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 expanded first-order arithmetic statement
forall F G n d. ~(exists pvs_factor_omit_guard. (n) = (d) * pvs_factor_omit_guard) -> ((((~((d)=0)) /\ (exists dc_quotient_omit_result dc_left_omit_result dc_right_omit_result. (((n)=(d)*dc_quotient_omit_result) /\ (((exists dst_positive_code_omit_resultleft dst_positive_scale_omit_resultleft dst_negative_code_omit_resultleft dst_negative_scale_omit_resultleft dst_positive_omit_resultleft dst_negative_omit_resultleft. (((F) = (((((dst_positive_code_omit_resultleft) + (dst_positive_scale_omit_resultleft)) * S ((dst_positive_code_omit_resultleft) + (dst_positive_scale_omit_resultleft)) + ((dst_positive_scale_omit_resultleft) + (dst_positive_scale_omit_resultleft))) + (((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) * S ((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) + ((dst_negative_scale_omit_resultleft) + (dst_negative_scale_omit_resultleft)))) * S ((((dst_positive_code_omit_resultleft) + (dst_positive_scale_omit_resultleft)) * S ((dst_positive_code_omit_resultleft) + (dst_positive_scale_omit_resultleft)) + ((dst_positive_scale_omit_resultleft) + (dst_positive_scale_omit_resultleft))) + (((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) * S ((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) + ((dst_negative_scale_omit_resultleft) + (dst_negative_scale_omit_resultleft)))) + ((((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) * S ((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) + ((dst_negative_scale_omit_resultleft) + (dst_negative_scale_omit_resultleft))) + (((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) * S ((dst_negative_code_omit_resultleft) + (dst_negative_scale_omit_resultleft)) + ((dst_negative_scale_omit_resultleft) + (dst_negative_scale_omit_resultleft)))))) /\ (((((exists ff_h_pvs_omit_resultleftpositive. ff_h_pvs_omit_resultleftpositive + S (dst_positive_omit_resultleft) = S ((S (d)) * dst_positive_scale_omit_resultleft)) /\ exists ff_q_pvs_omit_resultleftpositive. dst_positive_code_omit_resultleft = ff_q_pvs_omit_resultleftpositive * S ((S (d)) * dst_positive_scale_omit_resultleft) + (dst_positive_omit_resultleft))) /\ (((((exists ff_h_pvs_omit_resultleftnegative. ff_h_pvs_omit_resultleftnegative + S (dst_negative_omit_resultleft) = S ((S (d)) * dst_negative_scale_omit_resultleft)) /\ exists ff_q_pvs_omit_resultleftnegative. dst_negative_code_omit_resultleft = ff_q_pvs_omit_resultleftnegative * S ((S (d)) * dst_negative_scale_omit_resultleft) + (dst_negative_omit_resultleft))) /\ (exists ge_balance_positive_omit_resultleftvalue ge_balance_negative_omit_resultleftvalue. (((((dc_left_omit_result) = 2 * (ge_balance_positive_omit_resultleftvalue) /\ (ge_balance_negative_omit_resultleftvalue) = 0) \/ exists ge_signed_half_omit_resultleftvaluedecode. (((dc_left_omit_result) = 2 * ge_signed_half_omit_resultleftvaluedecode + 1 /\ (ge_balance_positive_omit_resultleftvalue) = 0) /\ (ge_balance_negative_omit_resultleftvalue) = S ge_signed_half_omit_resultleftvaluedecode))) /\ ((dst_positive_omit_resultleft) + ge_balance_negative_omit_resultleftvalue = (dst_negative_omit_resultleft) + ge_balance_positive_omit_resultleftvalue))))))))) /\ (((exists dst_positive_code_omit_resultright dst_positive_scale_omit_resultright dst_negative_code_omit_resultright dst_negative_scale_omit_resultright dst_positive_omit_resultright dst_negative_omit_resultright. (((G) = (((((dst_positive_code_omit_resultright) + (dst_positive_scale_omit_resultright)) * S ((dst_positive_code_omit_resultright) + (dst_positive_scale_omit_resultright)) + ((dst_positive_scale_omit_resultright) + (dst_positive_scale_omit_resultright))) + (((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) * S ((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) + ((dst_negative_scale_omit_resultright) + (dst_negative_scale_omit_resultright)))) * S ((((dst_positive_code_omit_resultright) + (dst_positive_scale_omit_resultright)) * S ((dst_positive_code_omit_resultright) + (dst_positive_scale_omit_resultright)) + ((dst_positive_scale_omit_resultright) + (dst_positive_scale_omit_resultright))) + (((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) * S ((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) + ((dst_negative_scale_omit_resultright) + (dst_negative_scale_omit_resultright)))) + ((((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) * S ((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) + ((dst_negative_scale_omit_resultright) + (dst_negative_scale_omit_resultright))) + (((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) * S ((dst_negative_code_omit_resultright) + (dst_negative_scale_omit_resultright)) + ((dst_negative_scale_omit_resultright) + (dst_negative_scale_omit_resultright)))))) /\ (((((exists ff_h_pvs_omit_resultrightpositive. ff_h_pvs_omit_resultrightpositive + S (dst_positive_omit_resultright) = S ((S (dc_quotient_omit_result)) * dst_positive_scale_omit_resultright)) /\ exists ff_q_pvs_omit_resultrightpositive. dst_positive_code_omit_resultright = ff_q_pvs_omit_resultrightpositive * S ((S (dc_quotient_omit_result)) * dst_positive_scale_omit_resultright) + (dst_positive_omit_resultright))) /\ (((((exists ff_h_pvs_omit_resultrightnegative. ff_h_pvs_omit_resultrightnegative + S (dst_negative_omit_resultright) = S ((S (dc_quotient_omit_result)) * dst_negative_scale_omit_resultright)) /\ exists ff_q_pvs_omit_resultrightnegative. dst_negative_code_omit_resultright = ff_q_pvs_omit_resultrightnegative * S ((S (dc_quotient_omit_result)) * dst_negative_scale_omit_resultright) + (dst_negative_omit_resultright))) /\ (exists ge_balance_positive_omit_resultrightvalue ge_balance_negative_omit_resultrightvalue. (((((dc_right_omit_result) = 2 * (ge_balance_positive_omit_resultrightvalue) /\ (ge_balance_negative_omit_resultrightvalue) = 0) \/ exists ge_signed_half_omit_resultrightvaluedecode. (((dc_right_omit_result) = 2 * ge_signed_half_omit_resultrightvaluedecode + 1 /\ (ge_balance_positive_omit_resultrightvalue) = 0) /\ (ge_balance_negative_omit_resultrightvalue) = S ge_signed_half_omit_resultrightvaluedecode))) /\ ((dst_positive_omit_resultright) + ge_balance_negative_omit_resultrightvalue = (dst_negative_omit_resultright) + ge_balance_positive_omit_resultrightvalue))))))))) /\ (exists sto_ap_omit_resultproduct sto_an_omit_resultproduct sto_bp_omit_resultproduct sto_bn_omit_resultproduct sto_cp_omit_resultproduct sto_cn_omit_resultproduct. (((((dc_left_omit_result) = 2 * (sto_ap_omit_resultproduct) /\ (sto_an_omit_resultproduct) = 0) \/ exists ge_signed_half_omit_resultproductleft. (((dc_left_omit_result) = 2 * ge_signed_half_omit_resultproductleft + 1 /\ (sto_ap_omit_resultproduct) = 0) /\ (sto_an_omit_resultproduct) = S ge_signed_half_omit_resultproductleft))) /\ ((((((dc_right_omit_result) = 2 * (sto_bp_omit_resultproduct) /\ (sto_bn_omit_resultproduct) = 0) \/ exists ge_signed_half_omit_resultproductright. (((dc_right_omit_result) = 2 * ge_signed_half_omit_resultproductright + 1 /\ (sto_bp_omit_resultproduct) = 0) /\ (sto_bn_omit_resultproduct) = S ge_signed_half_omit_resultproductright))) /\ ((((((0) = 2 * (sto_cp_omit_resultproduct) /\ (sto_cn_omit_resultproduct) = 0) \/ exists ge_signed_half_omit_resultproductoutput. (((0) = 2 * ge_signed_half_omit_resultproductoutput + 1 /\ (sto_cp_omit_resultproduct) = 0) /\ (sto_cn_omit_resultproduct) = S ge_signed_half_omit_resultproductoutput))) /\ ((sto_ap_omit_resultproduct * sto_bp_omit_resultproduct + sto_an_omit_resultproduct * sto_bn_omit_resultproduct) + sto_cn_omit_resultproduct = (sto_ap_omit_resultproduct * sto_bn_omit_resultproduct + sto_an_omit_resultproduct * sto_bp_omit_resultproduct) + sto_cp_omit_resultproduct))))))))))))))) \/ ((((d)=0 \/ ~(exists pvs_factor_omit_resultnondivisor. (n) = (d) * pvs_factor_omit_resultnondivisor)) /\ ((0)=0))))Constructive proof overview
Generated structural guide
A proved nondivisor contributes zero independently of both input tables.
The unchanged tactic script uses 0 declared prerequisites and contains 10 exact native proof lines.
Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
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.
01Fix variables and assumptionsL1–5
02Separate the logical casesL6–8
03Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
exact hd
04Calculate and transport equalitiesL10–10
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L10
refl