WS0006

signed_table_add_extensional_unique

Outputs of the same pointwise add operation agree in every represented value, not necessarily in their table codes or raw components.

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. ∀ K. ∀ l. ArithAdd(F,G,H,l)ArithAdd(F,G,K,l)ArithTableEqual(H,K,l)

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

Definition DAG

Actual proof prerequisites

Original expanded first-order statement
forall F G H K l. (((exists dst_positive_code_add_unique_firstleft_table dst_positive_scale_add_unique_firstleft_table dst_negative_code_add_unique_firstleft_table dst_negative_scale_add_unique_firstleft_table. (((F) = (((((dst_positive_code_add_unique_firstleft_table) + (dst_positive_scale_add_unique_firstleft_table)) * S ((dst_positive_code_add_unique_firstleft_table) + (dst_positive_scale_add_unique_firstleft_table)) + ((dst_positive_scale_add_unique_firstleft_table) + (dst_positive_scale_add_unique_firstleft_table))) + (((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) * S ((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) + ((dst_negative_scale_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)))) * S ((((dst_positive_code_add_unique_firstleft_table) + (dst_positive_scale_add_unique_firstleft_table)) * S ((dst_positive_code_add_unique_firstleft_table) + (dst_positive_scale_add_unique_firstleft_table)) + ((dst_positive_scale_add_unique_firstleft_table) + (dst_positive_scale_add_unique_firstleft_table))) + (((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) * S ((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) + ((dst_negative_scale_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)))) + ((((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) * S ((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) + ((dst_negative_scale_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table))) + (((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) * S ((dst_negative_code_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)) + ((dst_negative_scale_add_unique_firstleft_table) + (dst_negative_scale_add_unique_firstleft_table)))))) /\ (forall dst_index_add_unique_firstleft_table. (exists pvs_le_gap_add_unique_firstleft_tabledomain. pvs_le_gap_add_unique_firstleft_tabledomain + (dst_index_add_unique_firstleft_table) = (l)) -> exists dst_positive_add_unique_firstleft_table dst_negative_add_unique_firstleft_table dst_value_add_unique_firstleft_table. ((((exists ff_h_pvs_add_unique_firstleft_tableentrypositive. ff_h_pvs_add_unique_firstleft_tableentrypositive + S (dst_positive_add_unique_firstleft_table) = S ((S (dst_index_add_unique_firstleft_table)) * dst_positive_scale_add_unique_firstleft_table)) /\ exists ff_q_pvs_add_unique_firstleft_tableentrypositive. dst_positive_code_add_unique_firstleft_table = ff_q_pvs_add_unique_firstleft_tableentrypositive * S ((S (dst_index_add_unique_firstleft_table)) * dst_positive_scale_add_unique_firstleft_table) + (dst_positive_add_unique_firstleft_table))) /\ (((((exists ff_h_pvs_add_unique_firstleft_tableentrynegative. ff_h_pvs_add_unique_firstleft_tableentrynegative + S (dst_negative_add_unique_firstleft_table) = S ((S (dst_index_add_unique_firstleft_table)) * dst_negative_scale_add_unique_firstleft_table)) /\ exists ff_q_pvs_add_unique_firstleft_tableentrynegative. dst_negative_code_add_unique_firstleft_table = ff_q_pvs_add_unique_firstleft_tableentrynegative * S ((S (dst_index_add_unique_firstleft_table)) * dst_negative_scale_add_unique_firstleft_table) + (dst_negative_add_unique_firstleft_table))) /\ (exists ge_balance_positive_add_unique_firstleft_tableentryvalue ge_balance_negative_add_unique_firstleft_tableentryvalue. (((((dst_value_add_unique_firstleft_table) = 2 * (ge_balance_positive_add_unique_firstleft_tableentryvalue) /\ (ge_balance_negative_add_unique_firstleft_tableentryvalue) = 0) \/ exists ge_signed_half_add_unique_firstleft_tableentryvaluedecode. (((dst_value_add_unique_firstleft_table) = 2 * ge_signed_half_add_unique_firstleft_tableentryvaluedecode + 1 /\ (ge_balance_positive_add_unique_firstleft_tableentryvalue) = 0) /\ (ge_balance_negative_add_unique_firstleft_tableentryvalue) = S ge_signed_half_add_unique_firstleft_tableentryvaluedecode))) /\ ((dst_positive_add_unique_firstleft_table) + ge_balance_negative_add_unique_firstleft_tableentryvalue = (dst_negative_add_unique_firstleft_table) + ge_balance_positive_add_unique_firstleft_tableentryvalue))))))))) /\ (((exists dst_positive_code_add_unique_firstright_table dst_positive_scale_add_unique_firstright_table dst_negative_code_add_unique_firstright_table dst_negative_scale_add_unique_firstright_table. (((G) = (((((dst_positive_code_add_unique_firstright_table) + (dst_positive_scale_add_unique_firstright_table)) * S ((dst_positive_code_add_unique_firstright_table) + (dst_positive_scale_add_unique_firstright_table)) + ((dst_positive_scale_add_unique_firstright_table) + (dst_positive_scale_add_unique_firstright_table))) + (((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) * S ((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) + ((dst_negative_scale_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)))) * S ((((dst_positive_code_add_unique_firstright_table) + (dst_positive_scale_add_unique_firstright_table)) * S ((dst_positive_code_add_unique_firstright_table) + (dst_positive_scale_add_unique_firstright_table)) + ((dst_positive_scale_add_unique_firstright_table) + (dst_positive_scale_add_unique_firstright_table))) + (((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) * S ((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) + ((dst_negative_scale_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)))) + ((((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) * S ((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) + ((dst_negative_scale_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table))) + (((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) * S ((dst_negative_code_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)) + ((dst_negative_scale_add_unique_firstright_table) + (dst_negative_scale_add_unique_firstright_table)))))) /\ (forall dst_index_add_unique_firstright_table. (exists pvs_le_gap_add_unique_firstright_tabledomain. pvs_le_gap_add_unique_firstright_tabledomain + (dst_index_add_unique_firstright_table) = (l)) -> exists dst_positive_add_unique_firstright_table dst_negative_add_unique_firstright_table dst_value_add_unique_firstright_table. ((((exists ff_h_pvs_add_unique_firstright_tableentrypositive. ff_h_pvs_add_unique_firstright_tableentrypositive + S (dst_positive_add_unique_firstright_table) = S ((S (dst_index_add_unique_firstright_table)) * dst_positive_scale_add_unique_firstright_table)) /\ exists ff_q_pvs_add_unique_firstright_tableentrypositive. dst_positive_code_add_unique_firstright_table = ff_q_pvs_add_unique_firstright_tableentrypositive * S ((S (dst_index_add_unique_firstright_table)) * dst_positive_scale_add_unique_firstright_table) + (dst_positive_add_unique_firstright_table))) /\ (((((exists ff_h_pvs_add_unique_firstright_tableentrynegative. ff_h_pvs_add_unique_firstright_tableentrynegative + S (dst_negative_add_unique_firstright_table) = S ((S (dst_index_add_unique_firstright_table)) * dst_negative_scale_add_unique_firstright_table)) /\ exists ff_q_pvs_add_unique_firstright_tableentrynegative. dst_negative_code_add_unique_firstright_table = ff_q_pvs_add_unique_firstright_tableentrynegative * S ((S (dst_index_add_unique_firstright_table)) * dst_negative_scale_add_unique_firstright_table) + (dst_negative_add_unique_firstright_table))) /\ (exists ge_balance_positive_add_unique_firstright_tableentryvalue ge_balance_negative_add_unique_firstright_tableentryvalue. (((((dst_value_add_unique_firstright_table) = 2 * (ge_balance_positive_add_unique_firstright_tableentryvalue) /\ (ge_balance_negative_add_unique_firstright_tableentryvalue) = 0) \/ exists ge_signed_half_add_unique_firstright_tableentryvaluedecode. (((dst_value_add_unique_firstright_table) = 2 * ge_signed_half_add_unique_firstright_tableentryvaluedecode + 1 /\ (ge_balance_positive_add_unique_firstright_tableentryvalue) = 0) /\ (ge_balance_negative_add_unique_firstright_tableentryvalue) = S ge_signed_half_add_unique_firstright_tableentryvaluedecode))) /\ ((dst_positive_add_unique_firstright_table) + ge_balance_negative_add_unique_firstright_tableentryvalue = (dst_negative_add_unique_firstright_table) + ge_balance_positive_add_unique_firstright_tableentryvalue))))))))) /\ (((exists dst_positive_code_add_unique_firstoutput_table dst_positive_scale_add_unique_firstoutput_table dst_negative_code_add_unique_firstoutput_table