WS0007

signed_table_multiply_lookup

Every supplied canonical lookup value satisfies the actual multiply graph, by lookup functionality and the witnessed pointwise entries.

Alpha v34 checked-use · first admitted v31 · 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.

Operation tables contain actual beta-coded entries and compare represented signed values, not encodings. The strict sum window is i<l and the separately certified endpoint i=l is unused. Rectangular Fubini and full finite signed Möbius inversion are separate, now-admitted families.

Exact theorem in conservative defined notation

∀ F. ∀ G. ∀ H. ∀ l. ∀ i. ∀ a. ∀ b. ∀ c. ArithMul(F,G,H,l) → Lt(i,l) → ArithAt(F,i,a) → ArithAt(G,i,b) → ArithAt(H,i,c) → SignedMul(a,b,c)

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

Definition DAG

SignedMul(a,b,c) · 2ArithAt(F,i,z) · 6ArithMul(F,G,H,l) · 1Lt(a,b) · 1

Actual proof prerequisites

divisor_signed_table_at_functional · checked external prerequisite
Original expanded first-order statement 
forall F G H l i a b c. (((exists dst_positive_code_multiply_lookup_relationleft_table dst_positive_scale_multiply_lookup_relationleft_table dst_negative_code_multiply_lookup_relationleft_table dst_negative_scale_multiply_lookup_relationleft_table. (((F) = (((((dst_positive_code_multiply_lookup_relationleft_table) + (dst_positive_scale_multiply_lookup_relationleft_table)) * S ((dst_positive_code_multiply_lookup_relationleft_table) + (dst_positive_scale_multiply_lookup_relationleft_table)) + ((dst_positive_scale_multiply_lookup_relationleft_table) + (dst_positive_scale_multiply_lookup_relationleft_table))) + (((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) * S ((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) + ((dst_negative_scale_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)))) * S ((((dst_positive_code_multiply_lookup_relationleft_table) + (dst_positive_scale_multiply_lookup_relationleft_table)) * S ((dst_positive_code_multiply_lookup_relationleft_table) + (dst_positive_scale_multiply_lookup_relationleft_table)) + ((dst_positive_scale_multiply_lookup_relationleft_table) + (dst_positive_scale_multiply_lookup_relationleft_table))) + (((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) * S ((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) + ((dst_negative_scale_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)))) + ((((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) * S ((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) + ((dst_negative_scale_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table))) + (((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) * S ((dst_negative_code_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)) + ((dst_negative_scale_multiply_lookup_relationleft_table) + (dst_negative_scale_multiply_lookup_relationleft_table)))))) /\ (forall dst_index_multiply_lookup_relationleft_table. (exists pvs_le_gap_multiply_lookup_relationleft_tabledomain. pvs_le_gap_multiply_lookup_relationleft_tabledomain + (dst_index_multiply_lookup_relationleft_table) = (l)) -> exists dst_positive_multiply_lookup_relationleft_table dst_negative_multiply_lookup_relationleft_table dst_value_multiply_lookup_relationleft_table. ((((exists ff_h_pvs_multiply_lookup_relationleft_tableentrypositive. ff_h_pvs_multiply_lookup_relationleft_tableentrypositive + S (dst_positive_multiply_lookup_relationleft_table) = S ((S (dst_index_multiply_lookup_relationleft_table)) * dst_positive_scale_multiply_lookup_relationleft_table)) /\ exists ff_q_pvs_multiply_lookup_relationleft_tableentrypositive. dst_positive_code_multiply_lookup_relationleft_table = ff_q_pvs_multiply_lookup_relationleft_tableentrypositive * S ((S (dst_index_multiply_lookup_relationleft_table)) * dst_positive_scale_multiply_lookup_relationleft_table) + (dst_positive_multiply_lookup_relationleft_table))) /\ (((((exists ff_h_pvs_multiply_lookup_relationleft_tableentrynegative. ff_h_pvs_multiply_lookup_relationleft_tableentrynegative + S (dst_negative_multiply_lookup_relationleft_table) = S ((S (dst_index_multiply_lookup_relationleft_table)) * dst_negative_scale_multiply_lookup_relationleft_table)) /\ exists ff_q_pvs_multiply_lookup_relationleft_tableentrynegative. dst_negative_code_multiply_lookup_relationleft_table = ff_q_pvs_multiply_lookup_relationleft_tableentrynegative * S ((S (dst_index_multiply_lookup_relationleft_table)) * dst_negative_scale_multiply_lookup_relationleft_table) + (dst_negative_multiply_lookup_relationleft_table))) /\ (exists ge_balance_positive_multiply_lookup_relationleft_tableentryvalue ge_balance_negative_multiply_lookup_relationleft_tableentryvalue. (((((dst_value_multiply_lookup_relationleft_table) = 2 * (ge_balance_positive_multiply_lookup_relationleft_tableentryvalue) /\ (ge_balance_negative_multiply_lookup_relationleft_tableentryvalue) = 0) \/ exists ge_signed_half_multiply_lookup_relationleft_tableentryvaluedecode. (((dst_value_multiply_lookup_relationleft_table) = 2 * ge_signed_half_multiply_lookup_relationleft_tableentryvaluedecode + 1 /\ (ge_balance_positive_multiply_lookup_relationleft_tableentryvalue) = 0) /\ (ge_balance_negative_multiply_lookup_relationleft_tableentryvalue) = S ge_signed_half_multiply_lookup_relationleft_tableentryvaluedecode))) /\ ((dst_positive_multiply_lookup_relationleft_table) + ge_balance_negative_multiply_lookup_relationleft_tableentryvalue = (dst_negative_multiply_lookup_relationleft_table) + ge_balance_positive_multiply_lookup_relationleft_tableentryvalue))))))))) /\ (((exists dst_positive_code_multiply_lookup_relationright_table dst_positive_scale_multiply_lookup_relationright_table dst_negative_code_multiply_lookup_relationright_table dst_negative_scale_multiply_lookup_relationright_table. (((G) = (((((dst_positive_code_multiply_lookup_relationright_table) + (dst_positive_scale_multiply_lookup_relationright_table)) * S ((dst_positive_code_multiply_lookup_relationright_table) + (dst_positive_scale_multiply_lookup_relationright_table)) + ((dst_positive_scale_multiply_lookup_relationright_table) + (dst_positive_scale_multiply_lookup_relationright_table))) + (((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) * S ((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) + ((dst_negative_scale_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)))) * S ((((dst_positive_code_multiply_lookup_relationright_table) + (dst_positive_scale_multiply_lookup_relationright_table)) * S ((dst_positive_code_multiply_lookup_relationright_table) + (dst_positive_scale_multiply_lookup_relationright_table)) + ((dst_positive_scale_multiply_lookup_relationright_table) + (dst_positive_scale_multiply_lookup_relationright_table))) + (((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) * S ((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) + ((dst_negative_scale_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)))) + ((((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) * S ((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) + ((dst_negative_scale_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table))) + (((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) * S ((dst_negative_code_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)) + ((dst_negative_scale_multiply_lookup_relationright_table) + (dst_negative_scale_multiply_lookup_relationright_table)))))) /\ (forall dst_index_multiply_lookup_relationright_table. (exists pvs_le_gap_multiply_lookup_relationright_tabledomain. pvs_le_gap_multiply_lookup_relationright_tabledomain + (dst_index_multiply_lookup_relationright_table) = (l)) -> exists dst_positive_multiply_lookup_relationright_table dst_negative_multiply_lookup_relationright_table dst_value_multiply_lookup_relationright_table. ((((exists ff_h_pvs_multiply_lookup_relationright_tableentrypositive. ff_h_pvs_multiply_lookup_relationright_tableentrypositive + S (dst_positive_multiply_lookup_relationright_table) = S ((S (dst_index_multiply_lookup_relationright_table)) * dst_positive_scale_multiply_lookup_relationright_table)) /\ exists ff_q_pvs_multiply_lookup_relationright_tableentrypositive. dst_positive_code_multiply_lookup_relationright_table = ff_q_pvs_multiply_lookup_relationright_tableentrypositive * S ((S (dst_index_multiply_lookup_relationright_table)) * dst_positive_scale_multiply_lookup_relationright_table) + (dst_positive_multiply_lookup_relationright_table))) /\ (((((exists ff_h_pvs_multiply_lookup_relationright_tableentrynegative. ff_h_pvs_multiply_lookup_relationright_tableentrynegative + S (dst_negative_multiply_lookup_relationright_table) = S ((S (dst_index_multiply_lookup_relationright_table)) * dst_negative_scale_multiply_lookup_relationright_table)) /\ exists ff_q_pvs_multiply_lookup_relationright_tableentrynegative. dst_negative_code_multiply_lookup_relationright_table = ff_q_pvs_multiply_lookup_relationright_tableentrynegative * S ((S (dst_index_multiply_lookup_relationright_table)) * dst_negative_scale_multiply_lookup_relationright_table) + (dst_negative_multiply_lookup_relationright_table))) /\ (exists ge_balance_positive_multiply_lookup_relationright_tableentryvalue ge_balance_negative_multiply_lookup_relationright_tableentryvalue. (((((dst_value_multiply_lookup_relationright_table) = 2 * (ge_balance_positive_multiply_lookup_relationright_tableentryvalue) /\ (ge_balance_negative_multiply_lookup_relationright_tableentryvalue) = 0) \/ exists ge_signed_half_multiply_lookup_relationright_tableentryvaluedecode. (((dst_value_multiply_lookup_relationright_table) = 2 * ge_signed_half_multiply_lookup_relationright_tableentryvaluedecode + 1 /\ (ge_balance_positive_multiply_lookup_relationright_tableentryvalue) = 0) /\ (ge_balance_negative_multiply_lookup_relationright_tableentryvalue) = S ge_signed_half_multiply_lookup_relationright_tableentryvaluedecode))) /\ ((dst_positive_multiply_lookup_relationright_table) + ge_balance_negative_multiply_lookup_relationright_tableentryvalue = (dst_negative_multiply_lookup_relationright_table) + ge_balance_positive_multiply_lookup_relationright_tableentryvalue))))))))) /\ (((exists dst_positive_code_multiply_lookup_relationoutput_table dst_positive_scale_multiply_lookup_relationoutput_table dst_negative_code_multiply_lookup_relationoutput_table dst_negative_scale_multiply_lookup_relationoutput_table. (((H) = (((((dst_positive_code_multiply_lookup_relationoutput_table) + (dst_positive_scale_multiply_lookup_relationoutput_table)) * S ((dst_positive_code_multiply_lookup_relationoutput_table) + (dst_positive_scale_multiply_lookup_relationoutput_table)) + ((dst_positive_scale_multiply_lookup_relationoutput_table) + (dst_positive_scale_multiply_lookup_relationoutput_table))) + (((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) * S ((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) + ((dst_negative_scale_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)))) * S ((((dst_positive_code_multiply_lookup_relationoutput_table) + (dst_positive_scale_multiply_lookup_relationoutput_table)) * S ((dst_positive_code_multiply_lookup_relationoutput_table) + (dst_positive_scale_multiply_lookup_relationoutput_table)) + ((dst_positive_scale_multiply_lookup_relationoutput_table) + (dst_positive_scale_multiply_lookup_relationoutput_table))) + (((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) * S ((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) + ((dst_negative_scale_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)))) + ((((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) * S ((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) + ((dst_negative_scale_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table))) + (((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) * S ((dst_negative_code_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)) + ((dst_negative_scale_multiply_lookup_relationoutput_table) + (dst_negative_scale_multiply_lookup_relationoutput_table)))))) /\ (forall dst_index_multiply_lookup_relationoutput_table. (exists pvs_le_gap_multiply_lookup_relationoutput_tabledomain. pvs_le_gap_multiply_lookup_relationoutput_tabledomain + (dst_index_multiply_lookup_relationoutput_table) = (l)) -> exists dst_positive_multiply_lookup_relationoutput_table dst_negative_multiply_lookup_relationoutput_table dst_value_multiply_lookup_relationoutput_table. ((((exists ff_h_pvs_multiply_lookup_relationoutput_tableentrypositive. ff_h_pvs_multiply_lookup_relationoutput_tableentrypositive + S (dst_positive_multiply_lookup_relationoutput_table) = S ((S (dst_index_multiply_lookup_relationoutput_table)) * dst_positive_scale_multiply_lookup_relationoutput_table)) /\ exists ff_q_pvs_multiply_lookup_relationoutput_tableentrypositive. dst_positive_code_multiply_lookup_relationoutput_table = ff_q_pvs_multiply_lookup_relationoutput_tableentrypositive * S ((S (dst_index_multiply_lookup_relationoutput_table)) * dst_positive_scale_multiply_lookup_relationoutput_table) + (dst_positive_multiply_lookup_relationoutput_table))) /\ (((((exists ff_h_pvs_multiply_lookup_relationoutput_tableentrynegative. ff_h_pvs_multiply_lookup_relationoutput_tableentrynegative + S (dst_negative_multiply_lookup_relationoutput_table) = S ((S (dst_index_multiply_lookup_relationoutput_table)) * dst_negative_scale_multiply_lookup_relationoutput_table)) /\ exists ff_q_pvs_multiply_lookup_relationoutput_tableentrynegative. dst_negative_code_multiply_lookup_relationoutput_table = ff_q_pvs_multiply_lookup_relationoutput_tableentrynegative * S ((S (dst_index_multiply_lookup_relationoutput_table)) * dst_negative_scale_multiply_lookup_relationoutput_table) + (dst_negative_multiply_lookup_relationoutput_table))) /\ (exists ge_balance_positive_multiply_lookup_relationoutput_tableentryvalue ge_balance_negative_multiply_lookup_relationoutput_tableentryvalue. (((((dst_value_multiply_lookup_relationoutput_table) = 2 * (ge_balance_positive_multiply_lookup_relationoutput_tableentryvalue) /\ (ge_balance_negative_multiply_lookup_relationoutput_tableentryvalue) = 0) \/ exists ge_signed_half_multiply_lookup_relationoutput_tableentryvaluedecode. (((dst_value_multiply_lookup_relationoutput_table) = 2 * ge_signed_half_multiply_lookup_relationoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_multiply_lookup_relationoutput_tableentryvalue) = 0) /\ (ge_balance_negative_multiply_lookup_relationoutput_tableentryvalue) = S ge_signed_half_multiply_lookup_relationoutput_tableentryvaluedecode))) /\ ((dst_positive_multiply_lookup_relationoutput_table) + ge_balance_negative_multiply_lookup_relationoutput_tableentryvalue = (dst_negative_multiply_lookup_relationoutput_table) + ge_balance_positive_multiply_lookup_relationoutput_tableentryvalue))))))))) /\ (forall sto_index_multiply_lookup_relationentries. (exists pvs_gap_multiply_lookup_relationentriesbound. pvs_gap_multiply_lookup_relationentriesbound + S (sto_index_multiply_lookup_relationentries) = (l)) -> exists sto_left_multiply_lookup_relationentries sto_right_multiply_lookup_relationentries sto_output_multiply_lookup_relationentries. ((exists dst_positive_code_multiply_lookup_relationentriesentryleft dst_positive_scale_multiply_lookup_relationentriesentryleft dst_negative_code_multiply_lookup_relationentriesentryleft dst_negative_scale_multiply_lookup_relationentriesentryleft dst_positive_multiply_lookup_relationentriesentryleft dst_negative_multiply_lookup_relationentriesentryleft. (((F) = (((((dst_positive_code_multiply_lookup_relationentriesentryleft) + (dst_positive_scale_multiply_lookup_relationentriesentryleft)) * S ((dst_positive_code_multiply_lookup_relationentriesentryleft) + (dst_positive_scale_multiply_lookup_relationentriesentryleft)) + ((dst_positive_scale_multiply_lookup_relationentriesentryleft) + (dst_positive_scale_multiply_lookup_relationentriesentryleft))) + (((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) * S ((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) + ((dst_negative_scale_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)))) * S ((((dst_positive_code_multiply_lookup_relationentriesentryleft) + (dst_positive_scale_multiply_lookup_relationentriesentryleft)) * S ((dst_positive_code_multiply_lookup_relationentriesentryleft) + (dst_positive_scale_multiply_lookup_relationentriesentryleft)) + ((dst_positive_scale_multiply_lookup_relationentriesentryleft) + (dst_positive_scale_multiply_lookup_relationentriesentryleft))) + (((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) * S ((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) + ((dst_negative_scale_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)))) + ((((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) * S ((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) + ((dst_negative_scale_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft))) + (((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) * S ((dst_negative_code_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)) + ((dst_negative_scale_multiply_lookup_relationentriesentryleft) + (dst_negative_scale_multiply_lookup_relationentriesentryleft)))))) /\ (((((exists ff_h_pvs_multiply_lookup_relationentriesentryleftpositive. ff_h_pvs_multiply_lookup_relationentriesentryleftpositive + S (dst_positive_multiply_lookup_relationentriesentryleft) = S ((S (sto_index_multiply_lookup_relationentries)) * dst_positive_scale_multiply_lookup_relationentriesentryleft)) /\ exists ff_q_pvs_multiply_lookup_relationentriesentryleftpositive. dst_positive_code_multiply_lookup_relationentriesentryleft = ff_q_pvs_multiply_lookup_relationentriesentryleftpositive * S ((S (sto_index_multiply_lookup_relationentries)) * dst_positive_scale_multiply_lookup_relationentriesentryleft) + (dst_positive_multiply_lookup_relationentriesentryleft))) /\ (((((exists ff_h_pvs_multiply_lookup_relationentriesentryleftnegative. ff_h_pvs_multiply_lookup_relationentriesentryleftnegative + S (dst_negative_multiply_lookup_relationentriesentryleft) = S ((S (sto_index_multiply_lookup_relationentries)) * dst_negative_scale_multiply_lookup_relationentriesentryleft)) /\ exists ff_q_pvs_multiply_lookup_relationentriesentryleftnegative. dst_negative_code_multiply_lookup_relationentriesentryleft = ff_q_pvs_multiply_lookup_relationentriesentryleftnegative * S ((S (sto_index_multiply_lookup_relationentries)) * dst_negative_scale_multiply_lookup_relationentriesentryleft) + (dst_negative_multiply_lookup_relationentriesentryleft))) /\ (exists ge_balance_positive_multiply_lookup_relationentriesentryleftvalue ge_balance_negative_multiply_lookup_relationentriesentryleftvalue. (((((sto_left_multiply_lookup_relationentries) = 2 * (ge_balance_positive_multiply_lookup_relationentriesentryleftvalue) /\ (ge_balance_negative_multiply_lookup_relationentriesentryleftvalue) = 0) \/ exists ge_signed_half_multiply_lookup_relationentriesentryleftvaluedecode. (((sto_left_multiply_lookup_relationentries) = 2 * ge_signed_half_multiply_lookup_relationentriesentryleftvaluedecode + 1 /\ (ge_balance_positive_multiply_lookup_relationentriesentryleftvalue) = 0) /\ (ge_balance_negative_multiply_lookup_relationentriesentryleftvalue) = S ge_signed_half_multiply_lookup_relationentriesentryleftvaluedecode))) /\ ((dst_positive_multiply_lookup_relationentriesentryleft) + ge_balance_negative_multiply_lookup_relationentriesentryleftvalue = (dst_negative_multiply_lookup_relationentriesentryleft) + ge_balance_positive_multiply_lookup_relationentriesentryleftvalue))))))))) /\ (((exists dst_positive_code_multiply_lookup_relationentriesentryright dst_positive_scale_multiply_lookup_relationentriesentryright dst_negative_code_multiply_lookup_relationentriesentryright dst_negative_scale_multiply_lookup_relationentriesentryright dst_positive_multiply_lookup_relationentriesentryright dst_negative_multiply_lookup_relationentriesentryright. (((G) = (((((dst_positive_code_multiply_lookup_relationentriesentryright) + (dst_positive_scale_multiply_lookup_relationentriesentryright)) * S ((dst_positive_code_multiply_lookup_relationentriesentryright) + (dst_positive_scale_multiply_lookup_relationentriesentryright)) + ((dst_positive_scale_multiply_lookup_relationentriesentryright) + (dst_positive_scale_multiply_lookup_relationentriesentryright))) + (((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) * S ((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) + ((dst_negative_scale_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)))) * S ((((dst_positive_code_multiply_lookup_relationentriesentryright) + (dst_positive_scale_multiply_lookup_relationentriesentryright)) * S ((dst_positive_code_multiply_lookup_relationentriesentryright) + (dst_positive_scale_multiply_lookup_relationentriesentryright)) + ((dst_positive_scale_multiply_lookup_relationentriesentryright) + (dst_positive_scale_multiply_lookup_relationentriesentryright))) + (((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) * S ((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) + ((dst_negative_scale_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)))) + ((((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) * S ((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) + ((dst_negative_scale_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright))) + (((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) * S ((dst_negative_code_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)) + ((dst_negative_scale_multiply_lookup_relationentriesentryright) + (dst_negative_scale_multiply_lookup_relationentriesentryright)))))) /\ (((((exists ff_h_pvs_multiply_lookup_relationentriesentryrightpositive. ff_h_pvs_multiply_lookup_relationentriesentryrightpositive + S (dst_positive_multiply_lookup_relationentriesentryright) = S ((S (sto_index_multiply_lookup_relationentries)) * dst_positive_scale_multiply_lookup_relationentriesentryright)) /\ exists ff_q_pvs_multiply_lookup_relationentriesentryrightpositive. dst_positive_code_multiply_lookup_relationentriesentryright = ff_q_pvs_multiply_lookup_relationentriesentryrightpositive * S ((S (sto_index_multiply_lookup_relationentries)) * dst_positive_scale_multiply_lookup_relationentriesentryright) + (dst_positive_multiply_lookup_relationentriesentryright))) /\ (((((exists ff_h_pvs_multiply_lookup_relationentriesentryrightnegative. ff_h_pvs_multiply_lookup_relationentriesentryrightnegative + S (dst_negative_multiply_lookup_relationentriesentryright) = S ((S (sto_index_multiply_lookup_relationentries)) * dst_negative_scale_multiply_lookup_relationentriesentryright)) /\ exists ff_q_pvs_multiply_lookup_relationentriesentryrightnegative. dst_negative_code_multiply_lookup_relationentriesentryright = ff_q_pvs_multiply_lookup_relationentriesentryrightnegative * S ((S (sto_index_multiply_lookup_relationentries)) * dst_negative_scale_multiply_lookup_relationentriesentryright) + (dst_negative_multiply_lookup_relationentriesentryright))) /\ (exists ge_balance_positive_multiply_lookup_relationentriesentryrightvalue ge_balance_negative_multiply_lookup_relationentriesentryrightvalue. (((((sto_right_multiply_lookup_relationentries) = 2 * (ge_balance_positive_multiply_lookup_relationentriesentryrightvalue) /\ (ge_balance_negative_multiply_lookup_relationentriesentryrightvalue) = 0) \/ exists ge_signed_half_multiply_lookup_relationentriesentryrightvaluedecode. (((sto_right_multiply_lookup_relationentries) = 2 * ge_signed_half_multiply_lookup_relationentriesentryrightvaluedecode + 1 /\ (ge_balance_positive_multiply_lookup_relationentriesentryrightvalue) = 0) /\ (ge_balance_negative_multiply_lookup_relationentriesentryrightvalue) = S ge_signed_half_multiply_lookup_relationentriesentryrightvaluedecode))) /\ ((dst_positive_multiply_lookup_relationentriesentryright) + ge_balance_negative_multiply_lookup_relationentriesentryrightvalue = (dst_negative_multiply_lookup_relationentriesentryright) + ge_balance_positive_multiply_lookup_relationentriesentryrightvalue))))))))) /\ (((exists dst_positive_code_multiply_lookup_relationentriesentryoutput dst_positive_scale_multiply_lookup_relationentriesentryoutput dst_negative_code_multiply_lookup_relationentriesentryoutput dst_negative_scale_multiply_lookup_relationentriesentryoutput dst_positive_multiply_lookup_relationentriesentryoutput dst_negative_multiply_lookup_relationentriesentryoutput. (((H) = (((((dst_positive_code_multiply_lookup_relationentriesentryoutput) + (dst_positive_scale_multiply_lookup_relationentriesentryoutput)) * S ((dst_positive_code_multiply_lookup_relationentriesentryoutput) + (dst_positive_scale_multiply_lookup_relationentriesentryoutput)) + ((dst_positive_scale_multiply_lookup_relationentriesentryoutput) + (dst_positive_scale_multiply_lookup_relationentriesentryoutput))) + (((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) * S ((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) + ((dst_negative_scale_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)))) * S ((((dst_positive_code_multiply_lookup_relationentriesentryoutput) + (dst_positive_scale_multiply_lookup_relationentriesentryoutput)) * S ((dst_positive_code_multiply_lookup_relationentriesentryoutput) + (dst_positive_scale_multiply_lookup_relationentriesentryoutput)) + ((dst_positive_scale_multiply_lookup_relationentriesentryoutput) + (dst_positive_scale_multiply_lookup_relationentriesentryoutput))) + (((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) * S ((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) + ((dst_negative_scale_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)))) + ((((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) * S ((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) + ((dst_negative_scale_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput))) + (((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) * S ((dst_negative_code_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)) + ((dst_negative_scale_multiply_lookup_relationentriesentryoutput) + (dst_negative_scale_multiply_lookup_relationentriesentryoutput)))))) /\ (((((exists ff_h_pvs_multiply_lookup_relationentriesentryoutputpositive. ff_h_pvs_multiply_lookup_relationentriesentryoutputpositive + S (dst_positive_multiply_lookup_relationentriesentryoutput) = S ((S (sto_index_multiply_lookup_relationentries)) * dst_positive_scale_multiply_lookup_relationentriesentryoutput)) /\ exists ff_q_pvs_multiply_lookup_relationentriesentryoutputpositive. dst_positive_code_multiply_lookup_relationentriesentryoutput = ff_q_pvs_multiply_lookup_relationentriesentryoutputpositive * S ((S (sto_index_multiply_lookup_relationentries)) * dst_positive_scale_multiply_lookup_relationentriesentryoutput) + (dst_positive_multiply_lookup_relationentriesentryoutput))) /\ (((((exists ff_h_pvs_multiply_lookup_relationentriesentryoutputnegative. ff_h_pvs_multiply_lookup_relationentriesentryoutputnegative + S (dst_negative_multiply_lookup_relationentriesentryoutput) = S ((S (sto_index_multiply_lookup_relationentries)) * dst_negative_scale_multiply_lookup_relationentriesentryoutput)) /\ exists ff_q_pvs_multiply_lookup_relationentriesentryoutputnegative. dst_negative_code_multiply_lookup_relationentriesentryoutput = ff_q_pvs_multiply_lookup_relationentriesentryoutputnegative * S ((S (sto_index_multiply_lookup_relationentries)) * dst_negative_scale_multiply_lookup_relationentriesentryoutput) + (dst_negative_multiply_lookup_relationentriesentryoutput))) /\ (exists ge_balance_positive_multiply_lookup_relationentriesentryoutputvalue ge_balance_negative_multiply_lookup_relationentriesentryoutputvalue. (((((sto_output_multiply_lookup_relationentries) = 2 * (ge_balance_positive_multiply_lookup_relationentriesentryoutputvalue) /\ (ge_balance_negative_multiply_lookup_relationentriesentryoutputvalue) = 0) \/ exists ge_signed_half_multiply_lookup_relationentriesentryoutputvaluedecode. (((sto_output_multiply_lookup_relationentries) = 2 * ge_signed_half_multiply_lookup_relationentriesentryoutputvaluedecode + 1 /\ (ge_balance_positive_multiply_lookup_relationentriesentryoutputvalue) = 0) /\ (ge_balance_negative_multiply_lookup_relationentriesentryoutputvalue) = S ge_signed_half_multiply_lookup_relationentriesentryoutputvaluedecode))) /\ ((dst_positive_multiply_lookup_relationentriesentryoutput) + ge_balance_negative_multiply_lookup_relationentriesentryoutputvalue = (dst_negative_multiply_lookup_relationentriesentryoutput) + ge_balance_positive_multiply_lookup_relationentriesentryoutputvalue))))))))) /\ (exists sto_ap_multiply_lookup_relationentriesentryoperation sto_an_multiply_lookup_relationentriesentryoperation sto_bp_multiply_lookup_relationentriesentryoperation sto_bn_multiply_lookup_relationentriesentryoperation sto_cp_multiply_lookup_relationentriesentryoperation sto_cn_multiply_lookup_relationentriesentryoperation. (((((sto_left_multiply_lookup_relationentries) = 2 * (sto_ap_multiply_lookup_relationentriesentryoperation) /\ (sto_an_multiply_lookup_relationentriesentryoperation) = 0) \/ exists ge_signed_half_multiply_lookup_relationentriesentryoperationleft. (((sto_left_multiply_lookup_relationentries) = 2 * ge_signed_half_multiply_lookup_relationentriesentryoperationleft + 1 /\ (sto_ap_multiply_lookup_relationentriesentryoperation) = 0) /\ (sto_an_multiply_lookup_relationentriesentryoperation) = S ge_signed_half_multiply_lookup_relationentriesentryoperationleft))) /\ ((((((sto_right_multiply_lookup_relationentries) = 2 * (sto_bp_multiply_lookup_relationentriesentryoperation) /\ (sto_bn_multiply_lookup_relationentriesentryoperation) = 0) \/ exists ge_signed_half_multiply_lookup_relationentriesentryoperationright. (((sto_right_multiply_lookup_relationentries) = 2 * ge_signed_half_multiply_lookup_relationentriesentryoperationright + 1 /\ (sto_bp_multiply_lookup_relationentriesentryoperation) = 0) /\ (sto_bn_multiply_lookup_relationentriesentryoperation) = S