GF001E

gaussian_unit_valid

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

GF0007 gaussian_multiply_input_left_valid

Direct dependents

GF0056 gaussian_associate_transitive GF0067 gaussian_bezout_unit_divisor_cancel GF0092 gaussian_unit_empty_factorization GF00A5 gaussian_factor_associate_cancel_products

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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