GF0046

gaussian_one_divides

The actual Gaussian identity divides every valid Gaussian integer.

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

∀ z. ZPairValid(z) → GDvd(6,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_one_left
Original expanded first-order statement 
forall z. (exists ge_real_positive_one_divides_domain ge_real_negative_one_divides_domain ge_imaginary_positive_one_divides_domain ge_imaginary_negative_one_divides_domain. (exists ge_real_code_one_divides_domaindecode ge_imaginary_code_one_divides_domaindecode. (((z) = ((ge_real_code_one_divides_domaindecode) + (ge_imaginary_code_one_divides_domaindecode)) * S ((ge_real_code_one_divides_domaindecode) + (ge_imaginary_code_one_divides_domaindecode)) + ((ge_imaginary_code_one_divides_domaindecode) + (ge_imaginary_code_one_divides_domaindecode))) /\ (((((ge_real_code_one_divides_domaindecode) = 2 * (ge_real_positive_one_divides_domain) /\ (ge_real_negative_one_divides_domain) = 0) \/ exists ge_signed_half_ge_one_divides_domaindecode_real. (((ge_real_code_one_divides_domaindecode) = 2 * ge_signed_half_ge_one_divides_domaindecode_real + 1 /\ (ge_real_positive_one_divides_domain) = 0) /\ (ge_real_negative_one_divides_domain) = S ge_signed_half_ge_one_divides_domaindecode_real))) /\ ((((ge_imaginary_code_one_divides_domaindecode) = 2 * (ge_imaginary_positive_one_divides_domain) /\ (ge_imaginary_negative_one_divides_domain) = 0) \/ exists ge_signed_half_ge_one_divides_domaindecode_imaginary. (((ge_imaginary_code_one_divides_domaindecode) = 2 * ge_signed_half_ge_one_divides_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_one_divides_domain) = 0) /\ (ge_imaginary_negative_one_divides_domain) = S ge_signed_half_ge_one_divides_domaindecode_imaginary))))))) -> (exists gr_quotient_one_divides. (exists ge_first_rp_one_dividesproduct ge_first_rn_one_dividesproduct ge_first_ip_one_dividesproduct ge_first_in_one_dividesproduct ge_second_rp_one_dividesproduct ge_second_rn_one_dividesproduct ge_second_ip_one_dividesproduct ge_second_in_one_dividesproduct. ((exists ge_representation_real_code_one_dividesproductfirst ge_representation_imaginary_code_one_dividesproductfirst. (((6) = ((ge_representation_real_code_one_dividesproductfirst) + (ge_representation_imaginary_code_one_dividesproductfirst)) * S ((ge_representation_real_code_one_dividesproductfirst) + (ge_representation_imaginary_code_one_dividesproductfirst)) + ((ge_representation_imaginary_code_one_dividesproductfirst) + (ge_representation_imaginary_code_one_dividesproductfirst))) /\ ((exists ge_balance_positive_one_dividesproductfirstreal ge_balance_negative_one_dividesproductfirstreal. (((((ge_representation_real_code_one_dividesproductfirst) = 2 * (ge_balance_positive_one_dividesproductfirstreal) /\ (ge_balance_negative_one_dividesproductfirstreal) = 0) \/ exists ge_signed_half_one_dividesproductfirstrealdecode. (((ge_representation_real_code_one_dividesproductfirst) = 2 * ge_signed_half_one_dividesproductfirstrealdecode + 1 /\ (ge_balance_positive_one_dividesproductfirstreal) = 0) /\ (ge_balance_negative_one_dividesproductfirstreal) = S ge_signed_half_one_dividesproductfirstrealdecode))) /\ ((ge_first_rp_one_dividesproduct) + ge_balance_negative_one_dividesproductfirstreal = (ge_first_rn_one_dividesproduct) + ge_balance_positive_one_dividesproductfirstreal))) /\ (exists ge_balance_positive_one_dividesproductfirstimaginary ge_balance_negative_one_dividesproductfirstimaginary. (((((ge_representation_imaginary_code_one_dividesproductfirst) = 