GF001E

gaussian_unit_valid

The inverse-witness unit definition enforces the actual Gaussian carrier.

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. GUnit(z) → ZPairValid(z)

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

Definition DAG

ZPairValid(z) · 1GUnit(z) · 1

Actual proof prerequisites

gaussian_multiply_input_left_valid
Original expanded first-order statement 
forall z. (exists gr_inverse_unit_valid. (exists ge_first_rp_unit_valididentity ge_first_rn_unit_valididentity ge_first_ip_unit_valididentity ge_first_in_unit_valididentity ge_second_rp_unit_valididentity ge_second_rn_unit_valididentity ge_second_ip_unit_valididentity ge_second_in_unit_valididentity. ((exists ge_representation_real_code_unit_valididentityfirst ge_representation_imaginary_code_unit_valididentityfirst. (((z) = ((ge_representation_real_code_unit_valididentityfirst) + (ge_representation_imaginary_code_unit_valididentityfirst)) * S ((ge_representation_real_code_unit_valididentityfirst) + (ge_representation_imaginary_code_unit_valididentityfirst)) + ((ge_representation_imaginary_code_unit_valididentityfirst) + (ge_representation_imaginary_code_unit_valididentityfirst))) /\ ((exists ge_balance_positive_unit_valididentityfirstreal ge_balance_negative_unit_valididentityfirstreal. (((((ge_representation_real_code_unit_valididentityfirst) = 2 * (ge_balance_positive_unit_valididentityfirstreal) /\ (ge_balance_negative_unit_valididentityfirstreal) = 0) \/ exists ge_signed_half_unit_valididentityfirstrealdecode. (((ge_representation_real_code_unit_valididentityfirst) = 2 * ge_signed_half_unit_valididentityfirstrealdecode + 1 /\ (ge_balance_positive_unit_valididentityfirstreal) = 0) /\ (ge_balance_negative_unit_valididentityfirstreal) = S ge_signed_half_unit_valididentityfirstrealdecode))) /\ ((ge_first_rp_unit_valididentity) + ge_balance_negative_unit_valididentityfirstreal = (ge_first_rn_unit_valididentity) + ge_balance_positive_unit_valididentityfirstreal))) /\ (exists ge_balance_positive_unit_valididentityfirstimaginary ge_balance_negative_unit_valididentityfirstimaginary. (((((ge_representation_imaginary_code_unit_valididentityfirst) = 2 * (ge_balance_positive_unit_valididentityfirstimaginary) /\ (ge_balance_negative_unit_valididentityfirstimaginary) = 0) \/ exists ge_signed_half_unit_valididentityfirstimaginarydecode. (((ge_representation_imaginary_code_unit_valididentityfirst) = 2 * ge_signed_half_unit_valididentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unit_valididentityfirstimaginary) = 0) /\ (ge_balance_negative_unit_valididentityfirstimaginary) = S ge_signed_half_unit_valididentityfirstimaginarydecode))) /\ ((ge_first_ip_unit_valididentity) + ge_balance_negative_unit_valididentityfirstimaginary = (ge_first_in_unit_valididentity) + ge_balance_positive_unit_valididentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unit_valididentitysecond ge_representation_imaginary_code_unit_valididentitysecond. (((gr_inverse_unit_valid) = ((ge_representation_real_code_unit_valididentitysecond) + (ge_representation_imaginary_code_unit_valididentitysecond)) * S ((ge_representation_real_code_unit_valididentitysecond) + (ge_representation_imaginary_code_unit_valididentitysecond)) + ((ge_representation_imaginary_code_unit_valididentitysecond) + (ge_representation_imaginary_code_unit_valididentitysecond))) /\ ((exists ge_balance_positive_unit_valididentitysecondreal ge_balance_negative_unit_valididentitysecondreal. (((((ge_representation_real_code_unit_valididentitysecond) = 2 * (ge_balance_positive_unit_valididentitysecondreal) /\ (ge_balance_negative_unit_valididentitysecondreal) = 