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 d z. (exists gr_quotient_divisor_domain. (exists ge_first_rp_divisor_domainproduct ge_first_rn_divisor_domainproduct ge_first_ip_divisor_domainproduct ge_first_in_divisor_domainproduct ge_second_rp_divisor_domainproduct ge_second_rn_divisor_domainproduct ge_second_ip_divisor_domainproduct ge_second_in_divisor_domainproduct. ((exists ge_representation_real_code_divisor_domainproductfirst ge_representation_imaginary_code_divisor_domainproductfirst. (((d) = ((ge_representation_real_code_divisor_domainproductfirst) + (ge_representation_imaginary_code_divisor_domainproductfirst)) * S ((ge_representation_real_code_divisor_domainproductfirst) + (ge_representation_imaginary_code_divisor_domainproductfirst)) + ((ge_representation_imaginary_code_divisor_domainproductfirst) + (ge_representation_imaginary_code_divisor_domainproductfirst))) /\ ((exists ge_balance_positive_divisor_domainproductfirstreal ge_balance_negative_divisor_domainproductfirstreal. (((((ge_representation_real_code_divisor_domainproductfirst) = 2 * (ge_balance_positive_divisor_domainproductfirstreal) /\ (ge_balance_negative_divisor_domainproductfirstreal) = 0) \/ exists ge_signed_half_divisor_domainproductfirstrealdecode. (((ge_representation_real_code_divisor_domainproductfirst) = 2 * ge_signed_half_divisor_domainproductfirstrealdecode + 1 /\ (ge_balance_positive_divisor_domainproductfirstreal) = 0) /\ (ge_balance_negative_divisor_domainproductfirstreal) = S ge_signed_half_divisor_domainproductfirstrealdecode))) /\ ((ge_first_rp_divisor_domainproduct) + ge_balance_negative_divisor_domainproductfirstreal = (ge_first_rn_divisor_domainproduct) + ge_balance_positive_divisor_domainproductfirstreal))) /\ (exists ge_balance_positive_divisor_domainproductfirstimaginary ge_balance_negative_divisor_domainproductfirstimaginary. (((((ge_representation_imaginary_code_divisor_domainproductfirst) = 2 * (ge_balance_positive_divisor_domainproductfirstimaginary) /\ (ge_balance_negative_divisor_domainproductfirstimaginary) = 0) \/ exists ge_signed_half_divisor_domainproductfirstimaginarydecode. (((ge_representation_imaginary_code_divisor_domainproductfirst) = 2 * ge_signed_half_divisor_domainproductfirstimaginarydecode + 1 /\ (ge_balance_positive_divisor_domainproductfirstimaginary) = 0) /\ (ge_balance_negative_divisor_domainproductfirstimaginary) = S ge_signed_half_divisor_domainproductfirstimaginarydecode))) /\ ((ge_first_ip_divisor_domainproduct) + ge_balance_negative_divisor_domainproductfirstimaginary = (ge_first_in_divisor_domainproduct) + ge_balance_positive_divisor_domainproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_divisor_domainproductsecond ge_representation_imaginary_code_divisor_domainproductsecond. (((gr_quotient_divisor_domain) = ((ge_representation_real_code_divisor_domainproductsecond) + (ge_representation_imaginary_code_divisor_domainproductsecond)) * S ((ge_representation_real_code_divisor_domainproductsecond) + (ge_representation_imaginary_code_divisor_domainproductsecond)) + ((ge_representation_imaginary_code_divisor_domainproductsecond) + (ge_representation_imaginary_code_divisor_domainproductsecond))) /\ ((exists ge_balance_positive_divisor_domainproductsecondreal ge_balance_negative_divisor_domainproductsecondreal. (((((ge_representation_real_code_divisor_domainproductsecond) = 2 * (ge_balance_positive_divisor_domainproductsecondreal) /\ (ge_balance_negative_divisor_domainproductsecondreal) = 0) \/ exists ge_signed_half_divisor_domainproductsecondrealdecode. (((ge_representation_real_code_divisor_domainproductsecond) = 2 * ge_signed_half_divisor_domainproductsecondrealdecode + 1 /\ (ge_balance_positive_divisor_domainproductsecondreal) = 0) /\ (ge_balance_negative_divisor_domainproductsecondreal) = S ge_signed_half_divisor_domainproductsecondrealdecode))) /\ ((ge_second_rp_divisor_domainproduct) + ge_balance_negative_divisor_domainproductsecondreal = (ge_second_rn_divisor_domainproduct) + ge_balance_positive_divisor_domainproductsecondreal))) /\ (exists ge_balance_positive_divisor_domainproductsecondimaginary ge_balance_negative_divisor_domainproductsecondimaginary. (((((ge_representation_imaginary_code_divisor_domainproductsecond) = 2 * (ge_balance_positive_divisor_domainproductsecondimaginary) /\ (ge_balance_negative_divisor_domainproductsecondimaginary) = 0) \/ exists ge_signed_half_divisor_domainproductsecondimaginarydecode. (((ge_representation_imaginary_code_divisor_domainproductsecond) = 2 * ge_signed_half_divisor_domainproductsecondimaginarydecode + 1 /\ (ge_balance_positive_divisor_domainproductsecondimaginary) = 0) /\ (ge_balance_negative_divisor_domainproductsecondimaginary) = S ge_signed_half_divisor_domainproductsecondimaginarydecode))) /\ ((ge_second_ip_divisor_domainproduct) + ge_balance_negative_divisor_domainproductsecondimaginary = (ge_second_in_divisor_domainproduct) + ge_balance_positive_divisor_domainproductsecondimaginary)))))) /\ (exists ge_representation_real_code_divisor_domainproductoutput ge_representation_imaginary_code_divisor_domainproductoutput. (((z) = ((ge_representation_real_code_divisor_domainproductoutput) + (ge_representation_imaginary_code_divisor_domainproductoutput)) * S ((ge_representation_real_code_divisor_domainproductoutput) + (ge_representation_imaginary_code_divisor_domainproductoutput)) + ((ge_representation_imaginary_code_divisor_domainproductoutput) + (ge_representation_imaginary_code_divisor_domainproductoutput))) /\ ((exists ge_balance_positive_divisor_domainproductoutputreal ge_balance_negative_divisor_domainproductoutputreal. (((((ge_representation_real_code_divisor_domainproductoutput) = 2 * (ge_balance_positive_divisor_domainproductoutputreal) /\ (ge_balance_negative_divisor_domainproductoutputreal) = 0) \/ exists ge_signed_half_divisor_domainproductoutputrealdecode. (((ge_representation_real_code_divisor_domainproductoutput) = 2 * ge_signed_half_divisor_domainproductoutputrealdecode + 1 /\ (ge_balance_positive_divisor_domainproductoutputreal) = 0) /\ (ge_balance_negative_divisor_domainproductoutputreal) = S ge_signed_half_divisor_domainproductoutputrealdecode))) /\ ((((((((ge_first_rp_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))))))) + ge_balance_negative_divisor_domainproductoutputreal = (((((((ge_first_rp_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))))))) + ge_balance_positive_divisor_domainproductoutputreal))) /\ (exists ge_balance_positive_divisor_domainproductoutputimaginary ge_balance_negative_divisor_domainproductoutputimaginary. (((((ge_representation_imaginary_code_divisor_domainproductoutput) = 2 * (ge_balance_positive_divisor_domainproductoutputimaginary) /\ (ge_balance_negative_divisor_domainproductoutputimaginary) = 0) \/ exists ge_signed_half_divisor_domainproductoutputimaginarydecode. (((ge_representation_imaginary_code_divisor_domainproductoutput) = 2 * ge_signed_half_divisor_domainproductoutputimaginarydecode + 1 /\ (ge_balance_positive_divisor_domainproductoutputimaginary) = 0) /\ (ge_balance_negative_divisor_domainproductoutputimaginary) = S ge_signed_half_divisor_domainproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))))))) + ge_balance_negative_divisor_domainproductoutputimaginary = (((((((ge_first_rp_divisor_domainproduct) * (ge_second_in_divisor_domainproduct))) + (((ge_first_rn_divisor_domainproduct) * (ge_second_ip_divisor_domainproduct))))) + (((((ge_first_ip_divisor_domainproduct) * (ge_second_rn_divisor_domainproduct))) + (((ge_first_in_divisor_domainproduct) * (ge_second_rp_divisor_domainproduct))))))) + ge_balance_positive_divisor_domainproductoutputimaginary)))))))))) -> (exists ge_real_positive_divisor_valid ge_real_negative_divisor_valid ge_imaginary_positive_divisor_valid ge_imaginary_negative_divisor_valid. (exists ge_real_code_divisor_validdecode ge_imaginary_code_divisor_validdecode. (((d) = ((ge_real_code_divisor_validdecode) + (ge_imaginary_code_divisor_validdecode)) * S ((ge_real_code_divisor_validdecode) + (ge_imaginary_code_divisor_validdecode)) + ((ge_imaginary_code_divisor_validdecode) + (ge_imaginary_code_divisor_validdecode))) /\ (((((ge_real_code_divisor_validdecode) = 2 * (ge_real_positive_divisor_valid) /\ (ge_real_negative_divisor_valid) = 0) \/ exists ge_signed_half_ge_divisor_validdecode_real. (((ge_real_code_divisor_validdecode) = 2 * ge_signed_half_ge_divisor_validdecode_real + 1 /\ (ge_real_positive_divisor_valid) = 0) /\ (ge_real_negative_divisor_valid) = S ge_signed_half_ge_divisor_validdecode_real))) /\ ((((ge_imaginary_code_divisor_validdecode) = 2 * (ge_imaginary_positive_divisor_valid) /\ (ge_imaginary_negative_divisor_valid) = 0) \/ exists ge_signed_half_ge_divisor_validdecode_imaginary. (((ge_imaginary_code_divisor_validdecode) = 2 * ge_signed_half_ge_divisor_validdecode_imaginary + 1 /\ (ge_imaginary_positive_divisor_valid) = 0) /\ (ge_imaginary_negative_divisor_valid) = S ge_signed_half_ge_divisor_validdecode_imaginary)))))))Constructive proof overview
Generated structural guide
A witnessed Gaussian divisor belongs to the actual canonical Gaussian carrier.
The unchanged tactic script uses 1 declared prerequisite 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
none
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 (1)
01Fix variables and assumptionsL1–3
02Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
cases h
03Use earlier factsL5–9
Original exact command ledger · 9 lines
- 0001
intro d - 0002
intro z - 0003
intro h - 0004
cases h - 0005
specialize gaussian_multiply_input_left_valid (d) - 0006
specialize gaussian_multiply_input_left_valid (x) - 0007
specialize gaussian_multiply_input_left_valid (z) - 0008
apply gaussian_multiply_input_left_valid - 0009
exact h_witness