GF0029

gaussian_multiply_one_left

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

Commutativity supplies the actual left multiply one identity.

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_one_left_domain ge_real_negative_multiply_one_left_domain ge_imaginary_positive_multiply_one_left_domain ge_imaginary_negative_multiply_one_left_domain. (exists ge_real_code_multiply_one_left_domaindecode ge_imaginary_code_multiply_one_left_domaindecode. (((a) = ((ge_real_code_multiply_one_left_domaindecode) + (ge_imaginary_code_multiply_one_left_domaindecode)) * S ((ge_real_code_multiply_one_left_domaindecode) + (ge_imaginary_code_multiply_one_left_domaindecode)) + ((ge_imaginary_code_multiply_one_left_domaindecode) + (ge_imaginary_code_multiply_one_left_domaindecode))) /\ (((((ge_real_code_multiply_one_left_domaindecode) = 2 * (ge_real_positive_multiply_one_left_domain) /\ (ge_real_negative_multiply_one_left_domain) = 0) \/ exists ge_signed_half_ge_multiply_one_left_domaindecode_real. (((ge_real_code_multiply_one_left_domaindecode) = 2 * ge_signed_half_ge_multiply_one_left_domaindecode_real + 1 /\ (ge_real_positive_multiply_one_left_domain) = 0) /\ (ge_real_negative_multiply_one_left_domain) = S ge_signed_half_ge_multiply_one_left_domaindecode_real))) /\ ((((ge_imaginary_code_multiply_one_left_domaindecode) = 2 * (ge_imaginary_positive_multiply_one_left_domain) /\ (ge_imaginary_negative_multiply_one_left_domain) = 0) \/ exists ge_signed_half_ge_multiply_one_left_domaindecode_imaginary. (((ge_imaginary_code_multiply_one_left_domaindecode) = 2 * ge_signed_half_ge_multiply_one_left_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_multiply_one_left_domain) = 0) /\ (ge_imaginary_negative_multiply_one_left_domain) = S ge_signed_half_ge_multiply_one_left_domaindecode_imaginary))))))) -> (exists ge_first_rp_multiply_one_left ge_first_rn_multiply_one_left ge_first_ip_multiply_one_left ge_first_in_multiply_one_left ge_second_rp_multiply_one_left ge_second_rn_multiply_one_left ge_second_ip_multiply_one_left ge_second_in_multiply_one_left. ((exists ge_representation_real_code_multiply_one_leftfirst ge_representation_imaginary_code_multiply_one_leftfirst. (((6) = ((ge_representation_real_code_multiply_one_leftfirst) + (ge_representation_imaginary_code_multiply_one_leftfirst)) * S ((ge_representation_real_code_multiply_one_leftfirst) + (ge_representation_imaginary_code_multiply_one_leftfirst)) + ((ge_representation_imaginary_code_multiply_one_leftfirst) + (ge_representation_imaginary_code_multiply_one_leftfirst))) /\ ((exists ge_balance_positive_multiply_one_leftfirstreal ge_balance_negative_multiply_one_leftfirstreal. (((((ge_representation_real_code_multiply_one_leftfirst) = 2 * (ge_balance_positive_multiply_one_leftfirstreal) /\ (ge_balance_negative_multiply_one_leftfirstreal) = 0) \/ exists ge_signed_half_multiply_one_leftfirstrealdecode. (((ge_representation_real_code_multiply_one_leftfirst) = 2 * ge_signed_half_multiply_one_leftfirstrealdecode + 1 /\ (ge_balance_positive_multiply_one_leftfirstreal) = 0) /\ (ge_balance_negative_multiply_one_leftfirstreal) = S ge_signed_half_multiply_one_leftfirstrealdecode))) /\ ((ge_first_rp_multiply_one_left) + ge_balance_negative_multiply_one_leftfirstreal = (ge_first_rn_multiply_one_left) + ge_balance_positive_multiply_one_leftfirstreal))) /\ (exists ge_balance_positive_multiply_one_leftfirstimaginary ge_balance_negative_multiply_one_leftfirstimaginary. (((((ge_representation_imaginary_code_multiply_one_leftfirst) = 2 * (ge_balance_positive_multiply_one_leftfirstimaginary) /\ (ge_balance_negative_multiply_one_leftfirstimaginary) = 0) \/ exists ge_signed_half_multiply_one_leftfirstimaginarydecode. (((ge_representation_imaginary_code_multiply_one_leftfirst) = 2 * ge_signed_half_multiply_one_leftfirstimaginarydecode + 1 /\ (ge_balance_positive_multiply_one_leftfirstimaginary) = 0) /\ (ge_balance_negative_multiply_one_leftfirstimaginary) = S ge_signed_half_multiply_one_leftfirstimaginarydecode))) /\ ((ge_first_ip_multiply_one_left) + ge_balance_negative_multiply_one_leftfirstimaginary = (ge_first_in_multiply_one_left) + ge_balance_positive_multiply_one_leftfirstimaginary)))))) /\ ((exists ge_representation_real_code_multiply_one_leftsecond ge_representation_imaginary_code_multiply_one_leftsecond. (((a) = ((ge_representation_real_code_multiply_one_leftsecond) + (ge_representation_imaginary_code_multiply_one_leftsecond)) * S ((ge_representation_real_code_multiply_one_leftsecond) + (ge_representation_imaginary_code_multiply_one_leftsecond)) + ((ge_representation_imaginary_code_multiply_one_leftsecond) + (ge_representation_imaginary_code_multiply_one_leftsecond))) /\ ((exists ge_balance_positive_multiply_one_leftsecondreal