dst_negative_scale_add_unique_firstoutput_table. (((H) = (((((dst_positive_code_add_unique_firstoutput_table) + (dst_positive_scale_add_unique_firstoutput_table)) * S ((dst_positive_code_add_unique_firstoutput_table) + (dst_positive_scale_add_unique_firstoutput_table)) + ((dst_positive_scale_add_unique_firstoutput_table) + (dst_positive_scale_add_unique_firstoutput_table))) + (((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) * S ((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) + ((dst_negative_scale_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)))) * S ((((dst_positive_code_add_unique_firstoutput_table) + (dst_positive_scale_add_unique_firstoutput_table)) * S ((dst_positive_code_add_unique_firstoutput_table) + (dst_positive_scale_add_unique_firstoutput_table)) + ((dst_positive_scale_add_unique_firstoutput_table) + (dst_positive_scale_add_unique_firstoutput_table))) + (((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) * S ((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) + ((dst_negative_scale_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)))) + ((((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) * S ((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) + ((dst_negative_scale_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table))) + (((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) * S ((dst_negative_code_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)) + ((dst_negative_scale_add_unique_firstoutput_table) + (dst_negative_scale_add_unique_firstoutput_table)))))) /\ (forall dst_index_add_unique_firstoutput_table. (exists pvs_le_gap_add_unique_firstoutput_tabledomain. pvs_le_gap_add_unique_firstoutput_tabledomain + (dst_index_add_unique_firstoutput_table) = (l)) -> exists dst_positive_add_unique_firstoutput_table dst_negative_add_unique_firstoutput_table dst_value_add_unique_firstoutput_table. ((((exists ff_h_pvs_add_unique_firstoutput_tableentrypositive. ff_h_pvs_add_unique_firstoutput_tableentrypositive + S (dst_positive_add_unique_firstoutput_table) = S ((S (dst_index_add_unique_firstoutput_table)) * dst_positive_scale_add_unique_firstoutput_table)) /\ exists ff_q_pvs_add_unique_firstoutput_tableentrypositive. dst_positive_code_add_unique_firstoutput_table = ff_q_pvs_add_unique_firstoutput_tableentrypositive * S ((S (dst_index_add_unique_firstoutput_table)) * dst_positive_scale_add_unique_firstoutput_table) + (dst_positive_add_unique_firstoutput_table))) /\ (((((exists ff_h_pvs_add_unique_firstoutput_tableentrynegative. ff_h_pvs_add_unique_firstoutput_tableentrynegative + S (dst_negative_add_unique_firstoutput_table) = S ((S (dst_index_add_unique_firstoutput_table)) * dst_negative_scale_add_unique_firstoutput_table)) /\ exists ff_q_pvs_add_unique_firstoutput_tableentrynegative. dst_negative_code_add_unique_firstoutput_table = ff_q_pvs_add_unique_firstoutput_tableentrynegative * S ((S (dst_index_add_unique_firstoutput_table)) * dst_negative_scale_add_unique_firstoutput_table) + (dst_negative_add_unique_firstoutput_table))) /\ (exists ge_balance_positive_add_unique_firstoutput_tableentryvalue ge_balance_negative_add_unique_firstoutput_tableentryvalue. (((((dst_value_add_unique_firstoutput_table) = 2 * (ge_balance_positive_add_unique_firstoutput_tableentryvalue) /\ (ge_balance_negative_add_unique_firstoutput_tableentryvalue) = 0) \/ exists ge_signed_half_add_unique_firstoutput_tableentryvaluedecode. (((dst_value_add_unique_firstoutput_table) = 2 * ge_signed_half_add_unique_firstoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_add_unique_firstoutput_tableentryvalue) = 0) /\ (ge_balance_negative_add_unique_firstoutput_tableentryvalue) = S ge_signed_half_add_unique_firstoutput_tableentryvaluedecode))) /\ ((dst_positive_add_unique_firstoutput_table) + ge_balance_negative_add_unique_firstoutput_tableentryvalue = (dst_negative_add_unique_firstoutput_table) + ge_balance_positive_add_unique_firstoutput_tableentryvalue))))))))) /\ (forall sto_index_add_unique_firstentries. (exists pvs_gap_add_unique_firstentriesbound. pvs_gap_add_unique_firstentriesbound + S (sto_index_add_unique_firstentries) = (l)) -> exists sto_left_add_unique_firstentries sto_right_add_unique_firstentries sto_output_add_unique_firstentries. ((exists dst_positive_code_add_unique_firstentriesentryleft dst_positive_scale_add_unique_firstentriesentryleft dst_negative_code_add_unique_firstentriesentryleft dst_negative_scale_add_unique_firstentriesentryleft dst_positive_add_unique_firstentriesentryleft dst_negative_add_unique_firstentriesentryleft. (((F) = (((((dst_positive_code_add_unique_firstentriesentryleft) + (dst_positive_scale_add_unique_firstentriesentryleft)) * S ((dst_positive_code_add_unique_firstentriesentryleft) + (dst_positive_scale_add_unique_firstentriesentryleft)) + ((dst_positive_scale_add_unique_firstentriesentryleft) + (dst_positive_scale_add_unique_firstentriesentryleft))) + (((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) * S ((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) + ((dst_negative_scale_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)))) * S ((((dst_positive_code_add_unique_firstentriesentryleft) + (dst_positive_scale_add_unique_firstentriesentryleft)) * S ((dst_positive_code_add_unique_firstentriesentryleft) + (dst_positive_scale_add_unique_firstentriesentryleft)) + ((dst_positive_scale_add_unique_firstentriesentryleft) + (dst_positive_scale_add_unique_firstentriesentryleft))) + (((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) * S ((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) + ((dst_negative_scale_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)))) + ((((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) * S ((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) + ((dst_negative_scale_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft))) + (((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) * S ((dst_negative_code_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)) + ((dst_negative_scale_add_unique_firstentriesentryleft) + (dst_negative_scale_add_unique_firstentriesentryleft)))))) /\ (((((exists ff_h_pvs_add_unique_firstentriesentryleftpositive. ff_h_pvs_add_unique_firstentriesentryleftpositive + S (dst_positive_add_unique_firstentriesentryleft) = S ((S (sto_index_add_unique_firstentries)) * dst_positive_scale_add_unique_firstentriesentryleft)) /\ exists ff_q_pvs_add_unique_firstentriesentryleftpositive. dst_positive_code_add_unique_firstentriesentryleft = ff_q_pvs_add_unique_firstentriesentryleftpositive * S ((S (sto_index_add_unique_firstentries)) * dst_positive_scale_add_unique_firstentriesentryleft) + (dst_positive_add_unique_firstentriesentryleft))) /\ (((((exists ff_h_pvs_add_unique_firstentriesentryleftnegative. ff_h_pvs_add_unique_firstentriesentryleftnegative + S (dst_negative_add_unique_firstentriesentryleft) = S ((S (sto_index_add_unique_firstentries)) * dst_negative_scale_add_unique_firstentriesentryleft)) /\ exists ff_q_pvs_add_unique_firstentriesentryleftnegative. dst_negative_code_add_unique_firstentriesentryleft = ff_q_pvs_add_unique_firstentriesentryleftnegative * S ((S (sto_index_add_unique_firstentries)) * dst_negative_scale_add_unique_firstentriesentryleft) + (dst_negative_add_unique_firstentriesentryleft))) /\ (exists ge_balance_positive_add_unique_firstentriesentryleftvalue ge_balance_negative_add_unique_firstentriesentryleftvalue. (((((sto_left_add_unique_firstentries) = 2 * (ge_balance_positive_add_unique_firstentriesentryleftvalue) /\ (ge_balance_negative_add_unique_firstentriesentryleftvalue) = 0) \/ exists ge_signed_half_add_unique_firstentriesentryleftvaluedecode. (((sto_left_add_unique_firstentries) = 2 * ge_signed_half_add_unique_firstentriesentryleftvaluedecode + 1 /\ (ge_balance_positive_add_unique_firstentriesentryleftvalue) = 0) /\ (ge_balance_negative_add_unique_firstentriesentryleftvalue) = S ge_signed_half_add_unique_firstentriesentryleftvaluedecode))) /\ ((dst_positive_add_unique_firstentriesentryleft) + ge_balance_negative_add_unique_firstentriesentryleftvalue = (dst_negative_add_unique_firstentriesentryleft) + ge_balance_positive_add_unique_firstentriesentryleftvalue))))))))) /\ (((exists dst_positive_code_add_unique_firstentriesentryright dst_positive_scale_add_unique_firstentriesentryright dst_negative_code_add_unique_firstentriesentryright dst_negative_scale_add_unique_firstentriesentryright dst_positive_add_unique_firstentriesentryright dst_negative_add_unique_firstentriesentryright. (((G) = (((((dst_positive_code_add_unique_firstentriesentryright) + (dst_positive_scale_add_unique_firstentriesentryright)) * S ((dst_positive_code_add_unique_firstentriesentryright) + (dst_positive_scale_add_unique_firstentriesentryright)) + ((dst_positive_scale_add_unique_firstentriesentryright) + (dst_positive_scale_add_unique_firstentriesentryright))) + (((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) * S ((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) + ((dst_negative_scale_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)))) * S ((((dst_positive_code_add_unique_firstentriesentryright) + (dst_positive_scale_add_unique_firstentriesentryright)) * S ((dst_positive_code_add_unique_firstentriesentryright) + (dst_positive_scale_add_unique_firstentriesentryright)) + ((dst_positive_scale_add_unique_firstentriesentryright) + (dst_positive_scale_add_unique_firstentriesentryright))) + (((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) * S ((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) + ((dst_negative_scale_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)))) + ((((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) * S ((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) + ((dst_negative_scale_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright))) + (((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) * S ((dst_negative_code_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)) + ((dst_negative_scale_add_unique_firstentriesentryright) + (dst_negative_scale_add_unique_firstentriesentryright)))))) /\ (((((exists ff_h_pvs_add_unique_firstentriesentryrightpositive. ff_h_pvs_add_unique_firstentriesentryrightpositive + S (dst_positive_add_unique_firstentriesentryright) = S ((S (sto_index_add_unique_firstentries)) * dst_positive_scale_add_unique_firstentriesentryright)) /\ exists ff_q_pvs_add_unique_firstentriesentryrightpositive. dst_positive_code_add_unique_firstentriesentryright = ff_q_pvs_add_unique_firstentriesentryrightpositive * S ((S (sto_index_add_unique_firstentries)) * dst_positive_scale_add_unique_firstentriesentryright) + (dst_positive_add_unique_firstentriesentryright))) /\ (((((exists ff_h_pvs_add_unique_firstentriesentryrightnegative. ff_h_pvs_add_unique_firstentriesentryrightnegative + S (dst_negative_add_unique_firstentriesentryright) = S ((S (sto_index_add_unique_firstentries)) * dst_negative_scale_add_unique_firstentriesentryright)) /\ exists ff_q_pvs_add_unique_firstentriesentryrightnegative. dst_negative_code_add_unique_firstentriesentryright = ff_q_pvs_add_unique_firstentriesentryrightnegative * S ((S (sto_index_add_unique_firstentries)) * dst_negative_scale_add_unique_firstentriesentryright) + (dst_negative_add_unique_firstentriesentryright))) /\ (exists ge_balance_positive_add_unique_firstentriesentryrightvalue ge_balance_negative_add_unique_firstentriesentryrightvalue. (((((sto_right_add_unique_firstentries) = 2 * (ge_balance_positive_add_unique_firstentriesentryrightvalue) /\ (ge_balance_negative_add_unique_firstentriesentryrightvalue) = 0) \/ exists ge_signed_half_add_unique_firstentriesentryrightvaluedecode. (((sto_right_add_unique_firstentries) = 2 * ge_signed_half_add_unique_firstentriesentryrightvaluedecode + 1 /\ (ge_balance_positive_add_unique_firstentriesentryrightvalue) = 0) /\ (ge_balance_negative_add_unique_firstentriesentryrightvalue) = S ge_signed_half_add_unique_firstentriesentryrightvaluedecode))) /\ ((dst_positive_add_unique_firstentriesentryright) + ge_balance_negative_add_unique_firstentriesentryrightvalue = (dst_negative_add_unique_firstentriesentryright) + ge_balance_positive_add_unique_firstentriesentryrightvalue))))))))) /\ (((exists dst_positive_code_add_unique_firstentriesentryoutput dst_positive_scale_add_unique_firstentriesentryoutput dst_negative_code_add_unique_firstentriesentryoutput dst_negative_scale_add_unique_firstentriesentryoutput dst_positive_add_unique_firstentriesentryoutput dst_negative_add_unique_firstentriesentryoutput. (((H) = (((((dst_positive_code_add_unique_firstentriesentryoutput) + (dst_positive_scale_add_unique_firstentriesentryoutput)) * S ((dst_positive_code_add_unique_firstentriesentryoutput) + (dst_positive_scale_add_unique_firstentriesentryoutput)) + ((dst_positive_scale_add_unique_firstentriesentryoutput) + (dst_positive_scale_add_unique_firstentriesentryoutput))) + (((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) * S ((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) + ((dst_negative_scale_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)))) * S ((((dst_positive_code_add_unique_firstentriesentryoutput) + (dst_positive_scale_add_unique_firstentriesentryoutput)) * S ((dst_positive_code_add_unique_firstentriesentryoutput) + (dst_positive_scale_add_unique_firstentriesentryoutput)) + ((dst_positive_scale_add_unique_firstentriesentryoutput) + (dst_positive_scale_add_unique_firstentriesentryoutput))) + (((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) * S ((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) + ((dst_negative_scale_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)))) + ((((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) * S ((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) + ((dst_negative_scale_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput))) + (((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) * S ((dst_negative_code_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)) + ((dst_negative_scale_add_unique_firstentriesentryoutput) + (dst_negative_scale_add_unique_firstentriesentryoutput)))))) /\ (((((exists ff_h_pvs_add_unique_firstentriesentryoutputpositive. ff_h_pvs_add_unique_firstentriesentryoutputpositive + S (dst_positive_add_unique_firstentriesentryoutput) = S ((S (sto_index_add_unique_firstentries)) * dst_positive_scale_add_unique_firstentriesentryoutput)) /\ exists ff_q_pvs_add_unique_firstentriesentryoutputpositive. dst_positive_code_add_unique_firstentriesentryoutput = ff_q_pvs_add_unique_firstentriesentryoutputpositive * S ((S (sto_index_add_unique_firstentries)) * dst_positive_scale_add_unique_firstentriesentryoutput) + (dst_positive_add_unique_firstentriesentryoutput))) /\ (((((exists ff_h_pvs_add_unique_firstentriesentryoutputnegative. ff_h_pvs_add_unique_firstentriesentryoutputnegative + S (dst_negative_add_unique_firstentriesentryoutput) = S ((S (sto_index_add_unique_firstentries)) * dst_negative_scale_add_unique_firstentriesentryoutput)) /\ exists ff_q_pvs_add_unique_firstentriesentryoutputnegative. dst_negative_code_add_unique_firstentriesentryoutput = ff_q_pvs_add_unique_firstentriesentryoutputnegative * S ((S (sto_index_add_unique_firstentries)) * dst_negative_scale_add_unique_firstentriesentryoutput) + (dst_negative_add_unique_firstentriesentryoutput))) /\ (exists ge_balance_positive_add_unique_firstentriesentryoutputvalue ge_balance_negative_add_unique_firstentriesentryoutputvalue. (((((sto_output_add_unique_firstentries) = 2 * (ge_balance_positive_add_unique_firstentriesentryoutputvalue) /\ (ge_balance_negative_add_unique_firstentriesentryoutputvalue) = 0) \/ exists ge_signed_half_add_unique_firstentriesentryoutputvaluedecode. (((sto_output_add_unique_firstentries) = 2 * ge_signed_half_add_unique_firstentriesentryoutputvaluedecode + 1 /\ (ge_balance_positive_add_unique_firstentriesentryoutputvalue) = 0) /\ (ge_balance_negative_add_unique_firstentriesentryoutputvalue) = S ge_signed_half_add_unique_firstentriesentryoutputvaluedecode))) /\ ((dst_positive_add_unique_firstentriesentryoutput) + ge_balance_negative_add_unique_firstentriesentryoutputvalue = (dst_negative_add_unique_firstentriesentryoutput) + ge_balance_positive_add_unique_firstentriesentryoutputvalue))))))))) /\ (exists dsa_ap_add_unique_firstentriesentryoperation dsa_an_add_unique_firstentriesentryoperation dsa_bp_add_unique_firstentriesentryoperation dsa_bn_add_unique_firstentriesentryoperation dsa_cp_add_unique_firstentriesentryoperation dsa_cn_add_unique_firstentriesentryoperation. (((((sto_left_add_unique_firstentries) = 2 * (dsa_ap_add_unique_firstentriesentryoperation) /\ (dsa_an_add_unique_firstentriesentryoperation) = 0) \/ exists ge_signed_half_add_unique_firstentriesentryoperationleft. (((sto_left_add_unique_firstentries) = 2 * ge_signed_half_add_unique_firstentriesentryoperationleft + 1 /\ (dsa_ap_add_unique_firstentriesentryoperation) = 0) /\ (dsa_an_add_unique_firstentriesentryoperation) = S ge_signed_half_add_unique_firstentriesentryoperationleft))) /\ ((((((sto_right_add_unique_firstentries) = 2 * (dsa_bp_add_unique_firstentriesentryoperation) /\ (dsa_bn_add_unique_firstentriesentryoperation) = 0) \/ exists ge_signed_half_add_unique_firstentriesentryoperationright. (((sto_right_add_unique_firstentries) = 2 * ge_signed_half_add_unique_firstentriesentryoperationright + 1 /\ (dsa_bp_add_unique_firstentriesentryoperation) = 0) /\ (dsa_bn_add_unique_firstentriesentryoperation) = S ge_signed_half_add_unique_firstentriesentryoperationright))) /\ ((((((sto_output_add_unique_firstentries) = 2 * (dsa_cp_add_unique_firstentriesentryoperation) /\ (dsa_cn_add_unique_firstentriesentryoperation) = 0) \/ exists ge_signed_half_add_unique_firstentriesentryoperationoutput. (((sto_output_add_unique_firstentries) = 2 * ge_signed_half_add_unique_firstentriesentryoperationoutput + 1 /\ (dsa_cp_add_unique_firstentriesentryoperation) = 0) /\ (dsa_cn_add_unique_firstentriesentryoperation) = S ge_signed_half_add_unique_firstentriesentryoperationoutput))) /\ ((dsa_ap_add_unique_firstentriesentryoperation + dsa_bp_add_unique_firstentriesentryoperation) + dsa_cn_add_unique_firstentriesentryoperation = (dsa_an_add_unique_firstentriesentryoperation + dsa_bn_add_unique_firstentriesentryoperation) + dsa_cp_add_unique_firstentriesentryoperation))))))))))))))))))) -> (((exists dst_positive_code_add_unique_secondleft_table dst_positive_scale_add_unique_secondleft_table dst_negative_code_add_unique_secondleft_table dst_negative_scale_add_unique_secondleft_table. (((F) = (((((dst_positive_code_add_unique_secondleft_table) + (dst_positive_scale_add_unique_secondleft_table)) * S ((dst_positive_code_add_unique_secondleft_table) + (dst_positive_scale_add_unique_secondleft_table)) + ((dst_positive_scale_add_unique_secondleft_table) + (dst_positive_scale_add_unique_secondleft_table))) + (((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) * S ((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) + ((dst_negative_scale_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)))) * S ((((dst_positive_code_add_unique_secondleft_table) + (dst_positive_scale_add_unique_secondleft_table)) * S ((dst_positive_code_add_unique_secondleft_table) + (dst_positive_scale_add_unique_secondleft_table)) + ((dst_positive_scale_add_unique_secondleft_table) + (dst_positive_scale_add_unique_secondleft_table))) + (((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) * S ((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) + ((dst_negative_scale_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)))) + ((((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) * S ((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) + ((dst_negative_scale_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table))) + (((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) * S ((dst_negative_code_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)) + ((dst_negative_scale_add_unique_secondleft_table) + (dst_negative_scale_add_unique_secondleft_table)))))) /\ (forall dst_index_add_unique_secondleft_table. (exists pvs_le_gap_add_unique_secondleft_tabledomain. pvs_le_gap_add_unique_secondleft_tabledomain + (dst_index_add_unique_secondleft_table) = (l)) -> exists dst_positive_add_unique_secondleft_table dst_negative_add_unique_secondleft_table dst_value_add_unique_secondleft_table. ((((exists ff_h_pvs_add_unique_secondleft_tableentrypositive. ff_h_pvs_add_unique_secondleft_tableentrypositive + S (dst_positive_add_unique_secondleft_table) = S ((S (dst_index_add_unique_secondleft_table)) * dst_positive_scale_add_unique_secondleft_table)) /\ exists ff_q_pvs_add_unique_secondleft_tableentrypositive. dst_positive_code_add_unique_secondleft_table = ff_q_pvs_add_unique_secondleft_tableentrypositive * S ((S (dst_index_add_unique_secondleft_table)) * dst_positive_scale_add_unique_secondleft_table) + (dst_positive_add_unique_secondleft_table))) /\ (((((exists ff_h_pvs_add_unique_secondleft_tableentrynegative. ff_h_pvs_add_unique_secondleft_tableentrynegative + S (dst_negative_add_unique_secondleft_table) = S ((S (dst_index_add_unique_secondleft_table)) * dst_negative_scale_add_unique_secondleft_table)) /\ exists ff_q_pvs_add_unique_secondleft_tableentrynegative. dst_negative_code_add_unique_secondleft_table = ff_q_pvs_add_unique_secondleft_tableentrynegative * S ((S (dst_index_add_unique_secondleft_table)) * dst_negative_scale_add_unique_secondleft_table) + (dst_negative_add_unique_secondleft_table))) /\ (exists ge_balance_positive_add_unique_secondleft_tableentryvalue ge_balance_negative_add_unique_secondleft_tableentryvalue. (((((dst_value_add_unique_secondleft_table) = 2 * (ge_balance_positive_add_unique_secondleft_tableentryvalue) /\ (ge_balance_negative_add_unique_secondleft_tableentryvalue) = 0) \/ exists ge_signed_half_add_unique_secondleft_tableentryvaluedecode. (((dst_value_add_unique_secondleft_table) = 2 * ge_signed_half_add_unique_secondleft_tableentryvaluedecode + 1 /\ (ge_balance_positive_add_unique_secondleft_tableentryvalue) = 0) /\ (ge_balance_negative_add_unique_secondleft_tableentryvalue) = S ge_signed_half_add_unique_secondleft_tableentryvaluedecode))) /\ ((dst_positive_add_unique_secondleft_table) + ge_balance_negative_add_unique_secondleft_tableentryvalue = (dst_negative_add_unique_secondleft_table) + ge_balance_positive_add_unique_secondleft_tableentryvalue))))))))) /\ (((exists dst_positive_code_add_unique_secondright_table dst_positive_scale_add_unique_secondright_table dst_negative_code_add_unique_secondright_table dst_negative_scale_add_unique_secondright_table. (((G) = (((((dst_positive_code_add_unique_secondright_table) + (dst_positive_scale_add_unique_secondright_table)) * S ((dst_positive_code_add_unique_secondright_table) + (dst_positive_scale_add_unique_secondright_table)) + ((dst_positive_scale_add_unique_secondright_table) + (dst_positive_scale_add_unique_secondright_table))) + (((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) * S ((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) + ((dst_negative_scale_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)))) * S ((((dst_positive_code_add_unique_secondright_table) + (dst_positive_scale_add_unique_secondright_table)) * S ((dst_positive_code_add_unique_secondright_table) + (dst_positive_scale_add_unique_secondright_table)) + ((dst_positive_scale_add_unique_secondright_table) + (dst_positive_scale_add_unique_secondright_table))) + (((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) * S ((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) + ((dst_negative_scale_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)))) + ((((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) * S ((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) + ((dst_negative_scale_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table))) + (((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) * S ((dst_negative_code_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)) + ((dst_negative_scale_add_unique_secondright_table) + (dst_negative_scale_add_unique_secondright_table)))))) /\ (forall dst_index_add_unique_secondright_table. (exists pvs_le_gap_add_unique_secondright_tabledomain. pvs_le_gap_add_unique_secondright_tabledomain + (dst_index_add_unique_secondright_table) = (l)) -> exists dst_positive_add_unique_secondright_table dst_negative_add_unique_secondright_table dst_value_add_unique_secondright_table. ((((exists ff_h_pvs_add_unique_secondright_tableentrypositive. ff_h_pvs_add_unique_secondright_tableentrypositive + S (dst_positive_add_unique_secondright_table) = S ((S (dst_index_add_unique_secondright_table)) * dst_positive_scale_add_unique_secondright_table)) /\ exists ff_q_pvs_add_unique_secondright_tableentrypositive. dst_positive_code_add_unique_secondright_table = ff_q_pvs_add_unique_secondright_tableentrypositive * S ((S (dst_index_add_unique_secondright_table)) * dst_positive_scale_add_unique_secondright_table) + (dst_positive_add_unique_secondright_table))) /\ (((((exists ff_h_pvs_add_unique_secondright_tableentrynegative. ff_h_pvs_add_unique_secondright_tableentrynegative + S (dst_negative_add_unique_secondright_table) = S ((S (dst_index_add_unique_secondright_table)) * dst_negative_scale_add_unique_secondright_table)) /\ exists ff_q_pvs_add_unique_secondright_tableentrynegative. dst_negative_code_add_unique_secondright_table = ff_q_pvs_add_unique_secondright_tableentrynegative * S ((S (dst_index_add_unique_secondright_table)) * dst_negative_scale_add_unique_secondright_table) + (dst_negative_add_unique_secondright_table))) /\ (exists ge_balance_positive_add_unique_secondright_tableentryvalue ge_balance_negative_add_unique_secondright_tableentryvalue. (((((dst_value_add_unique_secondright_table) = 2 * (ge_balance_positive_add_unique_secondright_tableentryvalue) /\ (ge_balance_negative_add_unique_secondright_tableentryvalue) = 0) \/ exists ge_signed_half_add_unique_secondright_tableentryvaluedecode. (((dst_value_add_unique_secondright_table) = 2 * ge_signed_half_add_unique_secondright_tableentryvaluedecode + 1 /\ (ge_balance_positive_add_unique_secondright_tableentryvalue) = 0) /\ (ge_balance_negative_add_unique_secondright_tableentryvalue) = S