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_dividend_domain. (exists ge_first_rp_dividend_domainproduct ge_first_rn_dividend_domainproduct ge_first_ip_dividend_domainproduct ge_first_in_dividend_domainproduct ge_second_rp_dividend_domainproduct ge_second_rn_dividend_domainproduct ge_second_ip_dividend_domainproduct ge_second_in_dividend_domainproduct. ((exists ge_representation_real_code_dividend_domainproductfirst ge_representation_imaginary_code_dividend_domainproductfirst. (((d) = ((ge_representation_real_code_dividend_domainproductfirst) + (ge_representation_imaginary_code_dividend_domainproductfirst)) * S ((ge_representation_real_code_dividend_domainproductfirst) + (ge_representation_imaginary_code_dividend_domainproductfirst)) + ((ge_representation_imaginary_code_dividend_domainproductfirst) + (ge_representation_imaginary_code_dividend_domainproductfirst))) /\ ((exists ge_balance_positive_dividend_domainproductfirstreal ge_balance_negative_dividend_domainproductfirstreal. (((((ge_representation_real_code_dividend_domainproductfirst) = 2 * (ge_balance_positive_dividend_domainproductfirstreal) /\ (ge_balance_negative_dividend_domainproductfirstreal) = 0) \/ exists ge_signed_half_dividend_domainproductfirstrealdecode. (((ge_representation_real_code_dividend_domainproductfirst) = 2 * ge_signed_half_dividend_domainproductfirstrealdecode + 1 /\ (ge_balance_positive_dividend_domainproductfirstreal) = 0) /\ (ge_balance_negative_dividend_domainproductfirstreal) = S ge_signed_half_dividend_domainproductfirstrealdecode))) /\ ((ge_first_rp_dividend_domainproduct) + ge_balance_negative_dividend_domainproductfirstreal = (ge_first_rn_dividend_domainproduct) + ge_balance_positive_dividend_domainproductfirstreal))) /\ (exists ge_balance_positive_dividend_domainproductfirstimaginary ge_balance_negative_dividend_domainproductfirstimaginary. (((((ge_representation_imaginary_code_dividend_domainproductfirst) = 2 * (ge_balance_positive_dividend_domainproductfirstimaginary) /\ (ge_balance_negative_dividend_domainproductfirstimaginary) = 0) \/ exists ge_signed_half_dividend_domainproductfirstimaginarydecode. (((ge_representation_imaginary_code_dividend_domainproductfirst) = 2 * ge_signed_half_dividend_domainproductfirstimaginarydecode + 1 /\ (ge_balance_positive_dividend_domainproductfirstimaginary) = 0) /\ (ge_balance_negative_dividend_domainproductfirstimaginary) = S ge_signed_half_dividend_domainproductfirstimaginarydecode))) /\ ((ge_first_ip_dividend_domainproduct) + ge_balance_negative_dividend_domainproductfirstimaginary = (ge_first_in_dividend_domainproduct) + ge_balance_positive_dividend_domainproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_dividend_domainproductsecond ge_representation_imaginary_code_dividend_domainproductsecond. (((gr_quotient_dividend_domain) = ((ge_representation_real_code_dividend_domainproductsecond) + (ge_representation_imaginary_code_dividend_domainproductsecond)) * S ((ge_representation_real_code_dividend_domainproductsecond) + (ge_representation_imaginary_code_dividend_domainproductsecond)) + ((ge_representation_imaginary_code_dividend_domainproductsecond) + (ge_representation_imaginary_code_dividend_domainproductsecond))) /\ ((exists ge_balance_positive_dividend_domainproductsecondreal ge_balance_negative_dividend_domainproductsecondreal. (((((ge_representation_real_code_dividend_domainproductsecond) = 2 * (ge_balance_positive_dividend_domainproductsecondreal) /\ (ge_balance_negative_dividend_domainproductsecondreal) = 0) \/ exists ge_signed_half_dividend_domainproductsecondrealdecode. (((ge_representation_real_code_dividend_domainproductsecond) = 2 * ge_signed_half_dividend_domainproductsecondrealdecode + 1 /\ (ge_balance_positive_dividend_domainproductsecondreal) = 0) /\ (ge_balance_negative_dividend_domainproductsecondreal) = S ge_signed_half_dividend_domainproductsecondrealdecode))) /\ ((ge_second_rp_dividend_domainproduct) + ge_balance_negative_dividend_domainproductsecondreal = (ge_second_rn_dividend_domainproduct) + ge_balance_positive_dividend_domainproductsecondreal))) /\ (exists ge_balance_positive_dividend_domainproductsecondimaginary ge_balance_negative_dividend_domainproductsecondimaginary. (((((ge_representation_imaginary_code_dividend_domainproductsecond) = 2 * (ge_balance_positive_dividend_domainproductsecondimaginary) /\ (ge_balance_negative_dividend_domainproductsecondimaginary) = 0) \/ exists ge_signed_half_dividend_domainproductsecondimaginarydecode. (((ge_representation_imaginary_code_dividend_domainproductsecond) = 2 * ge_signed_half_dividend_domainproductsecondimaginarydecode + 1 /\ (ge_balance_positive_dividend_domainproductsecondimaginary) = 0) /\ (ge_balance_negative_dividend_domainproductsecondimaginary) = S