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 b c. (exists gr_product_trace_empty_product gr_product_scale_empty_product. ((((exists ff_h_gprod_empty_productstart. ff_h_gprod_empty_productstart + S (6) = S ((S (0)) * gr_product_scale_empty_product)) /\ exists ff_q_gprod_empty_productstart. gr_product_trace_empty_product = ff_q_gprod_empty_productstart * S ((S (0)) * gr_product_scale_empty_product) + (6))) /\ ((((exists ff_h_gprod_empty_productend. ff_h_gprod_empty_productend + S (6) = S ((S (0)) * gr_product_scale_empty_product)) /\ exists ff_q_gprod_empty_productend. gr_product_trace_empty_product = ff_q_gprod_empty_productend * S ((S (0)) * gr_product_scale_empty_product) + (6))) /\ (forall gr_product_index_empty_productsteps. (exists ge_gap_empty_productstepsindex_bound. ge_gap_empty_productstepsindex_bound + S (gr_product_index_empty_productsteps) = (0)) -> exists gr_product_factor_empty_productsteps gr_product_before_empty_productsteps gr_product_after_empty_productsteps. ((((exists ff_h_gprod_empty_productstepsfactor. ff_h_gprod_empty_productstepsfactor + S (gr_product_factor_empty_productsteps) = S ((S (gr_product_index_empty_productsteps)) * c)) /\ exists ff_q_gprod_empty_productstepsfactor. b = ff_q_gprod_empty_productstepsfactor * S ((S (gr_product_index_empty_productsteps)) * c) + (gr_product_factor_empty_productsteps))) /\ ((((exists ff_h_gprod_empty_productstepsbefore. ff_h_gprod_empty_productstepsbefore + S (gr_product_before_empty_productsteps) = S ((S (gr_product_index_empty_productsteps)) * gr_product_scale_empty_product)) /\ exists ff_q_gprod_empty_productstepsbefore. gr_product_trace_empty_product = ff_q_gprod_empty_productstepsbefore * S ((S (gr_product_index_empty_productsteps)) * gr_product_scale_empty_product) + (gr_product_before_empty_productsteps))) /\ ((((exists ff_h_gprod_empty_productstepsafter. ff_h_gprod_empty_productstepsafter + S (gr_product_after_empty_productsteps) = S ((S (S (gr_product_index_empty_productsteps))) * gr_product_scale_empty_product)) /\ exists ff_q_gprod_empty_productstepsafter. gr_product_trace_empty_product = ff_q_gprod_empty_productstepsafter * S ((S (S (gr_product_index_empty_productsteps))) * gr_product_scale_empty_product) + (gr_product_after_empty_productsteps))) /\ (exists ge_first_rp_empty_productstepsmultiply ge_first_rn_empty_productstepsmultiply ge_first_ip_empty_productstepsmultiply ge_first_in_empty_productstepsmultiply ge_second_rp_empty_productstepsmultiply ge_second_rn_empty_productstepsmultiply ge_second_ip_empty_productstepsmultiply ge_second_in_empty_productstepsmultiply. ((exists ge_representation_real_code_empty_productstepsmultiplyfirst ge_representation_imaginary_code_empty_productstepsmultiplyfirst. (((gr_product_before_empty_productsteps) = ((ge_representation_real_code_empty_productstepsmultiplyfirst) + (ge_representation_imaginary_code_empty_productstepsmultiplyfirst)) * S ((ge_representation_real_code_empty_productstepsmultiplyfirst) + (ge_representation_imaginary_code_empty_productstepsmultiplyfirst)) + ((ge_representation_imaginary_code_empty_productstepsmultiplyfirst) + (ge_representation_imaginary_code_empty_productstepsmultiplyfirst))) /\ ((exists ge_balance_positive_empty_productstepsmultiplyfirstreal ge_balance_negative_empty_productstepsmultiplyfirstreal. (((((ge_representation_real_code_empty_productstepsmultiplyfirst) = 2 * (ge_balance_positive_empty_productstepsmultiplyfirstreal) /\ (ge_balance_negative_empty_productstepsmultiplyfirstreal) = 0) \/ exists ge_signed_half_empty_productstepsmultiplyfirstrealdecode. (((ge_representation_real_code_empty_productstepsmultiplyfirst) = 2 * ge_signed_half_empty_productstepsmultiplyfirstrealdecode + 1 /\ (ge_balance_positive_empty_productstepsmultiplyfirstreal) = 0) /\ (ge_balance_negative_empty_productstepsmultiplyfirstreal) = S ge_signed_half_empty_productstepsmultiplyfirstrealdecode))) /\ ((ge_first_rp_empty_productstepsmultiply) + ge_balance_negative_empty_productstepsmultiplyfirstreal = (ge_first_rn_empty_productstepsmultiply) + ge_balance_positive_empty_productstepsmultiplyfirstreal))) /\ (exists ge_balance_positive_empty_productstepsmultiplyfirstimaginary ge_balance_negative_empty_productstepsmultiplyfirstimaginary. (((((ge_representation_imaginary_code_empty_productstepsmultiplyfirst) = 2 * (ge_balance_positive_empty_productstepsmultiplyfirstimaginary) /\ (ge_balance_negative_empty_productstepsmultiplyfirstimaginary) = 0) \/ exists