0) \/ exists ge_signed_half_unit_valididentitysecondrealdecode. (((ge_representation_real_code_unit_valididentitysecond) = 2 * ge_signed_half_unit_valididentitysecondrealdecode + 1 /\ (ge_balance_positive_unit_valididentitysecondreal) = 0) /\ (ge_balance_negative_unit_valididentitysecondreal) = S ge_signed_half_unit_valididentitysecondrealdecode))) /\ ((ge_second_rp_unit_valididentity) + ge_balance_negative_unit_valididentitysecondreal = (ge_second_rn_unit_valididentity) + ge_balance_positive_unit_valididentitysecondreal))) /\ (exists ge_balance_positive_unit_valididentitysecondimaginary ge_balance_negative_unit_valididentitysecondimaginary. (((((ge_representation_imaginary_code_unit_valididentitysecond) = 2 * (ge_balance_positive_unit_valididentitysecondimaginary) /\ (ge_balance_negative_unit_valididentitysecondimaginary) = 0) \/ exists ge_signed_half_unit_valididentitysecondimaginarydecode. (((ge_representation_imaginary_code_unit_valididentitysecond) = 2 * ge_signed_half_unit_valididentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unit_valididentitysecondimaginary) = 0) /\ (ge_balance_negative_unit_valididentitysecondimaginary) = S ge_signed_half_unit_valididentitysecondimaginarydecode))) /\ ((ge_second_ip_unit_valididentity) + ge_balance_negative_unit_valididentitysecondimaginary = (ge_second_in_unit_valididentity) + ge_balance_positive_unit_valididentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unit_valididentityoutput ge_representation_imaginary_code_unit_valididentityoutput. (((6) = ((ge_representation_real_code_unit_valididentityoutput) + (ge_representation_imaginary_code_unit_valididentityoutput)) * S ((ge_representation_real_code_unit_valididentityoutput) + (ge_representation_imaginary_code_unit_valididentityoutput)) + ((ge_representation_imaginary_code_unit_valididentityoutput) + (ge_representation_imaginary_code_unit_valididentityoutput))) /\ ((exists ge_balance_positive_unit_valididentityoutputreal ge_balance_negative_unit_valididentityoutputreal. (((((ge_representation_real_code_unit_valididentityoutput) = 2 * (ge_balance_positive_unit_valididentityoutputreal) /\ (ge_balance_negative_unit_valididentityoutputreal) = 0) \/ exists ge_signed_half_unit_valididentityoutputrealdecode. (((ge_representation_real_code_unit_valididentityoutput) = 2 * ge_signed_half_unit_valididentityoutputrealdecode + 1 /\ (ge_balance_positive_unit_valididentityoutputreal) = 0) /\ (ge_balance_negative_unit_valididentityoutputreal) = S ge_signed_half_unit_valididentityoutputrealdecode))) /\ ((((((((ge_first_rp_unit_valididentity) * (ge_second_rp_unit_valididentity))) + (((ge_first_rn_unit_valididentity) * (ge_second_rn_unit_valididentity))))) + (((((ge_first_ip_unit_valididentity) * (ge_second_in_unit_valididentity))) + (((ge_first_in_unit_valididentity) * (ge_second_ip_unit_valididentity))))))) + ge_balance_negative_unit_valididentityoutputreal = (((((((ge_first_rp_unit_valididentity) * (ge_second_rn_unit_valididentity))) + (((ge_first_rn_unit_valididentity) * (ge_second_rp_unit_valididentity))))) + (((((ge_first_ip_unit_valididentity) * (ge_second_ip_unit_valididentity))) + (((ge_first_in_unit_valididentity) * (ge_second_in_unit_valididentity))))))) + ge_balance_positive_unit_valididentityoutputreal))) /\ (exists ge_balance_positive_unit_valididentityoutputimaginary ge_balance_negative_unit_valididentityoutputimaginary. (((((ge_representation_imaginary_code_unit_valididentityoutput) = 2 * (ge_balance_positive_unit_valididentityoutputimaginary) /\ (ge_balance_negative_unit_valididentityoutputimaginary) = 0) \/ exists