2 * (ge_balance_positive_one_dividesproductfirstimaginary) /\ (ge_balance_negative_one_dividesproductfirstimaginary) = 0) \/ exists ge_signed_half_one_dividesproductfirstimaginarydecode. (((ge_representation_imaginary_code_one_dividesproductfirst) = 2 * ge_signed_half_one_dividesproductfirstimaginarydecode + 1 /\ (ge_balance_positive_one_dividesproductfirstimaginary) = 0) /\ (ge_balance_negative_one_dividesproductfirstimaginary) = S ge_signed_half_one_dividesproductfirstimaginarydecode))) /\ ((ge_first_ip_one_dividesproduct) + ge_balance_negative_one_dividesproductfirstimaginary = (ge_first_in_one_dividesproduct) + ge_balance_positive_one_dividesproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_one_dividesproductsecond ge_representation_imaginary_code_one_dividesproductsecond. (((gr_quotient_one_divides) = ((ge_representation_real_code_one_dividesproductsecond) + (ge_representation_imaginary_code_one_dividesproductsecond)) * S ((ge_representation_real_code_one_dividesproductsecond) + (ge_representation_imaginary_code_one_dividesproductsecond)) + ((ge_representation_imaginary_code_one_dividesproductsecond) + (ge_representation_imaginary_code_one_dividesproductsecond))) /\ ((exists ge_balance_positive_one_dividesproductsecondreal ge_balance_negative_one_dividesproductsecondreal. (((((ge_representation_real_code_one_dividesproductsecond) = 2 * (ge_balance_positive_one_dividesproductsecondreal) /\ (ge_balance_negative_one_dividesproductsecondreal) = 0) \/ exists ge_signed_half_one_dividesproductsecondrealdecode. (((ge_representation_real_code_one_dividesproductsecond) = 2 * ge_signed_half_one_dividesproductsecondrealdecode + 1 /\ (ge_balance_positive_one_dividesproductsecondreal) = 0) /\ (ge_balance_negative_one_dividesproductsecondreal) = S ge_signed_half_one_dividesproductsecondrealdecode))) /\ ((ge_second_rp_one_dividesproduct) + ge_balance_negative_one_dividesproductsecondreal = (ge_second_rn_one_dividesproduct) + ge_balance_positive_one_dividesproductsecondreal))) /\ (exists ge_balance_positive_one_dividesproductsecondimaginary ge_balance_negative_one_dividesproductsecondimaginary. (((((ge_representation_imaginary_code_one_dividesproductsecond) = 2 * (ge_balance_positive_one_dividesproductsecondimaginary) /\ (ge_balance_negative_one_dividesproductsecondimaginary) = 0) \/ exists ge_signed_half_one_dividesproductsecondimaginarydecode. (((ge_representation_imaginary_code_one_dividesproductsecond) = 2 * ge_signed_half_one_dividesproductsecondimaginarydecode + 1 /\ (ge_balance_positive_one_dividesproductsecondimaginary) = 0) /\ (ge_balance_negative_one_dividesproductsecondimaginary) = S ge_signed_half_one_dividesproductsecondimaginarydecode))) /\ ((ge_second_ip_one_dividesproduct) + ge_balance_negative_one_dividesproductsecondimaginary = (ge_second_in_one_dividesproduct) + ge_balance_positive_one_dividesproductsecondimaginary)))))) /\ (exists ge_representation_real_code_one_dividesproductoutput ge_representation_imaginary_code_one_dividesproductoutput. (((z) = ((ge_representation_real_code_one_dividesproductoutput) + (ge_representation_imaginary_code_one_dividesproductoutput)) * S ((ge_representation_real_code_one_dividesproductoutput) + (ge_representation_imaginary_code_one_dividesproductoutput)) + ((ge_representation_imaginary_code_one_dividesproductoutput) + (ge_representation_imaginary_code_one_dividesproductoutput))) /\ ((exists ge_balance_positive_one_dividesproductoutputreal