ge_signed_half_dividend_domainproductsecondimaginarydecode))) /\ ((ge_second_ip_dividend_domainproduct) + ge_balance_negative_dividend_domainproductsecondimaginary = (ge_second_in_dividend_domainproduct) + ge_balance_positive_dividend_domainproductsecondimaginary)))))) /\ (exists ge_representation_real_code_dividend_domainproductoutput ge_representation_imaginary_code_dividend_domainproductoutput. (((z) = ((ge_representation_real_code_dividend_domainproductoutput) + (ge_representation_imaginary_code_dividend_domainproductoutput)) * S ((ge_representation_real_code_dividend_domainproductoutput) + (ge_representation_imaginary_code_dividend_domainproductoutput)) + ((ge_representation_imaginary_code_dividend_domainproductoutput) + (ge_representation_imaginary_code_dividend_domainproductoutput))) /\ ((exists ge_balance_positive_dividend_domainproductoutputreal ge_balance_negative_dividend_domainproductoutputreal. (((((ge_representation_real_code_dividend_domainproductoutput) = 2 * (ge_balance_positive_dividend_domainproductoutputreal) /\ (ge_balance_negative_dividend_domainproductoutputreal) = 0) \/ exists ge_signed_half_dividend_domainproductoutputrealdecode. (((ge_representation_real_code_dividend_domainproductoutput) = 2 * ge_signed_half_dividend_domainproductoutputrealdecode + 1 /\ (ge_balance_positive_dividend_domainproductoutputreal) = 0) /\ (ge_balance_negative_dividend_domainproductoutputreal) = S ge_signed_half_dividend_domainproductoutputrealdecode))) /\ ((((((((ge_first_rp_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))))))) + ge_balance_negative_dividend_domainproductoutputreal = (((((((ge_first_rp_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))))))) + ge_balance_positive_dividend_domainproductoutputreal))) /\ (exists ge_balance_positive_dividend_domainproductoutputimaginary ge_balance_negative_dividend_domainproductoutputimaginary. (((((ge_representation_imaginary_code_dividend_domainproductoutput) = 2 * (ge_balance_positive_dividend_domainproductoutputimaginary) /\ (ge_balance_negative_dividend_domainproductoutputimaginary) = 0) \/ exists ge_signed_half_dividend_domainproductoutputimaginarydecode. (((ge_representation_imaginary_code_dividend_domainproductoutput) = 2 * ge_signed_half_dividend_domainproductoutputimaginarydecode + 1 /\ (ge_balance_positive_dividend_domainproductoutputimaginary) = 0) /\ (ge_balance_negative_dividend_domainproductoutputimaginary) = S ge_signed_half_dividend_domainproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))))))) + ge_balance_negative_dividend_domainproductoutputimaginary = (((((((ge_first_rp_dividend_domainproduct) * (ge_second_in_dividend_domainproduct))) + (((ge_first_rn_dividend_domainproduct) * (ge_second_ip_dividend_domainproduct))))) + (((((ge_first_ip_dividend_domainproduct) * (ge_second_rn_dividend_domainproduct))) + (((ge_first_in_dividend_domainproduct) * (ge_second_rp_dividend_domainproduct))))))) + ge_balance_positive_dividend_domainproductoutputimaginary)))))))))) -> (exists ge_real_positive_dividend_valid ge_real_negative_dividend_valid ge_imaginary_positive_dividend_valid ge_imaginary_negative_dividend_valid. (exists ge_real_code_dividend_validdecode ge_imaginary_code_dividend_validdecode. (((z) = ((ge_real_code_dividend_validdecode) + (ge_imaginary_code_dividend_validdecode)) * S ((ge_real_code_dividend_validdecode) + (ge_imaginary_code_dividend_validdecode)) + ((ge_imaginary_code_dividend_validdecode) + (ge_imaginary_code_dividend_validdecode))) /\ (((((ge_real_code_dividend_validdecode) = 2 * (ge_real_positive_dividend_valid) /\ (ge_real_negative_dividend_valid) = 0) \/ exists ge_signed_half_ge_dividend_validdecode_real. (((ge_real_code_dividend_validdecode) = 2 * ge_signed_half_ge_dividend_validdecode_real + 1 /\ (ge_real_positive_dividend_valid) = 0) /\ (ge_real_negative_dividend_valid) = S ge_signed_half_ge_dividend_validdecode_real))) /\ ((((ge_imaginary_code_dividend_validdecode) = 2 * (ge_imaginary_positive_dividend_valid) /\ (ge_imaginary_negative_dividend_valid) = 0) \/ exists ge_signed_half_ge_dividend_validdecode_imaginary. (((ge_imaginary_code_dividend_validdecode) = 2 * ge_signed_half_ge_dividend_validdecode_imaginary + 1 /\ (ge_imaginary_positive_dividend_valid) = 0) /\ (ge_imaginary_negative_dividend_valid) = S ge_signed_half_ge_dividend_validdecode_imaginary)))))))Constructive proof overview
Generated structural guide
An actually divisible Gaussian value has a valid canonical carrier code.
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_output_valid (d) - 0006
specialize gaussian_multiply_output_valid (x) - 0007
specialize gaussian_multiply_output_valid (z) - 0008
apply gaussian_multiply_output_valid - 0009
exact h_witness