainproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))))))) + ge_balance_negative_dividend_domainproductoutputimaginary = (((((((ge_first_rp_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))))))) + ge_balance_positive_dividend_domainproductoutputimaginary)))))))))) -> (exists ge_real_positive_dividend_valid ge_real_negative_dividend_valid ge_imaginary_positive_dividend_valid ge_imaginary_negative_dividend_valid. (exists ge_real_code_dividend_validdecode ge_imaginary_code_dividend_validdecode. (((z) = ((ge_real_code_dividend_validdecode) + (ge_imaginary_code_dividend_validdecode)) * S ((ge_real_code_dividend_validdecode) + (ge_imaginary_code_dividend_validdecode)) + ((ge_imaginary_code_dividend_validdecode) + (ge_imaginary_code_dividend_validdecode))) /\ (((((ge_real_code_dividend_validdecode) = 2 * (ge_real_positive_dividend_valid) /\ (ge_real_negative_dividend_valid) = 0) \/ exists ge_signed_half_ge_dividend_validdecode_real. (((ge_real_code_dividend_validdecode) = 2 * ge_signed_half_ge_dividend_validdecode_real + 1 /\ (ge_real_positive_dividend_valid) = 0) /\ (ge_real_negative_dividend_valid) = S ge_signed_half_ge_dividend_validdecode_real))) /\ ((((ge_imaginary_code_dividend_validdecode) = 2 * (ge_imaginary_positive_dividend_valid) /\ (ge_imaginary_negative_dividend_valid) = 0) \/ exists ge_signed_half_ge_dividend_validdecode_imaginary. (((ge_imaginary_code_dividend_validdecode) = 2 * ge_signed_half_ge_dividend_validdecode_imaginary + 1 /\ (ge_imaginary_positive_dividend_valid) = 0) /\ (ge_imaginary_negative_dividend_valid) = S ge_signed_half_ge_dividend_validdecode_imaginary)))))))

Complete tactic proof in conservative notation

All 9 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

9 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.

Named ingredients (1)
gaussian multiply output valid
01Fix variables and assumptionsL1–3

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

  1. L1
    intro d
  2. L2
    intro z
  3. L3
    intro h
02Separate the logical casesL4–4

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

  1. L4
    cases h
03Use earlier factsL5–9

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

  1. L5
    specialize gaussian_multiply_output_valid (d)
  2. L6
    specialize gaussian_multiply_output_valid (x)
  3. L7
    specialize gaussian_multiply_output_valid (z)
  4. L8
    apply gaussian_multiply_output_valid
  5. L9
    exact h_witness

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro d
  2. 0002intro z
  3. 0003intro h
  4. 0004cases h
  5. 0005specialize gaussian_multiply_output_valid (d)
  6. 0006specialize gaussian_multiply_output_valid (x)
  7. 0007specialize gaussian_multiply_output_valid (z)
  8. 0008apply gaussian_multiply_output_valid
  9. 0009exact h_witness