ge_signed_half_multiply_lookup_relationentriesentryoperationright))) /\ ((((((sto_output_multiply_lookup_relationentries) = 2 * (sto_cp_multiply_lookup_relationentriesentryoperation) /\ (sto_cn_multiply_lookup_relationentriesentryoperation) = 0) \/ exists ge_signed_half_multiply_lookup_relationentriesentryoperationoutput. (((sto_output_multiply_lookup_relationentries) = 2 * ge_signed_half_multiply_lookup_relationentriesentryoperationoutput + 1 /\ (sto_cp_multiply_lookup_relationentriesentryoperation) = 0) /\ (sto_cn_multiply_lookup_relationentriesentryoperation) = S ge_signed_half_multiply_lookup_relationentriesentryoperationoutput))) /\ ((sto_ap_multiply_lookup_relationentriesentryoperation * sto_bp_multiply_lookup_relationentriesentryoperation + sto_an_multiply_lookup_relationentriesentryoperation * sto_bn_multiply_lookup_relationentriesentryoperation) + sto_cn_multiply_lookup_relationentriesentryoperation = (sto_ap_multiply_lookup_relationentriesentryoperation * sto_bn_multiply_lookup_relationentriesentryoperation + sto_an_multiply_lookup_relationentriesentryoperation * sto_bp_multiply_lookup_relationentriesentryoperation) + sto_cp_multiply_lookup_relationentriesentryoperation))))))))))))))))))) -> (exists pvs_gap_multiply_lookup_bound. pvs_gap_multiply_lookup_bound + S (i) = (l)) -> (exists dst_positive_code_multiply_lookup_0 dst_positive_scale_multiply_lookup_0 dst_negative_code_multiply_lookup_0 dst_negative_scale_multiply_lookup_0 dst_positive_multiply_lookup_0 dst_negative_multiply_lookup_0. (((F) = (((((dst_positive_code_multiply_lookup_0) + (dst_positive_scale_multiply_lookup_0)) * S ((dst_positive_code_multiply_lookup_0) + (dst_positive_scale_multiply_lookup_0)) + ((dst_positive_scale_multiply_lookup_0) + (dst_positive_scale_multiply_lookup_0))) + (((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) * S ((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) + ((dst_negative_scale_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)))) * S ((((dst_positive_code_multiply_lookup_0) + (dst_positive_scale_multiply_lookup_0)) * S ((dst_positive_code_multiply_lookup_0) + (dst_positive_scale_multiply_lookup_0)) + ((dst_positive_scale_multiply_lookup_0) + (dst_positive_scale_multiply_lookup_0))) + (((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) * S ((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) + ((dst_negative_scale_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)))) + ((((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) * S ((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) + ((dst_negative_scale_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0))) + (((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) * S ((dst_negative_code_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)) + ((dst_negative_scale_multiply_lookup_0) + (dst_negative_scale_multiply_lookup_0)))))) /\ (((((exists ff_h_pvs_multiply_lookup_0positive. ff_h_pvs_multiply_lookup_0positive + S (dst_positive_multiply_lookup_0) = S ((S (i)) * dst_positive_scale_multiply_lookup_0)) /\ exists ff_q_pvs_multiply_lookup_0positive. dst_positive_code_multiply_lookup_0 = ff_q_pvs_multiply_lookup_0positive * S ((S (i)) * dst_positive_scale_multiply_lookup_0) + (dst_positive_multiply_lookup_0))) /\ (((((exists ff_h_pvs_multiply_lookup_0negative. ff_h_pvs_multiply_lookup_0negative + S (dst_negative_multiply_lookup_0) = S ((S (i)) * dst_negative_scale_multiply_lookup_0)) /\ exists ff_q_pvs_multiply_lookup_0negative. dst_negative_code_multiply_lookup_0 = ff_q_pvs_multiply_lookup_0negative * S ((S (i)) * dst_negative_scale_multiply_lookup_0) + (dst_negative_multiply_lookup_0))) /\ (exists ge_balance_positive_multiply_lookup_0value ge_balance_negative_multiply_lookup_0value. (((((a) = 2 * (ge_balance_positive_multiply_lookup_0value) /\ (ge_balance_negative_multiply_lookup_0value) = 0) \/ exists ge_signed_half_multiply_lookup_0valuedecode. (((a) = 2 * ge_signed_half_multiply_lookup_0valuedecode + 1 /\ (ge_balance_positive_multiply_lookup_0value) = 0) /\ (ge_balance_negative_multiply_lookup_0value) = S ge_signed_half_multiply_lookup_0valuedecode))) /\ ((dst_positive_multiply_lookup_0) + ge_balance_negative_multiply_lookup_0value = (dst_negative_multiply_lookup_0) + ge_balance_positive_multiply_lookup_0value))))))))) -> (exists dst_positive_code_multiply_lookup_1 dst_positive_scale_multiply_lookup_1 dst_negative_code_multiply_lookup_1 dst_negative_scale_multiply_lookup_1 dst_positive_multiply_lookup_1 dst_negative_multiply_lookup_1. (((G) = (((((dst_positive_code_multiply_lookup_1) + (dst_positive_scale_multiply_lookup_1)) * S ((dst_positive_code_multiply_lookup_1) + (dst_positive_scale_multiply_lookup_1)) + ((dst_positive_scale_multiply_lookup_1) + (dst_positive_scale_multiply_lookup_1))) + (((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) * S ((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) + ((dst_negative_scale_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)))) * S ((((dst_positive_code_multiply_lookup_1) + (dst_positive_scale_multiply_lookup_1)) * S ((dst_positive_code_multiply_lookup_1) + (dst_positive_scale_multiply_lookup_1)) + ((dst_positive_scale_multiply_lookup_1) + (dst_positive_scale_multiply_lookup_1))) + (((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) * S ((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) + ((dst_negative_scale_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)))) + ((((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) * S ((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) + ((dst_negative_scale_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1))) + (((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) * S ((dst_negative_code_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)) + ((dst_negative_scale_multiply_lookup_1) + (dst_negative_scale_multiply_lookup_1)))))) /\ (((((exists ff_h_pvs_multiply_lookup_1positive. ff_h_pvs_multiply_lookup_1positive + S (dst_positive_multiply_lookup_1) = S ((S (i)) * dst_positive_scale_multiply_lookup_1)) /\ exists ff_q_pvs_multiply_lookup_1positive. dst_positive_code_multiply_lookup_1 = ff_q_pvs_multiply_lookup_1positive * S ((S (i)) * dst_positive_scale_multiply_lookup_1) + (dst_positive_multiply_lookup_1))) /\ (((((exists ff_h_pvs_multiply_lookup_1negative. ff_h_pvs_multiply_lookup_1negative + S (dst_negative_multiply_lookup_1) = S ((S (i)) * dst_negative_scale_multiply_lookup_1)) /\ exists ff_q_pvs_multiply_lookup_1negative. dst_negative_code_multiply_lookup_1 = ff_q_pvs_multiply_lookup_1negative * S ((S (i)) * dst_negative_scale_multiply_lookup_1) + (dst_negative_multiply_lookup_1))) /\ (exists ge_balance_positive_multiply_lookup_1value ge_balance_negative_multiply_lookup_1value. (((((b) = 2 * (ge_balance_positive_multiply_lookup_1value) /\ (ge_balance_negative_multiply_lookup_1value) = 0) \/ exists ge_signed_half_multiply_lookup_1valuedecode. (((b) = 2 * ge_signed_half_multiply_lookup_1valuedecode + 1 /\ (ge_balance_positive_multiply_lookup_1value) = 0) /\ (ge_balance_negative_multiply_lookup_1value) = S ge_signed_half_multiply_lookup_1valuedecode))) /\ ((dst_positive_multiply_lookup_1) + ge_balance_negative_multiply_lookup_1value = (dst_negative_multiply_lookup_1) + ge_balance_positive_multiply_lookup_1value))))))))) -> (exists dst_positive_code_multiply_lookup_2 dst_positive_scale_multiply_lookup_2 dst_negative_code_multiply_lookup_2 dst_negative_scale_multiply_lookup_2 dst_positive_multiply_lookup_2 dst_negative_multiply_lookup_2. (((H) = (((((dst_positive_code_multiply_lookup_2) + (dst_positive_scale_multiply_lookup_2)) * S ((dst_positive_code_multiply_lookup_2) + (dst_positive_scale_multiply_lookup_2)) + ((dst_positive_scale_multiply_lookup_2) + (dst_positive_scale_multiply_lookup_2))) + (((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) * S ((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) + ((dst_negative_scale_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)))) * S ((((dst_positive_code_multiply_lookup_2) + (dst_positive_scale_multiply_lookup_2)) * S ((dst_positive_code_multiply_lookup_2) + (dst_positive_scale_multiply_lookup_2)) + ((dst_positive_scale_multiply_lookup_2) + (dst_positive_scale_multiply_lookup_2))) + (((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) * S ((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) + ((dst_negative_scale_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)))) + ((((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) * S ((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) + ((dst_negative_scale_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2))) + (((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) * S ((dst_negative_code_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)) + ((dst_negative_scale_multiply_lookup_2) + (dst_negative_scale_multiply_lookup_2)))))) /\ (((((exists ff_h_pvs_multiply_lookup_2positive. ff_h_pvs_multiply_lookup_2positive + S (dst_positive_multiply_lookup_2) = S ((S (i)) * dst_positive_scale_multiply_lookup_2)) /\ exists ff_q_pvs_multiply_lookup_2positive. dst_positive_code_multiply_lookup_2 = ff_q_pvs_multiply_lookup_2positive * S ((S (i)) * dst_positive_scale_multiply_lookup_2) + (dst_positive_multiply_lookup_2))) /\ (((((exists ff_h_pvs_multiply_lookup_2negative. ff_h_pvs_multiply_lookup_2negative + S (dst_negative_multiply_lookup_2) = S ((S (i)) * dst_negative_scale_multiply_lookup_2)) /\ exists ff_q_pvs_multiply_lookup_2negative. dst_negative_code_multiply_lookup_2 = ff_q_pvs_multiply_lookup_2negative * S ((S (i)) * dst_negative_scale_multiply_lookup_2) + (dst_negative_multiply_lookup_2))) /\ (exists ge_balance_positive_multiply_lookup_2value ge_balance_negative_multiply_lookup_2value. (((((c) = 2 * (ge_balance_positive_multiply_lookup_2value) /\ (ge_balance_negative_multiply_lookup_2value) = 0) \/ exists ge_signed_half_multiply_lookup_2valuedecode. (((c) = 2 * ge_signed_half_multiply_lookup_2valuedecode + 1 /\ (ge_balance_positive_multiply_lookup_2value) = 0) /\ (ge_balance_negative_multiply_lookup_2value) = S ge_signed_half_multiply_lookup_2valuedecode))) /\ ((dst_positive_multiply_lookup_2) + ge_balance_negative_multiply_lookup_2value = (dst_negative_multiply_lookup_2) + ge_balance_positive_multiply_lookup_2value))))))))) -> (exists sto_ap_multiply_lookup_operation sto_an_multiply_lookup_operation sto_bp_multiply_lookup_operation sto_bn_multiply_lookup_operation sto_cp_multiply_lookup_operation sto_cn_multiply_lookup_operation. (((((a) = 2 * (sto_ap_multiply_lookup_operation) /\ (sto_an_multiply_lookup_operation) = 0) \/ exists ge_signed_half_multiply_lookup_operationleft. (((a) = 2 * ge_signed_half_multiply_lookup_operationleft + 1 /\ (sto_ap_multiply_lookup_operation) = 0) /\ (sto_an_multiply_lookup_operation) = S ge_signed_half_multiply_lookup_operationleft))) /\ ((((((b) = 2 * (sto_bp_multiply_lookup_operation) /\ (sto_bn_multiply_lookup_operation) = 0) \/ exists ge_signed_half_multiply_lookup_operationright. (((b) = 2 * ge_signed_half_multiply_lookup_operationright + 1 /\ (sto_bp_multiply_lookup_operation) = 0) /\ (sto_bn_multiply_lookup_operation) = S ge_signed_half_multiply_lookup_operationright))) /\ ((((((c) = 2 * (sto_cp_multiply_lookup_operation) /\ (sto_cn_multiply_lookup_operation) = 0) \/ exists ge_signed_half_multiply_lookup_operationoutput. (((c) = 2 * ge_signed_half_multiply_lookup_operationoutput + 1 /\ (sto_cp_multiply_lookup_operation) = 0) /\ (sto_cn_multiply_lookup_operation) = S ge_signed_half_multiply_lookup_operationoutput))) /\ ((sto_ap_multiply_lookup_operation * sto_bp_multiply_lookup_operation + sto_an_multiply_lookup_operation * sto_bn_multiply_lookup_operation) + sto_cn_multiply_lookup_operation = (sto_ap_multiply_lookup_operation * sto_bn_multiply_lookup_operation + sto_an_multiply_lookup_operation * sto_bp_multiply_lookup_operation) + sto_cp_multiply_lookup_operation)))))))