ge_signed_half_add_unique_secondright_tableentryvaluedecode))) /\ ((dst_positive_add_unique_secondright_table) + ge_balance_negative_add_unique_secondright_tableentryvalue = (dst_negative_add_unique_secondright_table) + ge_balance_positive_add_unique_secondright_tableentryvalue))))))))) /\ (((exists dst_positive_code_add_unique_secondoutput_table dst_positive_scale_add_unique_secondoutput_table dst_negative_code_add_unique_secondoutput_table dst_negative_scale_add_unique_secondoutput_table. (((K) = (((((dst_positive_code_add_unique_secondoutput_table) + (dst_positive_scale_add_unique_secondoutput_table)) * S ((dst_positive_code_add_unique_secondoutput_table) + (dst_positive_scale_add_unique_secondoutput_table)) + ((dst_positive_scale_add_unique_secondoutput_table) + (dst_positive_scale_add_unique_secondoutput_table))) + (((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) * S ((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) + ((dst_negative_scale_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)))) * S ((((dst_positive_code_add_unique_secondoutput_table) + (dst_positive_scale_add_unique_secondoutput_table)) * S ((dst_positive_code_add_unique_secondoutput_table) + (dst_positive_scale_add_unique_secondoutput_table)) + ((dst_positive_scale_add_unique_secondoutput_table) + (dst_positive_scale_add_unique_secondoutput_table))) + (((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) * S ((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) + ((dst_negative_scale_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)))) + ((((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) * S ((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) + ((dst_negative_scale_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table))) + (((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) * S ((dst_negative_code_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)) + ((dst_negative_scale_add_unique_secondoutput_table) + (dst_negative_scale_add_unique_secondoutput_table)))))) /\ (forall dst_index_add_unique_secondoutput_table. (exists pvs_le_gap_add_unique_secondoutput_tabledomain. pvs_le_gap_add_unique_secondoutput_tabledomain + (dst_index_add_unique_secondoutput_table) = (l)) -> exists dst_positive_add_unique_secondoutput_table dst_negative_add_unique_secondoutput_table dst_value_add_unique_secondoutput_table. ((((exists ff_h_pvs_add_unique_secondoutput_tableentrypositive. ff_h_pvs_add_unique_secondoutput_tableentrypositive + S (dst_positive_add_unique_secondoutput_table) = S ((S (dst_index_add_unique_secondoutput_table)) * dst_positive_scale_add_unique_secondoutput_table)) /\ exists ff_q_pvs_add_unique_secondoutput_tableentrypositive. dst_positive_code_add_unique_secondoutput_table = ff_q_pvs_add_unique_secondoutput_tableentrypositive * S ((S (dst_index_add_unique_secondoutput_table)) * dst_positive_scale_add_unique_secondoutput_table) + (dst_positive_add_unique_secondoutput_table))) /\ (((((exists ff_h_pvs_add_unique_secondoutput_tableentrynegative. ff_h_pvs_add_unique_secondoutput_tableentrynegative + S (dst_negative_add_unique_secondoutput_table) = S ((S (dst_index_add_unique_secondoutput_table)) * dst_negative_scale_add_unique_secondoutput_table)) /\ exists ff_q_pvs_add_unique_secondoutput_tableentrynegative. dst_negative_code_add_unique_secondoutput_table = ff_q_pvs_add_unique_secondoutput_tableentrynegative * S ((S (dst_index_add_unique_secondoutput_table)) * dst_negative_scale_add_unique_secondoutput_table) + (dst_negative_add_unique_secondoutput_table))) /\ (exists ge_balance_positive_add_unique_secondoutput_tableentryvalue ge_balance_negative_add_unique_secondoutput_tableentryvalue. (((((dst_value_add_unique_secondoutput_table) = 2 * (ge_balance_positive_add_unique_secondoutput_tableentryvalue) /\ (ge_balance_negative_add_unique_secondoutput_tableentryvalue) = 0) \/ exists ge_signed_half_add_unique_secondoutput_tableentryvaluedecode. (((dst_value_add_unique_secondoutput_table) = 2 * ge_signed_half_add_unique_secondoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_add_unique_secondoutput_tableentryvalue) = 0) /\ (ge_balance_negative_add_unique_secondoutput_tableentryvalue) = S ge_signed_half_add_unique_secondoutput_tableentryvaluedecode))) /\ ((dst_positive_add_unique_secondoutput_table) + ge_balance_negative_add_unique_secondoutput_tableentryvalue = (dst_negative_add_unique_secondoutput_table) + ge_balance_positive_add_unique_secondoutput_tableentryvalue))))))))) /\ (forall sto_index_add_unique_secondentries. (exists pvs_gap_add_unique_secondentriesbound. pvs_gap_add_unique_secondentriesbound + S (sto_index_add_unique_secondentries) = (l)) -> exists sto_left_add_unique_secondentries sto_right_add_unique_secondentries sto_output_add_unique_secondentries. ((exists dst_positive_code_add_unique_secondentriesentryleft dst_positive_scale_add_unique_secondentriesentryleft dst_negative_code_add_unique_secondentriesentryleft dst_negative_scale_add_unique_secondentriesentryleft dst_positive_add_unique_secondentriesentryleft dst_negative_add_unique_secondentriesentryleft. (((F) = (((((dst_positive_code_add_unique_secondentriesentryleft) + (dst_positive_scale_add_unique_secondentriesentryleft)) * S ((dst_positive_code_add_unique_secondentriesentryleft) + (dst_positive_scale_add_unique_secondentriesentryleft)) + ((dst_positive_scale_add_unique_secondentriesentryleft) + (dst_positive_scale_add_unique_secondentriesentryleft))) + (((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) * S ((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) + ((dst_negative_scale_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)))) * S ((((dst_positive_code_add_unique_secondentriesentryleft) + (dst_positive_scale_add_unique_secondentriesentryleft)) * S ((dst_positive_code_add_unique_secondentriesentryleft) + (dst_positive_scale_add_unique_secondentriesentryleft)) + ((dst_positive_scale_add_unique_secondentriesentryleft) + (dst_positive_scale_add_unique_secondentriesentryleft))) + (((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) * S ((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) + ((dst_negative_scale_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)))) + ((((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) * S ((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) + ((dst_negative_scale_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft))) + (((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) * S ((dst_negative_code_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)) + ((dst_negative_scale_add_unique_secondentriesentryleft) + (dst_negative_scale_add_unique_secondentriesentryleft)))))) /\ (((((exists ff_h_pvs_add_unique_secondentriesentryleftpositive. ff_h_pvs_add_unique_secondentriesentryleftpositive + S (dst_positive_add_unique_secondentriesentryleft) = S ((S (sto_index_add_unique_secondentries)) * dst_positive_scale_add_unique_secondentriesentryleft)) /\ exists ff_q_pvs_add_unique_secondentriesentryleftpositive. dst_positive_code_add_unique_secondentriesentryleft = ff_q_pvs_add_unique_secondentriesentryleftpositive * S ((S (sto_index_add_unique_secondentries)) * dst_positive_scale_add_unique_secondentriesentryleft) + (dst_positive_add_unique_secondentriesentryleft))) /\ (((((exists ff_h_pvs_add_unique_secondentriesentryleftnegative. ff_h_pvs_add_unique_secondentriesentryleftnegative + S (dst_negative_add_unique_secondentriesentryleft) = S ((S (sto_index_add_unique_secondentries)) * dst_negative_scale_add_unique_secondentriesentryleft)) /\ exists ff_q_pvs_add_unique_secondentriesentryleftnegative. dst_negative_code_add_unique_secondentriesentryleft = ff_q_pvs_add_unique_secondentriesentryleftnegative * S ((S (sto_index_add_unique_secondentries)) * dst_negative_scale_add_unique_secondentriesentryleft) + (dst_negative_add_unique_secondentriesentryleft))) /\ (exists ge_balance_positive_add_unique_secondentriesentryleftvalue ge_balance_negative_add_unique_secondentriesentryleftvalue. (((((sto_left_add_unique_secondentries) = 2 * (ge_balance_positive_add_unique_secondentriesentryleftvalue) /\ (ge_balance_negative_add_unique_secondentriesentryleftvalue) = 0) \/ exists ge_signed_half_add_unique_secondentriesentryleftvaluedecode. (((sto_left_add_unique_secondentries) = 2 * ge_signed_half_add_unique_secondentriesentryleftvaluedecode + 1 /\ (ge_balance_positive_add_unique_secondentriesentryleftvalue) = 0) /\ (ge_balance_negative_add_unique_secondentriesentryleftvalue) = S ge_signed_half_add_unique_secondentriesentryleftvaluedecode))) /\ ((dst_positive_add_unique_secondentriesentryleft) + ge_balance_negative_add_unique_secondentriesentryleftvalue = (dst_negative_add_unique_secondentriesentryleft) + ge_balance_positive_add_unique_secondentriesentryleftvalue))))))))) /\ (((exists dst_positive_code_add_unique_secondentriesentryright dst_positive_scale_add_unique_secondentriesentryright dst_negative_code_add_unique_secondentriesentryright dst_negative_scale_add_unique_secondentriesentryright dst_positive_add_unique_secondentriesentryright dst_negative_add_unique_secondentriesentryright. (((G) = (((((dst_positive_code_add_unique_secondentriesentryright) + (dst_positive_scale_add_unique_secondentriesentryright)) * S ((dst_positive_code_add_unique_secondentriesentryright) + (dst_positive_scale_add_unique_secondentriesentryright)) + ((dst_positive_scale_add_unique_secondentriesentryright) + (dst_positive_scale_add_unique_secondentriesentryright))) + (((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) * S ((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) + ((dst_negative_scale_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)))) * S ((((dst_positive_code_add_unique_secondentriesentryright) + (dst_positive_scale_add_unique_secondentriesentryright)) * S ((dst_positive_code_add_unique_secondentriesentryright) + (dst_positive_scale_add_unique_secondentriesentryright)) + ((dst_positive_scale_add_unique_secondentriesentryright) + (dst_positive_scale_add_unique_secondentriesentryright))) + (((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) * S ((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) + ((dst_negative_scale_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)))) + ((((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) * S ((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) + ((dst_negative_scale_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright))) + (((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) * S ((dst_negative_code_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)) + ((dst_negative_scale_add_unique_secondentriesentryright) + (dst_negative_scale_add_unique_secondentriesentryright)))))) /\ (((((exists ff_h_pvs_add_unique_secondentriesentryrightpositive. ff_h_pvs_add_unique_secondentriesentryrightpositive + S (dst_positive_add_unique_secondentriesentryright) = S ((S (sto_index_add_unique_secondentries)) * dst_positive_scale_add_unique_secondentriesentryright)) /\ exists ff_q_pvs_add_unique_secondentriesentryrightpositive. dst_positive_code_add_unique_secondentriesentryright = ff_q_pvs_add_unique_secondentriesentryrightpositive * S ((S (sto_index_add_unique_secondentries)) * dst_positive_scale_add_unique_secondentriesentryright) + (dst_positive_add_unique_secondentriesentryright))) /\ (((((exists ff_h_pvs_add_unique_secondentriesentryrightnegative. ff_h_pvs_add_unique_secondentriesentryrightnegative + S (dst_negative_add_unique_secondentriesentryright) = S ((S (sto_index_add_unique_secondentries)) * dst_negative_scale_add_unique_secondentriesentryright)) /\ exists ff_q_pvs_add_unique_secondentriesentryrightnegative. dst_negative_code_add_unique_secondentriesentryright = ff_q_pvs_add_unique_secondentriesentryrightnegative * S ((S (sto_index_add_unique_secondentries)) * dst_negative_scale_add_unique_secondentriesentryright) + (dst_negative_add_unique_secondentriesentryright))) /\ (exists ge_balance_positive_add_unique_secondentriesentryrightvalue ge_balance_negative_add_unique_secondentriesentryrightvalue. (((((sto_right_add_unique_secondentries) = 2 * (ge_balance_positive_add_unique_secondentriesentryrightvalue) /\ (ge_balance_negative_add_unique_secondentriesentryrightvalue) = 0) \/ exists ge_signed_half_add_unique_secondentriesentryrightvaluedecode. (((sto_right_add_unique_secondentries) = 2 * ge_signed_half_add_unique_secondentriesentryrightvaluedecode + 1 /\ (ge_balance_positive_add_unique_secondentriesentryrightvalue) = 0) /\ (ge_balance_negative_add_unique_secondentriesentryrightvalue) = S ge_signed_half_add_unique_secondentriesentryrightvaluedecode))) /\ ((dst_positive_add_unique_secondentriesentryright) + ge_balance_negative_add_unique_secondentriesentryrightvalue = (dst_negative_add_unique_secondentriesentryright) + ge_balance_positive_add_unique_secondentriesentryrightvalue))))))))) /\ (((exists dst_positive_code_add_unique_secondentriesentryoutput dst_positive_scale_add_unique_secondentriesentryoutput dst_negative_code_add_unique_secondentriesentryoutput dst_negative_scale_add_unique_secondentriesentryoutput dst_positive_add_unique_secondentriesentryoutput dst_negative_add_unique_secondentriesentryoutput. (((K) = (((((dst_positive_code_add_unique_secondentriesentryoutput) + (dst_positive_scale_add_unique_secondentriesentryoutput)) * S ((dst_positive_code_add_unique_secondentriesentryoutput) + (dst_positive_scale_add_unique_secondentriesentryoutput)) + ((dst_positive_scale_add_unique_secondentriesentryoutput) + (dst_positive_scale_add_unique_secondentriesentryoutput))) + (((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) * S ((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) + ((dst_negative_scale_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)))) * S ((((dst_positive_code_add_unique_secondentriesentryoutput) + (dst_positive_scale_add_unique_secondentriesentryoutput)) * S ((dst_positive_code_add_unique_secondentriesentryoutput) + (dst_positive_scale_add_unique_secondentriesentryoutput)) + ((dst_positive_scale_add_unique_secondentriesentryoutput) + (dst_positive_scale_add_unique_secondentriesentryoutput))) + (((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) * S ((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) + ((dst_negative_scale_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)))) + ((((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) * S ((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) + ((dst_negative_scale_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput))) + (((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) * S ((dst_negative_code_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)) + ((dst_negative_scale_add_unique_secondentriesentryoutput) + (dst_negative_scale_add_unique_secondentriesentryoutput)))))) /\ (((((exists ff_h_pvs_add_unique_secondentriesentryoutputpositive. ff_h_pvs_add_unique_secondentriesentryoutputpositive + S (dst_positive_add_unique_secondentriesentryoutput) = S ((S (sto_index_add_unique_secondentries)) * dst_positive_scale_add_unique_secondentriesentryoutput)) /\ exists ff_q_pvs_add_unique_secondentriesentryoutputpositive. dst_positive_code_add_unique_secondentriesentryoutput = ff_q_pvs_add_unique_secondentriesentryoutputpositive * S ((S (sto_index_add_unique_secondentries)) * dst_positive_scale_add_unique_secondentriesentryoutput) + (dst_positive_add_unique_secondentriesentryoutput))) /\ (((((exists ff_h_pvs_add_unique_secondentriesentryoutputnegative. ff_h_pvs_add_unique_secondentriesentryoutputnegative + S (dst_negative_add_unique_secondentriesentryoutput) = S ((S (sto_index_add_unique_secondentries)) * dst_negative_scale_add_unique_secondentriesentryoutput)) /\ exists ff_q_pvs_add_unique_secondentriesentryoutputnegative. dst_negative_code_add_unique_secondentriesentryoutput = ff_q_pvs_add_unique_secondentriesentryoutputnegative * S ((S (sto_index_add_unique_secondentries)) * dst_negative_scale_add_unique_secondentriesentryoutput) + (dst_negative_add_unique_secondentriesentryoutput))) /\ (exists ge_balance_positive_add_unique_secondentriesentryoutputvalue ge_balance_negative_add_unique_secondentriesentryoutputvalue. (((((sto_output_add_unique_secondentries) = 2 * (ge_balance_positive_add_unique_secondentriesentryoutputvalue) /\ (ge_balance_negative_add_unique_secondentriesentryoutputvalue) = 0) \/ exists ge_signed_half_add_unique_secondentriesentryoutputvaluedecode. (((sto_output_add_unique_secondentries) = 2 * ge_signed_half_add_unique_secondentriesentryoutputvaluedecode + 1 /\ (ge_balance_positive_add_unique_secondentriesentryoutputvalue) = 0) /\ (ge_balance_negative_add_unique_secondentriesentryoutputvalue) = S ge_signed_half_add_unique_secondentriesentryoutputvaluedecode))) /\ ((dst_positive_add_unique_secondentriesentryoutput) + ge_balance_negative_add_unique_secondentriesentryoutputvalue = (dst_negative_add_unique_secondentriesentryoutput) + ge_balance_positive_add_unique_secondentriesentryoutputvalue))))))))) /\ (exists dsa_ap_add_unique_secondentriesentryoperation dsa_an_add_unique_secondentriesentryoperation dsa_bp_add_unique_secondentriesentryoperation dsa_bn_add_unique_secondentriesentryoperation dsa_cp_add_unique_secondentriesentryoperation dsa_cn_add_unique_secondentriesentryoperation. (((((sto_left_add_unique_secondentries) = 2 * (dsa_ap_add_unique_secondentriesentryoperation) /\ (dsa_an_add_unique_secondentriesentryoperation) = 0) \/ exists ge_signed_half_add_unique_secondentriesentryoperationleft. (((sto_left_add_unique_secondentries) = 