ge_signed_half_empty_productstepsmultiplyfirstimaginarydecode. (((ge_representation_imaginary_code_empty_productstepsmultiplyfirst) = 2 * ge_signed_half_empty_productstepsmultiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_empty_productstepsmultiplyfirstimaginary) = 0) /\ (ge_balance_negative_empty_productstepsmultiplyfirstimaginary) = S ge_signed_half_empty_productstepsmultiplyfirstimaginarydecode))) /\ ((ge_first_ip_empty_productstepsmultiply) + ge_balance_negative_empty_productstepsmultiplyfirstimaginary = (ge_first_in_empty_productstepsmultiply) + ge_balance_positive_empty_productstepsmultiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_empty_productstepsmultiplysecond ge_representation_imaginary_code_empty_productstepsmultiplysecond. (((gr_product_factor_empty_productsteps) = ((ge_representation_real_code_empty_productstepsmultiplysecond) + (ge_representation_imaginary_code_empty_productstepsmultiplysecond)) * S ((ge_representation_real_code_empty_productstepsmultiplysecond) + (ge_representation_imaginary_code_empty_productstepsmultiplysecond)) + ((ge_representation_imaginary_code_empty_productstepsmultiplysecond) + (ge_representation_imaginary_code_empty_productstepsmultiplysecond))) /\ ((exists ge_balance_positive_empty_productstepsmultiplysecondreal ge_balance_negative_empty_productstepsmultiplysecondreal. (((((ge_representation_real_code_empty_productstepsmultiplysecond) = 2 * (ge_balance_positive_empty_productstepsmultiplysecondreal) /\ (ge_balance_negative_empty_productstepsmultiplysecondreal) = 0) \/ exists ge_signed_half_empty_productstepsmultiplysecondrealdecode. (((ge_representation_real_code_empty_productstepsmultiplysecond) = 2 * ge_signed_half_empty_productstepsmultiplysecondrealdecode + 1 /\ (ge_balance_positive_empty_productstepsmultiplysecondreal) = 0) /\ (ge_balance_negative_empty_productstepsmultiplysecondreal) = S ge_signed_half_empty_productstepsmultiplysecondrealdecode))) /\ ((ge_second_rp_empty_productstepsmultiply) + ge_balance_negative_empty_productstepsmultiplysecondreal = (ge_second_rn_empty_productstepsmultiply) + ge_balance_positive_empty_productstepsmultiplysecondreal))) /\ (exists ge_balance_positive_empty_productstepsmultiplysecondimaginary ge_balance_negative_empty_productstepsmultiplysecondimaginary. (((((ge_representation_imaginary_code_empty_productstepsmultiplysecond) = 2 * (ge_balance_positive_empty_productstepsmultiplysecondimaginary) /\ (ge_balance_negative_empty_productstepsmultiplysecondimaginary) = 0) \/ exists ge_signed_half_empty_productstepsmultiplysecondimaginarydecode. (((ge_representation_imaginary_code_empty_productstepsmultiplysecond) = 2 * ge_signed_half_empty_productstepsmultiplysecondimaginarydecode + 1 /\ (ge_balance_positive_empty_productstepsmultiplysecondimaginary) = 0) /\ (ge_balance_negative_empty_productstepsmultiplysecondimaginary) = S ge_signed_half_empty_productstepsmultiplysecondimaginarydecode))) /\ ((ge_second_ip_empty_productstepsmultiply) + ge_balance_negative_empty_productstepsmultiplysecondimaginary = (ge_second_in_empty_productstepsmultiply) + ge_balance_positive_empty_productstepsmultiplysecondimaginary)))))) /\ (exists ge_representation_real_code_empty_productstepsmultiplyoutput ge_representation_imaginary_code_empty_productstepsmultiplyoutput. (((gr_product_after_empty_productsteps) = ((ge_representation_real_code_empty_productstepsmultiplyoutput) + (ge_representation_imaginary_code_empty_productstepsmultiplyoutput)) * S ((ge_representation_real_code_empty_productstepsmultiplyoutput) + (ge_representation_imaginary_code_empty_productstepsmultiplyoutput)) + ((ge_representation_imaginary_code_empty_productstepsmultiplyoutput) + (ge_representation_imaginary_code_empty_productstepsmultiplyoutput))) /\ ((exists ge_balance_positive_empty_productstepsmultiplyoutputreal ge_balance_negative_empty_productstepsmultiplyoutputreal. (((((ge_representation_real_code_empty_productstepsmultiplyoutput) = 2 * (ge_balance_positive_empty_productstepsmultiplyoutputreal) /\ (ge_balance_negative_empty_productstepsmultiplyoutputreal) = 0) \/ exists ge_signed_half_empty_productstepsmultiplyoutputrealdecode. (((ge_representation_real_code_empty_productstepsmultiplyoutput) = 2 * ge_signed_half_empty_productstepsmultiplyoutputrealdecode + 1 /\ (ge_balance_positive_empty_productstepsmultiplyoutputreal) = 0) /\ (ge_balance_negative_empty_productstepsmultiplyoutputreal) = S ge_signed_half_empty_productstepsmultiplyoutputrealdecode))) /\ ((((((((ge_first_rp_empty_productstepsmultiply) * (ge_second_rp_empty_productstepsmultiply))) + (((ge_first_rn_empty_productstepsmultiply) * (ge_second_rn_empty_productstepsmultiply))))) + (((((ge_first_ip_empty_productstepsmultiply) * (ge_second_in_empty_productstepsmultiply))) + (((ge_first_in_empty_productstepsmultiply) * (ge_second_ip_empty_productstepsmultiply))))))) + ge_balance_negative_empty_productstepsmultiplyoutputreal = (((((((ge_first_rp_empty_productstepsmultiply) * (ge_second_rn_empty_productstepsmultiply))) + (((ge_first_rn_empty_productstepsmultiply) * (ge_second_rp_empty_productstepsmultiply))))) + (((((ge_first_ip_empty_productstepsmultiply) * (ge_second_ip_empty_productstepsmultiply))) + (((ge_first_in_empty_productstepsmultiply) * (ge_second_in_empty_productstepsmultiply))))))) + ge_balance_positive_empty_productstepsmultiplyoutputreal))) /\ (exists ge_balance_positive_empty_productstepsmultiplyoutputimaginary ge_balance_negative_empty_productstepsmultiplyoutputimaginary. (((((ge_representation_imaginary_code_empty_productstepsmultiplyoutput) = 2 * (ge_balance_positive_empty_productstepsmultiplyoutputimaginary) /\ (ge_balance_negative_empty_productstepsmultiplyoutputimaginary) = 0) \/ exists ge_signed_half_empty_productstepsmultiplyoutputimaginarydecode. (((ge_representation_imaginary_code_empty_productstepsmultiplyoutput) = 2 * ge_signed_half_empty_productstepsmultiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_empty_productstepsmultiplyoutputimaginary) = 0) /\ (ge_balance_negative_empty_productstepsmultiplyoutputimaginary) = S ge_signed_half_empty_productstepsmultiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_empty_productstepsmultiply) * (ge_second_ip_empty_productstepsmultiply))) + (((ge_first_rn_empty_productstepsmultiply) * (ge_second_in_empty_productstepsmultiply))))) + (((((ge_first_ip_empty_productstepsmultiply) * (ge_second_rp_empty_productstepsmultiply))) + (((ge_first_in_empty_productstepsmultiply) * (ge_second_rn_empty_productstepsmultiply))))))) + ge_balance_negative_empty_productstepsmultiplyoutputimaginary = (((((((ge_first_rp_empty_productstepsmultiply) * (ge_second_in_empty_productstepsmultiply))) + (((ge_first_rn_empty_productstepsmultiply) * (ge_second_ip_empty_productstepsmultiply))))) + (((((ge_first_ip_empty_productstepsmultiply) * (ge_second_rn_empty_productstepsmultiply))) + (((ge_first_in_empty_productstepsmultiply) * (ge_second_rp_empty_productstepsmultiply))))))) + ge_balance_positive_empty_productstepsmultiplyoutputimaginary))))))))))))))))Constructive proof overview
Generated structural guide
Every empty Gaussian factor prefix has a genuine constant product trace with canonical value six, not natural one.
The unchanged tactic script uses 1 declared prerequisite and contains 22 exact native proof lines.
Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
Direct dependents
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 assumptionsL1–2
02Construct an explicit witnessL3–4
03Separate the logical casesL5–6
04Construct an explicit witnessL7–7
Supply the displayed value, then prove that it has the required property.
- L7
exists 0
05Calculate and transport equalitiesL8–8
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L8
norm_num
06Construct an explicit witnessL9–9
Supply the displayed value, then prove that it has the required property.
- L9
exists 0
07Calculate and transport equalitiesL10–10
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L10
norm_num
08Separate the logical casesL11–12
09Construct an explicit witnessL13–13
Supply the displayed value, then prove that it has the required property.
- L13
exists 0
10Calculate and transport equalitiesL14–14
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L14
norm_num
11Construct an explicit witnessL15–15
Supply the displayed value, then prove that it has the required property.
- L15
exists 0
12Calculate and transport equalitiesL16–16
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L16
norm_num
13Fix variables and assumptionsL17–18
14Separate the logical casesL19–19
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L19
exfalso
Original exact command ledger · 22 lines
- 0001
intro b - 0002
intro c - 0003
exists (6) - 0004
exists (6) - 0005
split - 0006
split - 0007
exists 0 - 0008
norm_num - 0009
exists 0 - 0010
norm_num - 0011
split - 0012
split - 0013
exists 0 - 0014
norm_num - 0015
exists 0 - 0016
norm_num - 0017
intro i - 0018
intro hi - 0019
exfalso - 0020
specialize gaussian_search_no_index_below_zero (i) - 0021
apply gaussian_search_no_index_below_zero - 0022
exact hi