ge_balance_negative_multiply_one_leftsecondreal. (((((ge_representation_real_code_multiply_one_leftsecond) = 2 * (ge_balance_positive_multiply_one_leftsecondreal) /\ (ge_balance_negative_multiply_one_leftsecondreal) = 0) \/ exists ge_signed_half_multiply_one_leftsecondrealdecode. (((ge_representation_real_code_multiply_one_leftsecond) = 2 * ge_signed_half_multiply_one_leftsecondrealdecode + 1 /\ (ge_balance_positive_multiply_one_leftsecondreal) = 0) /\ (ge_balance_negative_multiply_one_leftsecondreal) = S ge_signed_half_multiply_one_leftsecondrealdecode))) /\ ((ge_second_rp_multiply_one_left) + ge_balance_negative_multiply_one_leftsecondreal = (ge_second_rn_multiply_one_left) + ge_balance_positive_multiply_one_leftsecondreal))) /\ (exists ge_balance_positive_multiply_one_leftsecondimaginary ge_balance_negative_multiply_one_leftsecondimaginary. (((((ge_representation_imaginary_code_multiply_one_leftsecond) = 2 * (ge_balance_positive_multiply_one_leftsecondimaginary) /\ (ge_balance_negative_multiply_one_leftsecondimaginary) = 0) \/ exists ge_signed_half_multiply_one_leftsecondimaginarydecode. (((ge_representation_imaginary_code_multiply_one_leftsecond) = 2 * ge_signed_half_multiply_one_leftsecondimaginarydecode + 1 /\ (ge_balance_positive_multiply_one_leftsecondimaginary) = 0) /\ (ge_balance_negative_multiply_one_leftsecondimaginary) = S ge_signed_half_multiply_one_leftsecondimaginarydecode))) /\ ((ge_second_ip_multiply_one_left) + ge_balance_negative_multiply_one_leftsecondimaginary = (ge_second_in_multiply_one_left) + ge_balance_positive_multiply_one_leftsecondimaginary)))))) /\ (exists ge_representation_real_code_multiply_one_leftoutput ge_representation_imaginary_code_multiply_one_leftoutput. (((a) = ((ge_representation_real_code_multiply_one_leftoutput) + (ge_representation_imaginary_code_multiply_one_leftoutput)) * S ((ge_representation_real_code_multiply_one_leftoutput) + (ge_representation_imaginary_code_multiply_one_leftoutput)) + ((ge_representation_imaginary_code_multiply_one_leftoutput) + (ge_representation_imaginary_code_multiply_one_leftoutput))) /\ ((exists ge_balance_positive_multiply_one_leftoutputreal ge_balance_negative_multiply_one_leftoutputreal. (((((ge_representation_real_code_multiply_one_leftoutput) = 2 * (ge_balance_positive_multiply_one_leftoutputreal) /\ (ge_balance_negative_multiply_one_leftoutputreal) = 0) \/ exists ge_signed_half_multiply_one_leftoutputrealdecode. (((ge_representation_real_code_multiply_one_leftoutput) = 2 * ge_signed_half_multiply_one_leftoutputrealdecode + 1 /\ (ge_balance_positive_multiply_one_leftoutputreal) = 0) /\ (ge_balance_negative_multiply_one_leftoutputreal) = S ge_signed_half_multiply_one_leftoutputrealdecode))) /\ ((((((((ge_first_rp_multiply_one_left) * (ge_second_rp_multiply_one_left))) + (((ge_first_rn_multiply_one_left) * (ge_second_rn_multiply_one_left))))) + (((((ge_first_ip_multiply_one_left) * (ge_second_in_multiply_one_left))) + (((ge_first_in_multiply_one_left) * (ge_second_ip_multiply_one_left))))))) + ge_balance_negative_multiply_one_leftoutputreal = (((((((ge_first_rp_multiply_one_left) * (ge_second_rn_multiply_one_left))) + (((ge_first_rn_multiply_one_left) * (ge_second_rp_multiply_one_left))))) + (((((ge_first_ip_multiply_one_left) * (ge_second_ip_multiply_one_left))) + (((ge_first_in_multiply_one_left) * (ge_second_in_multiply_one_left))))))) + ge_balance_positive_multiply_one_leftoutputreal))) /\ (exists ge_balance_positive_multiply_one_leftoutputimaginary ge_balance_negative_multiply_one_leftoutputimaginary. (((((ge_representation_imaginary_code_multiply_one_leftoutput) = 2 * (ge_balance_positive_multiply_one_leftoutputimaginary) /\ (ge_balance_negative_multiply_one_leftoutputimaginary) = 0) \/ exists ge_signed_half_multiply_one_leftoutputimaginarydecode. (((ge_representation_imaginary_code_multiply_one_leftoutput) = 2 * ge_signed_half_multiply_one_leftoutputimaginarydecode + 1 /\ (ge_balance_positive_multiply_one_leftoutputimaginary) = 0) /\ (ge_balance_negative_multiply_one_leftoutputimaginary) = S ge_signed_half_multiply_one_leftoutputimaginarydecode))) /\ ((((((((ge_first_rp_multiply_one_left) * (ge_second_ip_multiply_one_left))) + (((ge_first_rn_multiply_one_left) * (ge_second_in_multiply_one_left))))) + (((((ge_first_ip_multiply_one_left) * (ge_second_rp_multiply_one_left))) + (((ge_first_in_multiply_one_left) * (ge_second_rn_multiply_one_left))))))) + ge_balance_negative_multiply_one_leftoutputimaginary = (((((((ge_first_rp_multiply_one_left) * (ge_second_in_multiply_one_left))) + (((ge_first_rn_multiply_one_left) * (ge_second_ip_multiply_one_left))))) + (((((ge_first_ip_multiply_one_left) * (ge_second_rn_multiply_one_left))) + (((ge_first_in_multiply_one_left) * (ge_second_rp_multiply_one_left))))))) + ge_balance_positive_multiply_one_leftoutputimaginary)))))))))