2 * ge_signed_half_add_unique_secondentriesentryoperationleft + 1 /\ (dsa_ap_add_unique_secondentriesentryoperation) = 0) /\ (dsa_an_add_unique_secondentriesentryoperation) = S ge_signed_half_add_unique_secondentriesentryoperationleft))) /\ ((((((sto_right_add_unique_secondentries) = 2 * (dsa_bp_add_unique_secondentriesentryoperation) /\ (dsa_bn_add_unique_secondentriesentryoperation) = 0) \/ exists ge_signed_half_add_unique_secondentriesentryoperationright. (((sto_right_add_unique_secondentries) = 2 * ge_signed_half_add_unique_secondentriesentryoperationright + 1 /\ (dsa_bp_add_unique_secondentriesentryoperation) = 0) /\ (dsa_bn_add_unique_secondentriesentryoperation) = S ge_signed_half_add_unique_secondentriesentryoperationright))) /\ ((((((sto_output_add_unique_secondentries) = 2 * (dsa_cp_add_unique_secondentriesentryoperation) /\ (dsa_cn_add_unique_secondentriesentryoperation) = 0) \/ exists ge_signed_half_add_unique_secondentriesentryoperationoutput. (((sto_output_add_unique_secondentries) = 2 * ge_signed_half_add_unique_secondentriesentryoperationoutput + 1 /\ (dsa_cp_add_unique_secondentriesentryoperation) = 0) /\ (dsa_cn_add_unique_secondentriesentryoperation) = S ge_signed_half_add_unique_secondentriesentryoperationoutput))) /\ ((dsa_ap_add_unique_secondentriesentryoperation + dsa_bp_add_unique_secondentriesentryoperation) + dsa_cn_add_unique_secondentriesentryoperation = (dsa_an_add_unique_secondentriesentryoperation + dsa_bn_add_unique_secondentriesentryoperation) + dsa_cp_add_unique_secondentriesentryoperation))))))))))))))))))) -> (forall dst_index_add_unique_result dst_first_add_unique_result dst_second_add_unique_result. (exists pvs_gap_add_unique_resultbound. pvs_gap_add_unique_resultbound + S (dst_index_add_unique_result) = (l)) -> (exists dst_positive_code_add_unique_resultfirst dst_positive_scale_add_unique_resultfirst dst_negative_code_add_unique_resultfirst dst_negative_scale_add_unique_resultfirst dst_positive_add_unique_resultfirst dst_negative_add_unique_resultfirst. (((H) = (((((dst_positive_code_add_unique_resultfirst) + (dst_positive_scale_add_unique_resultfirst)) * S ((dst_positive_code_add_unique_resultfirst) + (dst_positive_scale_add_unique_resultfirst)) + ((dst_positive_scale_add_unique_resultfirst) + (dst_positive_scale_add_unique_resultfirst))) + (((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) * S ((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) + ((dst_negative_scale_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)))) * S ((((dst_positive_code_add_unique_resultfirst) + (dst_positive_scale_add_unique_resultfirst)) * S ((dst_positive_code_add_unique_resultfirst) + (dst_positive_scale_add_unique_resultfirst)) + ((dst_positive_scale_add_unique_resultfirst) + (dst_positive_scale_add_unique_resultfirst))) + (((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) * S ((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) + ((dst_negative_scale_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)))) + ((((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) * S ((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) + ((dst_negative_scale_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst))) + (((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) * S ((dst_negative_code_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)) + ((dst_negative_scale_add_unique_resultfirst) + (dst_negative_scale_add_unique_resultfirst)))))) /\ (((((exists ff_h_pvs_add_unique_resultfirstpositive. ff_h_pvs_add_unique_resultfirstpositive + S (dst_positive_add_unique_resultfirst) = S ((S (dst_index_add_unique_result)) * dst_positive_scale_add_unique_resultfirst)) /\ exists ff_q_pvs_add_unique_resultfirstpositive. dst_positive_code_add_unique_resultfirst = ff_q_pvs_add_unique_resultfirstpositive * S ((S (dst_index_add_unique_result)) * dst_positive_scale_add_unique_resultfirst) + (dst_positive_add_unique_resultfirst))) /\ (((((exists ff_h_pvs_add_unique_resultfirstnegative. ff_h_pvs_add_unique_resultfirstnegative + S (dst_negative_add_unique_resultfirst) = S ((S (dst_index_add_unique_result)) * dst_negative_scale_add_unique_resultfirst)) /\ exists ff_q_pvs_add_unique_resultfirstnegative. dst_negative_code_add_unique_resultfirst = ff_q_pvs_add_unique_resultfirstnegative * S ((S (dst_index_add_unique_result)) * dst_negative_scale_add_unique_resultfirst) + (dst_negative_add_unique_resultfirst))) /\ (exists ge_balance_positive_add_unique_resultfirstvalue ge_balance_negative_add_unique_resultfirstvalue. (((((dst_first_add_unique_result) = 2 * (ge_balance_positive_add_unique_resultfirstvalue) /\ (ge_balance_negative_add_unique_resultfirstvalue) = 0) \/ exists ge_signed_half_add_unique_resultfirstvaluedecode. (((dst_first_add_unique_result) = 2 * ge_signed_half_add_unique_resultfirstvaluedecode + 1 /\ (ge_balance_positive_add_unique_resultfirstvalue) = 0) /\ (ge_balance_negative_add_unique_resultfirstvalue) = S ge_signed_half_add_unique_resultfirstvaluedecode))) /\ ((dst_positive_add_unique_resultfirst) + ge_balance_negative_add_unique_resultfirstvalue = (dst_negative_add_unique_resultfirst) + ge_balance_positive_add_unique_resultfirstvalue))))))))) -> (exists dst_positive_code_add_unique_resultsecond dst_positive_scale_add_unique_resultsecond dst_negative_code_add_unique_resultsecond dst_negative_scale_add_unique_resultsecond dst_positive_add_unique_resultsecond dst_negative_add_unique_resultsecond. (((K) = (((((dst_positive_code_add_unique_resultsecond) + (dst_positive_scale_add_unique_resultsecond)) * S ((dst_positive_code_add_unique_resultsecond) + (dst_positive_scale_add_unique_resultsecond)) + ((dst_positive_scale_add_unique_resultsecond) + (dst_positive_scale_add_unique_resultsecond))) + (((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) * S ((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) + ((dst_negative_scale_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)))) * S ((((dst_positive_code_add_unique_resultsecond) + (dst_positive_scale_add_unique_resultsecond)) * S ((dst_positive_code_add_unique_resultsecond) + (dst_positive_scale_add_unique_resultsecond)) + ((dst_positive_scale_add_unique_resultsecond) + (dst_positive_scale_add_unique_resultsecond))) + (((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) * S ((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) + ((dst_negative_scale_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)))) + ((((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) * S ((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) + ((dst_negative_scale_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond))) + (((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) * S ((dst_negative_code_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)) + ((dst_negative_scale_add_unique_resultsecond) + (dst_negative_scale_add_unique_resultsecond)))))) /\ (((((exists ff_h_pvs_add_unique_resultsecondpositive. ff_h_pvs_add_unique_resultsecondpositive + S (dst_positive_add_unique_resultsecond) = S ((S (dst_index_add_unique_result)) * dst_positive_scale_add_unique_resultsecond)) /\ exists ff_q_pvs_add_unique_resultsecondpositive. dst_positive_code_add_unique_resultsecond = ff_q_pvs_add_unique_resultsecondpositive * S ((S (dst_index_add_unique_result)) * dst_positive_scale_add_unique_resultsecond) + (dst_positive_add_unique_resultsecond))) /\ (((((exists ff_h_pvs_add_unique_resultsecondnegative. ff_h_pvs_add_unique_resultsecondnegative + S (dst_negative_add_unique_resultsecond) = S ((S (dst_index_add_unique_result)) * dst_negative_scale_add_unique_resultsecond)) /\ exists ff_q_pvs_add_unique_resultsecondnegative. dst_negative_code_add_unique_resultsecond = ff_q_pvs_add_unique_resultsecondnegative * S ((S (dst_index_add_unique_result)) * dst_negative_scale_add_unique_resultsecond) + (dst_negative_add_unique_resultsecond))) /\ (exists ge_balance_positive_add_unique_resultsecondvalue ge_balance_negative_add_unique_resultsecondvalue. (((((dst_second_add_unique_result) = 2 * (ge_balance_positive_add_unique_resultsecondvalue) /\ (ge_balance_negative_add_unique_resultsecondvalue) = 0) \/ exists ge_signed_half_add_unique_resultsecondvaluedecode. (((dst_second_add_unique_result) = 2 * ge_signed_half_add_unique_resultsecondvaluedecode + 1 /\ (ge_balance_positive_add_unique_resultsecondvalue) = 0) /\ (ge_balance_negative_add_unique_resultsecondvalue) = S ge_signed_half_add_unique_resultsecondvaluedecode))) /\ ((dst_positive_add_unique_resultsecond) + ge_balance_negative_add_unique_resultsecondvalue = (dst_negative_add_unique_resultsecond) + ge_balance_positive_add_unique_resultsecondvalue))))))))) -> dst_first_add_unique_result = dst_second_add_unique_result)