Complete tactic proof in conservative notation

All 57 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

57 script commands · 9 reading checkpoints · 4 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.

01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro F
  2. L2
    intro G
  3. L3
    intro H
  4. L4
    intro l
  5. L5
    intro i
  6. L6
    intro a
  7. L7
    intro b
  8. L8
    intro c
  9. L9
    intro hop
  10. L10
    intro hi
02Fix variables and assumptionsL11–13

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro h0
  2. L12
    intro h1
  3. L13
    intro h2
03Separate the logical casesL14–16

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    cases hop
  2. L15
    cases hop_right
  3. L16
    cases hop_right_right
04Establish heL17–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hop right right right.

  1. L17
    have he : ∃ a. ∃ b. ∃ c. ArithAt(F,i,a) ∧ (ArithAt(G,i,b) ∧ (ArithAt(H,i,c) ∧ SignedMul(a,b,c)))Definitions: ArithAt(F,i,a)ArithAt(G,i,b)ArithAt(H,i,c)SignedMul(a,b,c)Original native command in the exact edition
  2. L18
    specialize hop_right_right_right (i)
  3. L19
    apply hop_right_right_right
  4. L20
    exact hi
05Separate the logical casesL21–26

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L21
    cases he
  2. L22
    cases he_witness
  3. L23
    cases he_witness_witness
  4. L24
    cases he_witness_witness_witness
  5. L25
    cases he_witness_witness_witness_right
  6. L26
    cases he_witness_witness_witness_right_right
06Establish heq0L27–36

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed table at functional.

  1. L27
    have heq0 : x = a
  2. L28
    specialize divisor_signed_table_at_functional (F)
  3. L29
    specialize divisor_signed_table_at_functional (i)
  4. L30
    specialize divisor_signed_table_at_functional (x)
  5. L31
    specialize divisor_signed_table_at_functional (a)
  6. L32
    apply divisor_signed_table_at_functional
  7. L33
    exact he_witness_witness_witness_left
  8. L34
    exact h0
  9. L35
    rewrite heq0 at he_witness_witness_witness_right_right_right
  10. L36
    rewrite heq0 at he_witness_witness_witness_right_right_right
07Establish heq1L37–46

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed table at functional.

  1. L37
    have heq1 : x1 = b
  2. L38
    specialize divisor_signed_table_at_functional (G)
  3. L39
    specialize divisor_signed_table_at_functional (i)
  4. L40
    specialize divisor_signed_table_at_functional (x1)
  5. L41
    specialize divisor_signed_table_at_functional (b)
  6. L42
    apply divisor_signed_table_at_functional
  7. L43
    exact he_witness_witness_witness_right_left
  8. L44
    exact h1
  9. L45
    rewrite heq1 at he_witness_witness_witness_right_right_right
  10. L46
    rewrite heq1 at he_witness_witness_witness_right_right_right
08Establish heq2L47–56

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed table at functional.

  1. L47
    have heq2 : x2 = c
  2. L48
    specialize divisor_signed_table_at_functional (H)
  3. L49
    specialize divisor_signed_table_at_functional (i)
  4. L50
    specialize divisor_signed_table_at_functional (x2)
  5. L51
    specialize divisor_signed_table_at_functional (c)
  6. L52
    apply divisor_signed_table_at_functional
  7. L53
    exact he_witness_witness_witness_right_right_left
  8. L54
    exact h2
  9. L55
    rewrite heq2 at he_witness_witness_witness_right_right_right
  10. L56
    rewrite heq2 at he_witness_witness_witness_right_right_right
09Use earlier factsL57–57

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L57
    exact he_witness_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 57 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro H
  4. 0004intro l
  5. 0005intro i
  6. 0006intro a
  7. 0007intro b
  8. 0008intro c
  9. 0009intro hop
  10. 0010intro hi
  11. 0011intro h0
  12. 0012intro h1
  13. 0013intro h2
  14. 0014cases hop
  15. 0015cases hop_right
  16. 0016cases hop_right_right
  17. 0017have he : ∃ a. ∃ b. ∃ c. ArithAt(F,i,a) ∧ (ArithAt(G,i,b) ∧ (ArithAt(H,i,c) ∧ SignedMul(a,b,c)))
  18. 0018specialize hop_right_right_right (i)
  19. 0019apply hop_right_right_right
  20. 0020exact hi
  21. 0021cases he
  22. 0022cases he_witness
  23. 0023cases he_witness_witness
  24. 0024cases he_witness_witness_witness
  25. 0025cases he_witness_witness_witness_right
  26. 0026cases he_witness_witness_witness_right_right
  27. 0027have heq0 : x = a
  28. 0028specialize divisor_signed_table_at_functional (F)
  29. 0029specialize divisor_signed_table_at_functional (i)
  30. 0030specialize divisor_signed_table_at_functional (x)
  31. 0031specialize divisor_signed_table_at_functional (a)
  32. 0032apply divisor_signed_table_at_functional
  33. 0033exact he_witness_witness_witness_left
  34. 0034exact h0
  35. 0035rewrite heq0 at he_witness_witness_witness_right_right_right
  36. 0036rewrite heq0 at he_witness_witness_witness_right_right_right
  37. 0037have heq1 : x1 = b
  38. 0038specialize divisor_signed_table_at_functional (G)
  39. 0039specialize divisor_signed_table_at_functional (i)
  40. 0040specialize divisor_signed_table_at_functional (x1)
  41. 0041specialize divisor_signed_table_at_functional (b)
  42. 0042apply divisor_signed_table_at_functional
  43. 0043exact he_witness_witness_witness_right_left
  44. 0044exact h1
  45. 0045rewrite heq1 at he_witness_witness_witness_right_right_right
  46. 0046rewrite heq1 at he_witness_witness_witness_right_right_right
  47. 0047have heq2 : x2 = c
  48. 0048specialize divisor_signed_table_at_functional (H)
  49. 0049specialize divisor_signed_table_at_functional (i)
  50. 0050specialize divisor_signed_table_at_functional (x2)
  51. 0051specialize divisor_signed_table_at_functional (c)
  52. 0052apply divisor_signed_table_at_functional
  53. 0053exact he_witness_witness_witness_right_right_left
  54. 0054exact h2
  55. 0055rewrite heq2 at he_witness_witness_witness_right_right_right
  56. 0056rewrite heq2 at he_witness_witness_witness_right_right_right
  57. 0057exact he_witness_witness_witness_right_right_right