GF0027

gaussian_add_zero_left

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

Commutativity supplies the actual left add zero 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_add_zero_left_domain ge_real_negative_add_zero_left_domain ge_imaginary_positive_add_zero_left_domain ge_imaginary_negative_add_zero_left_domain. (exists ge_real_code_add_zero_left_domaindecode ge_imaginary_code_add_zero_left_domaindecode. (((a) = ((ge_real_code_add_zero_left_domaindecode) + (ge_imaginary_code_add_zero_left_domaindecode)) * S ((ge_real_code_add_zero_left_domaindecode) + (ge_imaginary_code_add_zero_left_domaindecode)) + ((ge_imaginary_code_add_zero_left_domaindecode) + (ge_imaginary_code_add_zero_left_domaindecode))) /\ (((((ge_real_code_add_zero_left_domaindecode) = 2 * (ge_real_positive_add_zero_left_domain) /\ (ge_real_negative_add_zero_left_domain) = 0) \/ exists ge_signed_half_ge_add_zero_left_domaindecode_real. (((ge_real_code_add_zero_left_domaindecode) = 2 * ge_signed_half_ge_add_zero_left_domaindecode_real + 1 /\ (ge_real_positive_add_zero_left_domain) = 0) /\ (ge_real_negative_add_zero_left_domain) = S ge_signed_half_ge_add_zero_left_domaindecode_real))) /\ ((((ge_imaginary_code_add_zero_left_domaindecode) = 2 * (ge_imaginary_positive_add_zero_left_domain) /\ (ge_imaginary_negative_add_zero_left_domain) = 0) \/ exists ge_signed_half_ge_add_zero_left_domaindecode_imaginary. (((ge_imaginary_code_add_zero_left_domaindecode) = 2 * ge_signed_half_ge_add_zero_left_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_add_zero_left_domain) = 0) /\ (ge_imaginary_negative_add_zero_left_domain) = S ge_signed_half_ge_add_zero_left_domaindecode_imaginary))))))) -> (exists ge_first_rp_add_zero_left ge_first_rn_add_zero_left ge_first_ip_add_zero_left ge_first_in_add_zero_left ge_second_rp_add_zero_left ge_second_rn_add_zero_left ge_second_ip_add_zero_left ge_second_in_add_zero_left. ((exists ge_representation_real_code_add_zero_leftfirst ge_representation_imaginary_code_add_zero_leftfirst. (((0) = ((ge_representation_real_code_add_zero_leftfirst) + (ge_representation_imaginary_code_add_zero_leftfirst)) * S ((ge_representation_real_code_add_zero_leftfirst) + (ge_representation_imaginary_code_add_zero_leftfirst)) + ((ge_representation_imaginary_code_add_zero_leftfirst) + (ge_representation_imaginary_code_add_zero_leftfirst))) /\ ((exists ge_balance_positive_add_zero_leftfirstreal ge_balance_negative_add_zero_leftfirstreal. (((((ge_representation_real_code_add_zero_leftfirst) = 2 * (ge_balance_positive_add_zero_leftfirstreal) /\ (ge_balance_negative_add_zero_leftfirstreal) = 0) \/ exists ge_signed_half_add_zero_leftfirstrealdecode. (((ge_representation_real_code_add_zero_leftfirst) = 2 * ge_signed_half_add_zero_leftfirstrealdecode + 1 /\ (ge_balance_positive_add_zero_leftfirstreal) = 0) /\ (ge_balance_negative_add_zero_leftfirstreal) = S ge_signed_half_add_zero_leftfirstrealdecode))) /\ ((ge_first_rp_add_zero_left) + ge_balance_negative_add_zero_leftfirstreal = (ge_first_rn_add_zero_left) + ge_balance_positive_add_zero_leftfirstreal))) /\ (exists ge_balance_positive_add_zero_leftfirstimaginary ge_balance_negative_add_zero_leftfirstimaginary. (((((ge_representation_imaginary_code_add_zero_leftfirst) = 2 * (ge_balance_positive_add_zero_leftfirstimaginary) /\ (ge_balance_negative_add_zero_leftfirstimaginary) = 0) \/ exists ge_signed_half_add_zero_leftfirstimaginarydecode. (((ge_representation_imaginary_code_add_zero_leftfirst) = 2 * ge_signed_half_add_zero_leftfirstimaginarydecode + 1 /\ (ge_balance_positive_add_zero_leftfirstimaginary) = 0) /\ (ge_balance_negative_add_zero_leftfirstimaginary) = S ge_signed_half_add_zero_leftfirstimaginarydecode))) /\ ((ge_first_ip_add_zero_left) + ge_balance_negative_add_zero_leftfirstimaginary = (ge_first_in_add_zero_left) + ge_balance_positive_add_zero_leftfirstimaginary)))))) /\ ((exists ge_representation_real_code_add_zero_leftsecond ge_representation_imaginary_code_add_zero_leftsecond. (((a) = ((ge_representation_real_code_add_zero_leftsecond) + (ge_representation_imaginary_code_add_zero_leftsecond)) * S ((ge_representation_real_code_add_zero_leftsecond) + (ge_representation_imaginary_code_add_zero_leftsecond)) + ((ge_representation_imaginary_code_add_zero_leftsecond) + (ge_representation_imaginary_code_add_zero_leftsecond))) /\ ((exists ge_balance_positive_add_zero_leftsecondreal ge_balance_negative_add_zero_leftsecondreal. (((((ge_representation_real_code_add_zero_leftsecond) = 2 * (ge_balance_positive_add_zero_leftsecondreal) /\ (ge_balance_negative_add_zero_leftsecondreal) = 0) \/ exists ge_signed_half_add_zero_leftsecondrealdecode. (((ge_representation_real_code_add_zero_leftsecond) = 2 * ge_signed_half_add_zero_leftsecondrealdecode + 1 /\ (ge_balance_positive_add_zero_leftsecondreal) = 0) /\ (ge_balance_negative_add_zero_leftsecondreal) = S ge_signed_half_add_zero_leftsecondrealdecode))) /\ ((ge_second_rp_add_zero_left) + ge_balance_negative_add_zero_leftsecondreal = (ge_second_rn_add_zero_left) + ge_balance_positive_add_zero_leftsecondreal))) /\ (exists ge_balance_positive_add_zero_leftsecondimaginary ge_balance_negative_add_zero_leftsecondimaginary. (((((ge_representation_imaginary_code_add_zero_leftsecond) = 2 * (ge_balance_positive_add_zero_leftsecondimaginary) /\ (ge_balance_negative_add_zero_leftsecondimaginary) = 0) \/ exists ge_signed_half_add_zero_leftsecondimaginarydecode. (((ge_representation_imaginary_code_add_zero_leftsecond) = 2 * ge_signed_half_add_zero_leftsecondimaginarydecode + 1 /\ (ge_balance_positive_add_zero_leftsecondimaginary) = 0) /\ (ge_balance_negative_add_zero_leftsecondimaginary) = S ge_signed_half_add_zero_leftsecondimaginarydecode))) /\ ((ge_second_ip_add_zero_left) + ge_balance_negative_add_zero_leftsecondimaginary = (ge_second_in_add_zero_left) + ge_balance_positive_add_zero_leftsecondimaginary)))))) /\ (exists ge_representation_real_code_add_zero_leftoutput ge_representation_imaginary_code_add_zero_leftoutput. (((a) = ((ge_representation_real_code_add_zero_leftoutput) + (ge_representation_imaginary_code_add_zero_leftoutput)) * S ((ge_representation_real_code_add_zero_leftoutput) + (ge_representation_imaginary_code_add_zero_leftoutput)) + ((ge_representation_imaginary_code_add_zero_leftoutput) + (ge_representation_imaginary_code_add_zero_leftoutput))) /\ ((exists ge_balance_positive_add_zero_leftoutputreal ge_balance_negative_add_zero_leftoutputreal. (((((ge_representation_real_code_add_zero_leftoutput) = 2 * (ge_balance_positive_add_zero_leftoutputreal) /\ (ge_balance_negative_add_zero_leftoutputreal) = 0) \/ exists ge_signed_half_add_zero_leftoutputrealdecode. (((ge_representation_real_code_add_zero_leftoutput) = 2 * ge_signed_half_add_zero_leftoutputrealdecode + 1 /\ (ge_balance_positive_add_zero_leftoutputreal) = 0) /\ (ge_balance_negative_add_zero_leftoutputreal) = S ge_signed_half_add_zero_leftoutputrealdecode))) /\ ((((ge_first_rp_add_zero_left) + (ge_second_rp_add_zero_left))) + ge_balance_negative_add_zero_leftoutputreal = (((ge_first_rn_add_zero_left) + (ge_second_rn_add_zero_left))) + ge_balance_positive_add_zero_leftoutputreal))) /\ (exists ge_balance_positive_add_zero_leftoutputimaginary ge_balance_negative_add_zero_leftoutputimaginary. (((((ge_representation_imaginary_code_add_zero_leftoutput) = 2 * (ge_balance_positive_add_zero_leftoutputimaginary) /\ (ge_balance_negative_add_zero_leftoutputimaginary) = 0) \/ exists ge_signed_half_add_zero_leftoutputimaginarydecode. (((ge_representation_imaginary_code_add_zero_leftoutput) = 2 * ge_signed_half_add_zero_leftoutputimaginarydecode + 1 /\ (ge_balance_positive_add_zero_leftoutputimaginary) = 0) /\ (ge_balance_negative_add_zero_leftoutputimaginary) = S ge_signed_half_add_zero_leftoutputimaginarydecode))) /\ ((((ge_first_ip_add_zero_left) + (ge_second_ip_add_zero_left))) + ge_balance_negative_add_zero_leftoutputimaginary = (((ge_first_in_add_zero_left) + (ge_second_in_add_zero_left))) + ge_balance_positive_add_zero_leftoutputimaginary)))))))))

Constructive proof overview

Generated structural guide

Commutativity supplies the actual left add 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

GF0026 gaussian_add_zero_right GF0012 gaussian_add_commutative

Direct dependents

GF003B gaussian_multiply_cancel_left

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 add zero rightgaussian add 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_add_commutative (a)
  2. L4
    specialize gaussian_add_commutative (0)
  3. L5
    specialize gaussian_add_commutative (a)
  4. L6
    apply gaussian_add_commutative
  5. L7
    specialize gaussian_add_zero_right (a)
  6. L8
    apply gaussian_add_zero_right
  7. L9
    exact hv

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002intro hv
  3. 0003specialize gaussian_add_commutative (a)
  4. 0004specialize gaussian_add_commutative (0)
  5. 0005specialize gaussian_add_commutative (a)
  6. 0006apply gaussian_add_commutative
  7. 0007specialize gaussian_add_zero_right (a)
  8. 0008apply gaussian_add_zero_right
  9. 0009exact hv