Complete tactic proof in conservative notation

All 70 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

70 script commands · 16 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.

Named ingredients (2)
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 K
  5. L5
    intro l
  6. L6
    intro hop
  7. L7
    intro hother
  8. L8
    intro i
  9. L9
    intro u
  10. L10
    intro v
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro hu
  3. L13
    intro hv
03Establish ht0L14–14

Establish this local claim before using it. It is not an additional assumption.

  1. L14
04Separate the logical casesL15–17

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

  1. L15
    cases hop
  2. L16
    cases hop_right
  3. L17
    cases hop_right_right
05Use earlier factsL18–18

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

  1. L18
    exact hop_left
06Establish he0L19–24

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

  1. L19
    have he0 : ∃ z. ArithAt(F,i,z)Definitions: ArithAt(F,i,z)Original native command in the exact edition
  2. L20
    specialize signed_table_lookup_any (l)
  3. L21
    specialize signed_table_lookup_any (F)
  4. L22
    specialize signed_table_lookup_any (i)
  5. L23
    apply signed_table_lookup_any
  6. L24
    exact ht0
07Separate the logical casesL25–25

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

  1. L25
    cases he0
08Establish ht1L26–26

Establish this local claim before using it. It is not an additional assumption.

  1. L26
09Separate the logical casesL27–29

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

  1. L27
    cases hop
  2. L28
    cases hop_right
  3. L29
    cases hop_right_right
10Use earlier factsL30–30

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

  1. L30
    exact hop_right_left
11Establish he1L31–36

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

  1. L31
    have he1 : ∃ z. ArithAt(G,i,z)Definitions: ArithAt(G,i,z)Original native command in the exact edition
  2. L32
    specialize signed_table_lookup_any (l)
  3. L33
    specialize signed_table_lookup_any (G)
  4. L34
    specialize signed_table_lookup_any (i)
  5. L35
    apply signed_table_lookup_any
  6. L36
    exact ht1
12Separate the logical casesL37–37

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

  1. L37
    cases he1
13Use earlier factsL38–47

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

  1. L38
    specialize signed_add_functional (x)
  2. L39
    specialize signed_add_functional (x1)
  3. L40
    specialize signed_add_functional (u)
  4. L41
    specialize signed_add_functional (v)
  5. L42
    apply signed_add_functional
  6. L43
    specialize signed_table_add_lookup (F)
  7. L44
    specialize signed_table_add_lookup (G)
  8. L45
    specialize signed_table_add_lookup (H)
  9. L46
    specialize signed_table_add_lookup (l)
  10. L47
    specialize signed_table_add_lookup (i)
14Use earlier factsL48–57

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

  1. L48
    specialize signed_table_add_lookup (x)
  2. L49
    specialize signed_table_add_lookup (x1)
  3. L50
    specialize signed_table_add_lookup (u)
  4. L51
    apply signed_table_add_lookup
  5. L52
    exact hop
  6. L53
    exact hi
  7. L54
    exact he0_witness
  8. L55
    exact he1_witness
  9. L56
    exact hu
  10. L57
    specialize signed_table_add_lookup (F)
15Use earlier factsL58–67

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

  1. L58
    specialize signed_table_add_lookup (G)
  2. L59
    specialize signed_table_add_lookup (K)
  3. L60
    specialize signed_table_add_lookup (l)
  4. L61
    specialize signed_table_add_lookup (i)
  5. L62
    specialize signed_table_add_lookup (x)
  6. L63
    specialize signed_table_add_lookup (x1)
  7. L64
    specialize signed_table_add_lookup (v)
  8. L65
    apply signed_table_add_lookup
  9. L66
    exact hother
  10. L67
    exact hi
16Use earlier factsL68–70

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

  1. L68
    exact he0_witness
  2. L69
    exact he1_witness
  3. L70
    exact hv

Library-wide reading audit

Original defined command ledger · 70 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro H
  4. 0004intro K
  5. 0005intro l
  6. 0006intro hop
  7. 0007intro hother
  8. 0008intro i
  9. 0009intro u
  10. 0010intro v
  11. 0011intro hi
  12. 0012intro hu
  13. 0013intro hv
  14. 0014have ht0 : ArithTable(l,F)
  15. 0015cases hop
  16. 0016cases hop_right
  17. 0017cases hop_right_right
  18. 0018exact hop_left
  19. 0019have he0 : ∃ z. ArithAt(F,i,z)
  20. 0020specialize signed_table_lookup_any (l)
  21. 0021specialize signed_table_lookup_any (F)
  22. 0022specialize signed_table_lookup_any (i)
  23. 0023apply signed_table_lookup_any
  24. 0024exact ht0
  25. 0025cases he0
  26. 0026have ht1 : ArithTable(l,G)
  27. 0027cases hop
  28. 0028cases hop_right
  29. 0029cases hop_right_right
  30. 0030exact hop_right_left
  31. 0031have he1 : ∃ z. ArithAt(G,i,z)
  32. 0032specialize signed_table_lookup_any (l)
  33. 0033specialize signed_table_lookup_any (G)
  34. 0034specialize signed_table_lookup_any (i)
  35. 0035apply signed_table_lookup_any
  36. 0036exact ht1
  37. 0037cases he1
  38. 0038specialize signed_add_functional (x)
  39. 0039specialize signed_add_functional (x1)
  40. 0040specialize signed_add_functional (u)
  41. 0041specialize signed_add_functional (v)
  42. 0042apply signed_add_functional
  43. 0043specialize signed_table_add_lookup (F)
  44. 0044specialize signed_table_add_lookup (G)
  45. 0045specialize signed_table_add_lookup (H)
  46. 0046specialize signed_table_add_lookup (l)
  47. 0047specialize signed_table_add_lookup (i)
  48. 0048specialize signed_table_add_lookup (x)
  49. 0049specialize signed_table_add_lookup (x1)
  50. 0050specialize signed_table_add_lookup (u)
  51. 0051apply signed_table_add_lookup
  52. 0052exact hop
  53. 0053exact hi
  54. 0054exact he0_witness
  55. 0055exact he1_witness
  56. 0056exact hu
  57. 0057specialize signed_table_add_lookup (F)
  58. 0058specialize signed_table_add_lookup (G)
  59. 0059specialize signed_table_add_lookup (K)
  60. 0060specialize signed_table_add_lookup (l)
  61. 0061specialize signed_table_add_lookup (i)
  62. 0062specialize signed_table_add_lookup (x)
  63. 0063specialize signed_table_add_lookup (x1)
  64. 0064specialize signed_table_add_lookup (v)
  65. 0065apply signed_table_add_lookup
  66. 0066exact hother
  67. 0067exact hi
  68. 0068exact he0_witness
  69. 0069exact he1_witness
  70. 0070exact hv