GF0029

gaussian_multiply_one_left

Commutativity supplies the actual left multiply one identity.

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

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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

∀ a. ZPairValid(a) → GMul(6,a,a)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

ZPairValid(z) · 1GMul(a,b,c) · 1

Actual proof prerequisites

gaussian_multiply_one_rightgaussian_multiply_commutative
Original expanded first-order 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)))))))))

Complete tactic proof in conservative notation

All 9 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
gaussian multiply commutativegaussian multiply one right
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 defined 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