ge_signed_half_unit_valididentityoutputimaginarydecode. (((ge_representation_imaginary_code_unit_valididentityoutput) = 2 * ge_signed_half_unit_valididentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unit_valididentityoutputimaginary) = 0) /\ (ge_balance_negative_unit_valididentityoutputimaginary) = S ge_signed_half_unit_valididentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unit_valididentity) * (ge_second_ip_unit_valididentity))) + (((ge_first_rn_unit_valididentity) * (ge_second_in_unit_valididentity))))) + (((((ge_first_ip_unit_valididentity) * (ge_second_rp_unit_valididentity))) + (((ge_first_in_unit_valididentity) * (ge_second_rn_unit_valididentity))))))) + ge_balance_negative_unit_valididentityoutputimaginary = (((((((ge_first_rp_unit_valididentity) * (ge_second_in_unit_valididentity))) + (((ge_first_rn_unit_valididentity) * (ge_second_ip_unit_valididentity))))) + (((((ge_first_ip_unit_valididentity) * (ge_second_rn_unit_valididentity))) + (((ge_first_in_unit_valididentity) * (ge_second_rp_unit_valididentity))))))) + ge_balance_positive_unit_valididentityoutputimaginary)))))))))) -> (exists ge_real_positive_unit_valid_carrier ge_real_negative_unit_valid_carrier ge_imaginary_positive_unit_valid_carrier ge_imaginary_negative_unit_valid_carrier. (exists ge_real_code_unit_valid_carrierdecode ge_imaginary_code_unit_valid_carrierdecode. (((z) = ((ge_real_code_unit_valid_carrierdecode) + (ge_imaginary_code_unit_valid_carrierdecode)) * S ((ge_real_code_unit_valid_carrierdecode) + (ge_imaginary_code_unit_valid_carrierdecode)) + ((ge_imaginary_code_unit_valid_carrierdecode) + (ge_imaginary_code_unit_valid_carrierdecode))) /\ (((((ge_real_code_unit_valid_carrierdecode) = 2 * (ge_real_positive_unit_valid_carrier) /\ (ge_real_negative_unit_valid_carrier) = 0) \/ exists ge_signed_half_ge_unit_valid_carrierdecode_real. (((ge_real_code_unit_valid_carrierdecode) = 2 * ge_signed_half_ge_unit_valid_carrierdecode_real + 1 /\ (ge_real_positive_unit_valid_carrier) = 0) /\ (ge_real_negative_unit_valid_carrier) = S ge_signed_half_ge_unit_valid_carrierdecode_real))) /\ ((((ge_imaginary_code_unit_valid_carrierdecode) = 2 * (ge_imaginary_positive_unit_valid_carrier) /\ (ge_imaginary_negative_unit_valid_carrier) = 0) \/ exists ge_signed_half_ge_unit_valid_carrierdecode_imaginary. (((ge_imaginary_code_unit_valid_carrierdecode) = 2 * ge_signed_half_ge_unit_valid_carrierdecode_imaginary + 1 /\ (ge_imaginary_positive_unit_valid_carrier) = 0) /\ (ge_imaginary_negative_unit_valid_carrier) = S ge_signed_half_ge_unit_valid_carrierdecode_imaginary)))))))

Complete tactic proof in conservative notation

All 8 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

8 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 input left valid
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro hu
02Separate the logical casesL3–3

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

  1. L3
    cases hu
03Use earlier factsL4–8

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

  1. L4
    specialize gaussian_multiply_input_left_valid (z)
  2. L5
    specialize gaussian_multiply_input_left_valid (x)
  3. L6
    specialize gaussian_multiply_input_left_valid (6)
  4. L7
    apply gaussian_multiply_input_left_valid
  5. L8
    exact hu_witness

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro z
  2. 0002intro hu
  3. 0003cases hu
  4. 0004specialize gaussian_multiply_input_left_valid (z)
  5. 0005specialize gaussian_multiply_input_left_valid (x)
  6. 0006specialize gaussian_multiply_input_left_valid (6)
  7. 0007apply gaussian_multiply_input_left_valid
  8. 0008exact hu_witness