Constructive proof overview

Generated structural guide

Commutativity supplies the actual left multiply one 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

GF0028 gaussian_multiply_one_right GF0014 gaussian_multiply_commutative

Direct dependents

GF0046 gaussian_one_divides GF0054 gaussian_associate_reflexive GF0055 gaussian_associate_symmetric GF005A gaussian_mutual_divisibility_associate GF0067 gaussian_bezout_unit_divisor_cancel

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

9 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 rightgaussian multiply commutative
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro hv
02Use earlier factsL3–9

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

  1. L3
    specialize gaussian_multiply_commutative (a)
  2. L4
    specialize gaussian_multiply_commutative (6)
  3. L5
    specialize gaussian_multiply_commutative (a)
  4. L6
    apply gaussian_multiply_commutative
  5. L7
    specialize gaussian_multiply_one_right (a)
  6. L8
    apply gaussian_multiply_one_right
  7. L9
    exact hv

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002intro hv
  3. 0003specialize gaussian_multiply_commutative (a)
  4. 0004specialize gaussian_multiply_commutative (6)
  5. 0005specialize gaussian_multiply_commutative (a)
  6. 0006apply gaussian_multiply_commutative
  7. 0007specialize gaussian_multiply_one_right (a)
  8. 0008apply gaussian_multiply_one_right
  9. 0009exact hv