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 a. (exists ge_real_positive_multiply_zero_left_domain ge_real_negative_multiply_zero_left_domain ge_imaginary_positive_multiply_zero_left_domain ge_imaginary_negative_multiply_zero_left_domain. (exists ge_real_code_multiply_zero_left_domaindecode ge_imaginary_code_multiply_zero_left_domaindecode. (((a) = ((ge_real_code_multiply_zero_left_domaindecode) + (ge_imaginary_code_multiply_zero_left_domaindecode)) * S ((ge_real_code_multiply_zero_left_domaindecode) + (ge_imaginary_code_multiply_zero_left_domaindecode)) + ((ge_imaginary_code_multiply_zero_left_domaindecode) + (ge_imaginary_code_multiply_zero_left_domaindecode))) /\ (((((ge_real_code_multiply_zero_left_domaindecode) = 2 * (ge_real_positive_multiply_zero_left_domain) /\ (ge_real_negative_multiply_zero_left_domain) = 0) \/ exists ge_signed_half_ge_multiply_zero_left_domaindecode_real. (((ge_real_code_multiply_zero_left_domaindecode) = 2 * ge_signed_half_ge_multiply_zero_left_domaindecode_real + 1 /\ (ge_real_positive_multiply_zero_left_domain) = 0) /\ (ge_real_negative_multiply_zero_left_domain) = S ge_signed_half_ge_multiply_zero_left_domaindecode_real))) /\ ((((ge_imaginary_code_multiply_zero_left_domaindecode) = 2 * (ge_imaginary_positive_multiply_zero_left_domain) /\ (ge_imaginary_negative_multiply_zero_left_domain) = 0) \/ exists ge_signed_half_ge_multiply_zero_left_domaindecode_imaginary. (((ge_imaginary_code_multiply_zero_left_domaindecode) = 2 * ge_signed_half_ge_multiply_zero_left_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_multiply_zero_left_domain) = 0) /\ (ge_imaginary_negative_multiply_zero_left_domain) = S ge_signed_half_ge_multiply_zero_left_domaindecode_imaginary))))))) -> (exists ge_first_rp_multiply_zero_left ge_first_rn_multiply_zero_left ge_first_ip_multiply_zero_left ge_first_in_multiply_zero_left ge_second_rp_multiply_zero_left ge_second_rn_multiply_zero_left ge_second_ip_multiply_zero_left ge_second_in_multiply_zero_left. ((exists ge_representation_real_code_multiply_zero_leftfirst ge_representation_imaginary_code_multiply_zero_leftfirst. (((0) = ((ge_representation_real_code_multiply_zero_leftfirst) + (ge_representation_imaginary_code_multiply_zero_leftfirst)) * S ((ge_representation_real_code_multiply_zero_leftfirst) + (ge_representation_imaginary_code_multiply_zero_leftfirst)) + ((ge_representation_imaginary_code_multiply_zero_leftfirst) + (ge_representation_imaginary_code_multiply_zero_leftfirst))) /\ ((exists ge_balance_positive_multiply_zero_leftfirstreal ge_balance_negative_multiply_zero_leftfirstreal. (((((ge_representation_real_code_multiply_zero_leftfirst) = 2 * (ge_balance_positive_multiply_zero_leftfirstreal) /\ (ge_balance_negative_multiply_zero_leftfirstreal) = 0) \/ exists ge_signed_half_multiply_zero_leftfirstrealdecode. (((ge_representation_real_code_multiply_zero_leftfirst) = 2 * ge_signed_half_multiply_zero_leftfirstrealdecode + 1 /\ (ge_balance_positive_multiply_zero_leftfirstreal) = 0) /\ (ge_balance_negative_multiply_zero_leftfirstreal) = S ge_signed_half_multiply_zero_leftfirstrealdecode))) /\ ((ge_first_rp_multiply_zero_left) + ge_balance_negative_multiply_zero_leftfirstreal = (ge_first_rn_multiply_zero_left) + ge_balance_positive_multiply_zero_leftfirstreal))) /\ (exists ge_balance_positive_multiply_zero_leftfirstimaginary ge_balance_negative_multiply_zero_leftfirstimaginary. (((((ge_representation_imaginary_code_multiply_zero_leftfirst) = 2 * (ge_balance_positive_multiply_zero_leftfirstimaginary) /\ (ge_balance_negative_multiply_zero_leftfirstimaginary) = 0) \/ exists ge_signed_half_multiply_zero_leftfirstimaginarydecode. (((ge_representation_imaginary_code_multiply_zero_leftfirst) = 2 * ge_signed_half_multiply_zero_leftfirstimaginarydecode + 1 /\ (ge_balance_positive_multiply_zero_leftfirstimaginary) = 0) /\ (ge_balance_negative_multiply_zero_leftfirstimaginary) = S ge_signed_half_multiply_zero_leftfirstimaginarydecode))) /\ ((ge_first_ip_multiply_zero_left) + ge_balance_negative_multiply_zero_leftfirstimaginary = (ge_first_in_multiply_zero_left) + ge_balance_positive_multiply_zero_leftfirstimaginary)))))) /\ ((exists ge_representation_real_code_multiply_zero_leftsecond ge_representation_imaginary_code_multiply_zero_leftsecond. (((a) = ((ge_representation_real_code_multiply_zero_leftsecond) + (ge_representation_imaginary_code_multiply_zero_leftsecond)) * S ((ge_representation_real_code_multiply_zero_leftsecond) + (ge_representation_imaginary_code_multiply_zero_leftsecond)) + ((ge_representation_imaginary_code_multiply_zero_leftsecond) + (ge_representation_imaginary_code_multiply_zero_leftsecond))) /\ ((exists ge_balance_positive_multiply_zero_leftsecondreal ge_balance_negative_multiply_zero_leftsecondreal. (((((ge_representation_real_code_multiply_zero_leftsecond) = 2 * (ge_balance_positive_multiply_zero_leftsecondreal) /\ (ge_balance_negative_multiply_zero_leftsecondreal) = 0) \/ exists ge_signed_half_multiply_zero_leftsecondrealdecode. (((ge_representation_real_code_multiply_zero_leftsecond) = 2 * ge_signed_half_multiply_zero_leftsecondrealdecode + 1 /\ (ge_balance_positive_multiply_zero_leftsecondreal) = 0) /\ (ge_balance_negative_multiply_zero_leftsecondreal) = S ge_signed_half_multiply_zero_leftsecondrealdecode))) /\ ((ge_second_rp_multiply_zero_left) + ge_balance_negative_multiply_zero_leftsecondreal = (ge_second_rn_multiply_zero_left) + ge_balance_positive_multiply_zero_leftsecondreal))) /\ (exists ge_balance_positive_multiply_zero_leftsecondimaginary ge_balance_negative_multiply_zero_leftsecondimaginary. (((((ge_representation_imaginary_code_multiply_zero_leftsecond) = 2 * (ge_balance_positive_multiply_zero_leftsecondimaginary) /\ (ge_balance_negative_multiply_zero_leftsecondimaginary) = 0) \/ exists ge_signed_half_multiply_zero_leftsecondimaginarydecode. (((ge_representation_imaginary_code_multiply_zero_leftsecond) = 2 * ge_signed_half_multiply_zero_leftsecondimaginarydecode + 1 /\ (ge_balance_positive_multiply_zero_leftsecondimaginary) = 0) /\ (ge_balance_negative_multiply_zero_leftsecondimaginary) = S ge_signed_half_multiply_zero_leftsecondimaginarydecode))) /\ ((ge_second_ip_multiply_zero_left) + ge_balance_negative_multiply_zero_leftsecondimaginary = (ge_second_in_multiply_zero_left) + ge_balance_positive_multiply_zero_leftsecondimaginary)))))) /\ (exists ge_representation_real_code_multiply_zero_leftoutput ge_representation_imaginary_code_multiply_zero_leftoutput. (((0) = ((ge_representation_real_code_multiply_zero_leftoutput) + (ge_representation_imaginary_code_multiply_zero_leftoutput)) * S ((ge_representation_real_code_multiply_zero_leftoutput) + (ge_representation_imaginary_code_multiply_zero_leftoutput)) + ((ge_representation_imaginary_code_multiply_zero_leftoutput) + (ge_representation_imaginary_code_multiply_zero_leftoutput))) /\ ((exists ge_balance_positive_multiply_zero_leftoutputreal ge_balance_negative_multiply_zero_leftoutputreal. (((((ge_representation_real_code_multiply_zero_leftoutput) = 2 * (ge_balance_positive_multiply_zero_leftoutputreal) /\ (ge_balance_negative_multiply_zero_leftoutputreal) = 0) \/ exists ge_signed_half_multiply_zero_leftoutputrealdecode. (((ge_representation_real_code_multiply_zero_leftoutput) = 2 * ge_signed_half_multiply_zero_leftoutputrealdecode + 1 /\ (ge_balance_positive_multiply_zero_leftoutputreal) = 0) /\ (ge_balance_negative_multiply_zero_leftoutputreal) = S ge_signed_half_multiply_zero_leftoutputrealdecode))) /\ ((((((((ge_first_rp_multiply_zero_left) * (ge_second_rp_multiply_zero_left))) + (((ge_first_rn_multiply_zero_left) * (ge_second_rn_multiply_zero_left))))) + (((((ge_first_ip_multiply_zero_left) * (ge_second_in_multiply_zero_left))) + (((ge_first_in_multiply_zero_left) * (ge_second_ip_multiply_zero_left))))))) + ge_balance_negative_multiply_zero_leftoutputreal = (((((((ge_first_rp_multiply_zero_left) * (ge_second_rn_multiply_zero_left))) + (((ge_first_rn_multiply_zero_left) * (ge_second_rp_multiply_zero_left))))) + (((((ge_first_ip_multiply_zero_left) * (ge_second_ip_multiply_zero_left))) + (((ge_first_in_multiply_zero_left) * (ge_second_in_multiply_zero_left))))))) + ge_balance_positive_multiply_zero_leftoutputreal))) /\ (exists ge_balance_positive_multiply_zero_leftoutputimaginary ge_balance_negative_multiply_zero_leftoutputimaginary. (((((ge_representation_imaginary_code_multiply_zero_leftoutput) = 2 * (ge_balance_positive_multiply_zero_leftoutputimaginary) /\ (ge_balance_negative_multiply_zero_leftoutputimaginary) = 0) \/ exists ge_signed_half_multiply_zero_leftoutputimaginarydecode. (((ge_representation_imaginary_code_multiply_zero_leftoutput) = 2 * ge_signed_half_multiply_zero_leftoutputimaginarydecode + 1 /\ (ge_balance_positive_multiply_zero_leftoutputimaginary) = 0) /\ (ge_balance_negative_multiply_zero_leftoutputimaginary) = S ge_signed_half_multiply_zero_leftoutputimaginarydecode))) /\ ((((((((ge_first_rp_multiply_zero_left) * (ge_second_ip_multiply_zero_left))) + (((ge_first_rn_multiply_zero_left) * (ge_second_in_multiply_zero_left))))) + (((((ge_first_ip_multiply_zero_left) * (ge_second_rp_multiply_zero_left))) + (((ge_first_in_multiply_zero_left) * (ge_second_rn_multiply_zero_left))))))) + ge_balance_negative_multiply_zero_leftoutputimaginary = (((((((ge_first_rp_multiply_zero_left) * (ge_second_in_multiply_zero_left))) + (((ge_first_rn_multiply_zero_left) * (ge_second_ip_multiply_zero_left))))) + (((((ge_first_ip_multiply_zero_left) * (ge_second_rn_multiply_zero_left))) + (((ge_first_in_multiply_zero_left) * (ge_second_rp_multiply_zero_left))))))) + ge_balance_positive_multiply_zero_leftoutputimaginary)))))))))Constructive proof overview
Generated structural guide
Commutativity supplies the actual left multiply zero identity.
The unchanged tactic script uses 2 declared prerequisites and contains 9 exact native proof lines.
Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
Direct dependents
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
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)
01Fix variables and assumptionsL1–2
02Use earlier factsL3–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
Original exact command ledger · 9 lines
- 0001
intro a - 0002
intro hv - 0003
specialize gaussian_multiply_commutative (a) - 0004
specialize gaussian_multiply_commutative (0) - 0005
specialize gaussian_multiply_commutative (0) - 0006
apply gaussian_multiply_commutative - 0007
specialize gaussian_multiply_zero_right (a) - 0008
apply gaussian_multiply_zero_right - 0009
exact hv