ge_balance_negative_one_dividesproductoutputreal. (((((ge_representation_real_code_one_dividesproductoutput) = 2 * (ge_balance_positive_one_dividesproductoutputreal) /\ (ge_balance_negative_one_dividesproductoutputreal) = 0) \/ exists ge_signed_half_one_dividesproductoutputrealdecode. (((ge_representation_real_code_one_dividesproductoutput) = 2 * ge_signed_half_one_dividesproductoutputrealdecode + 1 /\ (ge_balance_positive_one_dividesproductoutputreal) = 0) /\ (ge_balance_negative_one_dividesproductoutputreal) = S ge_signed_half_one_dividesproductoutputrealdecode))) /\ ((((((((ge_first_rp_one_dividesproduct) * (ge_second_rp_one_dividesproduct))) + (((ge_first_rn_one_dividesproduct) * (ge_second_rn_one_dividesproduct))))) + (((((ge_first_ip_one_dividesproduct) * (ge_second_in_one_dividesproduct))) + (((ge_first_in_one_dividesproduct) * (ge_second_ip_one_dividesproduct))))))) + ge_balance_negative_one_dividesproductoutputreal = (((((((ge_first_rp_one_dividesproduct) * (ge_second_rn_one_dividesproduct))) + (((ge_first_rn_one_dividesproduct) * (ge_second_rp_one_dividesproduct))))) + (((((ge_first_ip_one_dividesproduct) * (ge_second_ip_one_dividesproduct))) + (((ge_first_in_one_dividesproduct) * (ge_second_in_one_dividesproduct))))))) + ge_balance_positive_one_dividesproductoutputreal))) /\ (exists ge_balance_positive_one_dividesproductoutputimaginary ge_balance_negative_one_dividesproductoutputimaginary. (((((ge_representation_imaginary_code_one_dividesproductoutput) = 2 * (ge_balance_positive_one_dividesproductoutputimaginary) /\ (ge_balance_negative_one_dividesproductoutputimaginary) = 0) \/ exists ge_signed_half_one_dividesproductoutputimaginarydecode. (((ge_representation_imaginary_code_one_dividesproductoutput) = 2 * ge_signed_half_one_dividesproductoutputimaginarydecode + 1 /\ (ge_balance_positive_one_dividesproductoutputimaginary) = 0) /\ (ge_balance_negative_one_dividesproductoutputimaginary) = S ge_signed_half_one_dividesproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_one_dividesproduct) * (ge_second_ip_one_dividesproduct))) + (((ge_first_rn_one_dividesproduct) * (ge_second_in_one_dividesproduct))))) + (((((ge_first_ip_one_dividesproduct) * (ge_second_rp_one_dividesproduct))) + (((ge_first_in_one_dividesproduct) * (ge_second_rn_one_dividesproduct))))))) + ge_balance_negative_one_dividesproductoutputimaginary = (((((((ge_first_rp_one_dividesproduct) * (ge_second_in_one_dividesproduct))) + (((ge_first_rn_one_dividesproduct) * (ge_second_ip_one_dividesproduct))))) + (((((ge_first_ip_one_dividesproduct) * (ge_second_rn_one_dividesproduct))) + (((ge_first_in_one_dividesproduct) * (ge_second_rp_one_dividesproduct))))))) + ge_balance_positive_one_dividesproductoutputimaginary))))))))))

Complete tactic proof in conservative notation

All 6 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

6 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 one left
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro h
02Construct an explicit witnessL3–3

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

  1. L3
    exists (z)
03Use earlier factsL4–6

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

  1. L4
    specialize gaussian_multiply_one_left (z)
  2. L5
    apply gaussian_multiply_one_left
  3. L6
    exact h

Library-wide reading audit

Original defined command ledger · 6 lines
  1. 0001intro z
  2. 0002intro h
  3. 0003exists (z)
  4. 0004specialize gaussian_multiply_one_left (z)
  5. 0005apply gaussian_multiply_one_left
  6. 0006exact h