GF003D — gaussian_one_unit

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

exists gr_inverse_one_unit. (exists ge_first_rp_one_unitidentity ge_first_rn_one_unitidentity ge_first_ip_one_unitidentity ge_first_in_one_unitidentity ge_second_rp_one_unitidentity ge_second_rn_one_unitidentity ge_second_ip_one_unitidentity ge_second_in_one_unitidentity. ((exists ge_representation_real_code_one_unitidentityfirst ge_representation_imaginary_code_one_unitidentityfirst. (((6) = ((ge_representation_real_code_one_unitidentityfirst) + (ge_representation_imaginary_code_one_unitidentityfirst)) * S ((ge_representation_real_code_one_unitidentityfirst) + (ge_representation_imaginary_code_one_unitidentityfirst)) + ((ge_representation_imaginary_code_one_unitidentityfirst) + (ge_representation_imaginary_code_one_unitidentityfirst))) /\ ((exists ge_balance_positive_one_unitidentityfirstreal ge_balance_negative_one_unitidentityfirstreal. (((((ge_representation_real_code_one_unitidentityfirst) = 2 * (ge_balance_positive_one_unitidentityfirstreal) /\ (ge_balance_negative_one_unitidentityfirstreal) = 0) \/ exists ge_signed_half_one_unitidentityfirstrealdecode. (((ge_representation_real_code_one_unitidentityfirst) = 2 * ge_signed_half_one_unitidentityfirstrealdecode + 1 /\ (ge_balance_positive_one_unitidentityfirstreal) = 0) /\ (ge_balance_negative_one_unitidentityfirstreal) = S ge_signed_half_one_unitidentityfirstrealdecode))) /\ ((ge_first_rp_one_unitidentity) + ge_balance_negative_one_unitidentityfirstreal = (ge_first_rn_one_unitidentity) + ge_balance_positive_one_unitidentityfirstreal))) /\ (exists ge_balance_positive_one_unitidentityfirstimaginary ge_balance_negative_one_unitidentityfirstimaginary. (((((ge_representation_imaginary_code_one_unitidentityfirst) = 2 * (ge_balance_positive_one_unitidentityfirstimaginary) /\ (ge_balance_negative_one_unitidentityfirstimaginary) = 0) \/ exists ge_signed_half_one_unitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_one_unitidentityfirst) = 2 * ge_signed_half_one_unitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_one_unitidentityfirstimaginary) = 0) /\ (ge_balance_negative_one_unitidentityfirstimaginary) = S ge_signed_half_one_unitidentityfirstimaginarydecode))) /\ ((ge_first_ip_one_unitidentity) + ge_balance_negative_one_unitidentityfirstimaginary = (ge_first_in_one_unitidentity) + ge_balance_positive_one_unitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_one_unitidentitysecond ge_representation_imaginary_code_one_unitidentitysecond. (((gr_inverse_one_unit) = ((ge_representation_real_code_one_unitidentitysecond) + (ge_representation_imaginary_code_one_unitidentitysecond)) * S ((ge_representation_real_code_one_unitidentitysecond) + (ge_representation_imaginary_code_one_unitidentitysecond)) + ((ge_representation_imaginary_code_one_unitidentitysecond) + (ge_representation_imaginary_code_one_unitidentitysecond))) /\ ((exists ge_balance_positive_one_unitidentitysecondreal ge_balance_negative_one_unitidentitysecondreal. (((((ge_representation_real_code_one_unitidentitysecond) = 2 * (ge_balance_positive_one_unitidentitysecondreal) /\ (ge_balance_negative_one_unitidentitysecondreal) = 0) \/ exists ge_signed_half_one_unitidentitysecondrealdecode. (((ge_representation_real_code_one_unitidentitysecond) = 2 * ge_signed_half_one_unitidentitysecondrealdecode + 1 /\ (ge_balance_positive_one_unitidentitysecondreal) = 0) /\ (ge_balance_negative_one_unitidentitysecondreal) = S ge_signed_half_one_unitidentitysecondrealdecode))) /\ ((ge_second_rp_one_unitidentity) + ge_balance_negative_one_unitidentitysecondreal = (ge_second_rn_one_unitidentity) + ge_balance_positive_one_unitidentitysecondreal))) /\ (exists ge_balance_positive_one_unitidentitysecondimaginary ge_balance_negative_one_unitidentitysecondimaginary. (((((ge_representation_imaginary_code_one_unitidentitysecond) = 2 * (ge_balance_positive_one_unitidentitysecondimaginary) /\ (ge_balance_negative_one_unitidentitysecondimaginary) = 0) \/ exists ge_signed_half_one_unitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_one_unitidentitysecond) = 2 * ge_signed_half_one_unitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_one_unitidentitysecondimaginary) = 0) /\ (ge_balance_negative_one_unitidentitysecondimaginary) = S ge_signed_half_one_unitidentitysecondimaginarydecode))) /\ ((ge_second_ip_one_unitidentity) + ge_balance_negative_one_unitidentitysecondimaginary = (ge_second_in_one_unitidentity) + ge_balance_positive_one_unitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_one_unitidentityoutput ge_representation_imaginary_code_one_unitidentityoutput. (((6) = ((ge_representation_real_code_one_unitidentityoutput) + (ge_representation_imaginary_code_one_unitidentityoutput)) * S ((ge_representation_real_code_one_unitidentityoutput) + (ge_representation_imaginary_code_one_unitidentityoutput)) + ((ge_representation_imaginary_code_one_unitidentityoutput) + (ge_representation_imaginary_code_one_unitidentityoutput))) /\ ((exists ge_balance_positive_one_unitidentityoutputreal ge_balance_negative_one_unitidentityoutputreal. (((((ge_representation_real_code_one_unitidentityoutput) = 2 * (ge_balance_positive_one_unitidentityoutputreal) /\ (ge_balance_negative_one_unitidentityoutputreal) = 0) \/ exists ge_signed_half_one_unitidentityoutputrealdecode. (((ge_representation_real_code_one_unitidentityoutput) = 2 * ge_signed_half_one_unitidentityoutputrealdecode + 1 /\ (ge_balance_positive_one_unitidentityoutputreal) = 0) /\ (ge_balance_negative_one_unitidentityoutputreal) = S ge_signed_half_one_unitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_one_unitidentity) * (ge_second_rp_one_unitidentity))) + (((ge_first_rn_one_unitidentity) * (ge_second_rn_one_unitidentity))))) + (((((ge_first_ip_one_unitidentity) * (ge_second_in_one_unitidentity))) + (((ge_first_in_one_unitidentity) * (ge_second_ip_one_unitidentity))))))) + ge_balance_negative_one_unitidentityoutputreal = (((((((ge_first_rp_one_unitidentity) * (ge_second_rn_one_unitidentity))) + (((ge_first_rn_one_unitidentity) * (ge_second_rp_one_unitidentity))))) + (((((ge_first_ip_one_unitidentity) * (ge_second_ip_one_unitidentity))) + (((ge_first_in_one_unitidentity) * (ge_second_in_one_unitidentity))))))) + ge_balance_positive_one_unitidentityoutputreal))) /\ (exists ge_balance_positive_one_unitidentityoutputimaginary ge_balance_negative_one_unitidentityoutputimaginary. (((((ge_representation_imaginary_code_one_unitidentityoutput) = 2 * (ge_balance_positive_one_unitidentityoutputimaginary) /\ (ge_balance_negative_one_unitidentityoutputimaginary) = 0) \/ exists ge_signed_half_one_unitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_one_unitidentityoutput) = 2 * ge_signed_half_one_unitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_one_unitidentityoutputimaginary) = 0) /\ (ge_balance_negative_one_unitidentityoutputimaginary) = S ge_signed_half_one_unitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_one_unitidentity) * (ge_second_ip_one_unitidentity))) + (((ge_first_rn_one_unitidentity) * (ge_second_in_one_unitidentity))))) + (((((ge_first_ip_one_unitidentity) * (ge_second_rp_one_unitidentity))) + (((ge_first_in_one_unitidentity) * (ge_second_rn_one_unitidentity))))))) + ge_balance_negative_one_unitidentityoutputimaginary = (((((((ge_first_rp_one_unitidentity) * (ge_second_in_one_unitidentity))) + (((ge_first_rn_one_unitidentity) * (ge_second_ip_one_unitidentity))))) + (((((ge_first_ip_one_unitidentity) * (ge_second_rn_one_unitidentity))) + (((ge_first_in_one_unitidentity) * (ge_second_rp_one_unitidentity))))))) + ge_balance_positive_one_unitidentityoutputimaginary)))))))))

Constructive proof overview

Generated structural guide

The actual canonical Gaussian identity code six is a unit with itself as inverse.

The unchanged tactic script uses 2 declared prerequisites and contains 4 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GF0028 gaussian_multiply_one_right GF000F gaussian_one_valid

Direct dependents

GF0054 gaussian_associate_reflexive GF005A gaussian_mutual_divisibility_associate GF009F gaussian_irreducible_divisor_product_member GF00AF gaussian_irreducible_products_associate_unique

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

4 script commands · 2 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 (2)

gaussian multiply one right gaussian one valid

01Construct an explicit witnessL1–1

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

L1
exists (6)

02Use earlier factsL2–4

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

L2
specialize gaussian_multiply_one_right (6)
L3
apply gaussian_multiply_one_right
L4
exact gaussian_one_valid

Library-wide reading audit

Original exact command ledger · 4 lines

0001exists (6)
0002specialize gaussian_multiply_one_right (6)
0003apply gaussian_multiply_one_right
0004exact gaussian_one_valid