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
∀ l. ∀ F. ∀ G. ∀ H. ArithAdd(F,G,H,l) → ∃ x. ∃ y. ∃ z. SignedPrefixSum(F,l,x) ∧ (SignedPrefixSum(G,l,y) ∧ (SignedPrefixSum(H,l,z) ∧ SignedAdd(x,y,z)))Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall l F G H. (((exists dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphleft_table dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table. (((F) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)))))) /\ (forall dst_index_signed_prefix_sum_pointwise_add_construct_graphleft_table. (exists pvs_le_gap_signed_prefix_sum_pointwise_add_construct_graphleft_tabledomain. pvs_le_gap_signed_prefix_sum_pointwise_add_construct_graphleft_tabledomain + (dst_index_signed_prefix_sum_pointwise_add_construct_graphleft_table) = (l)) -> exists dst_positive_signed_prefix_sum_pointwise_add_construct_graphleft_table dst_negative_signed_prefix_sum_pointwise_add_construct_graphleft_table dst_value_signed_prefix_sum_pointwise_add_construct_graphleft_table. ((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrypositive. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrypositive + S (dst_positive_signed_prefix_sum_pointwise_add_construct_graphleft_table) = S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrypositive. dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphleft_table = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrypositive * S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_positive_signed_prefix_sum_pointwise_add_construct_graphleft_table))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrynegative. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrynegative + S (dst_negative_signed_prefix_sum_pointwise_add_construct_graphleft_table) = S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrynegative. dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphleft_table = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphleft_tableentrynegative * S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphleft_table)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphleft_table) + (dst_negative_signed_prefix_sum_pointwise_add_construct_graphleft_table))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue. (((((dst_value_signed_prefix_sum_pointwise_add_construct_graphleft_table) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvaluedecode. (((dst_value_signed_prefix_sum_pointwise_add_construct_graphleft_table) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvaluedecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvaluedecode))) /\ ((dst_positive_signed_prefix_sum_pointwise_add_construct_graphleft_table) + ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue = (dst_negative_signed_prefix_sum_pointwise_add_construct_graphleft_table) + ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphleft_tableentryvalue))))))))) /\ (((exists dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphright_table dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table. (((G) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)))))) /\ (forall dst_index_signed_prefix_sum_pointwise_add_construct_graphright_table. (exists pvs_le_gap_signed_prefix_sum_pointwise_add_construct_graphright_tabledomain. pvs_le_gap_signed_prefix_sum_pointwise_add_construct_graphright_tabledomain + (dst_index_signed_prefix_sum_pointwise_add_construct_graphright_table) = (l)) -> exists dst_positive_signed_prefix_sum_pointwise_add_construct_graphright_table dst_negative_signed_prefix_sum_pointwise_add_construct_graphright_table dst_value_signed_prefix_sum_pointwise_add_construct_graphright_table. ((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrypositive. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrypositive + S (dst_positive_signed_prefix_sum_pointwise_add_construct_graphright_table) = S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphright_table)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrypositive. dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphright_table = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrypositive * S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphright_table)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_positive_signed_prefix_sum_pointwise_add_construct_graphright_table))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrynegative. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrynegative + S (dst_negative_signed_prefix_sum_pointwise_add_construct_graphright_table) = S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphright_table)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrynegative. dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphright_table = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphright_tableentrynegative * S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphright_table)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphright_table) + (dst_negative_signed_prefix_sum_pointwise_add_construct_graphright_table))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue. (((((dst_value_signed_prefix_sum_pointwise_add_construct_graphright_table) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvaluedecode. (((dst_value_signed_prefix_sum_pointwise_add_construct_graphright_table) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvaluedecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvaluedecode))) /\ ((dst_positive_signed_prefix_sum_pointwise_add_construct_graphright_table) + ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue = (dst_negative_signed_prefix_sum_pointwise_add_construct_graphright_table) + ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphright_tableentryvalue))))))))) /\ (((exists dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table. (((H) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)))))) /\ (forall dst_index_signed_prefix_sum_pointwise_add_construct_graphoutput_table. (exists pvs_le_gap_signed_prefix_sum_pointwise_add_construct_graphoutput_tabledomain. pvs_le_gap_signed_prefix_sum_pointwise_add_construct_graphoutput_tabledomain + (dst_index_signed_prefix_sum_pointwise_add_construct_graphoutput_table) = (l)) -> exists dst_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_table dst_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_table dst_value_signed_prefix_sum_pointwise_add_construct_graphoutput_table. ((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrypositive. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrypositive + S (dst_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_table) = S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrypositive. dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrypositive * S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_table))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrynegative. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrynegative + S (dst_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_table) = S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrynegative. dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphoutput_table = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentrynegative * S ((S (dst_index_signed_prefix_sum_pointwise_add_construct_graphoutput_table)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + (dst_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_table))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue. (((((dst_value_signed_prefix_sum_pointwise_add_construct_graphoutput_table) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvaluedecode. (((dst_value_signed_prefix_sum_pointwise_add_construct_graphoutput_table) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvaluedecode))) /\ ((dst_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue = (dst_negative_signed_prefix_sum_pointwise_add_construct_graphoutput_table) + ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphoutput_tableentryvalue))))))))) /\ (forall sto_index_signed_prefix_sum_pointwise_add_construct_graphentries. (exists pvs_gap_signed_prefix_sum_pointwise_add_construct_graphentriesbound. pvs_gap_signed_prefix_sum_pointwise_add_construct_graphentriesbound + S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries) = (l)) -> exists sto_left_signed_prefix_sum_pointwise_add_construct_graphentries sto_right_signed_prefix_sum_pointwise_add_construct_graphentries sto_output_signed_prefix_sum_pointwise_add_construct_graphentries. ((exists dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft. (((F) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)))))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftpositive. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftpositive + S (dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) = S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftpositive. dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftpositive * S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftnegative. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftnegative + S (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) = S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftnegative. dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftnegative * S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue. (((((sto_left_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvaluedecode. (((sto_left_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvaluedecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvaluedecode))) /\ ((dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue = (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryleft) + ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryleftvalue))))))))) /\ (((exists dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryright dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryright. (((G) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)))))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightpositive. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightpositive + S (dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) = S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightpositive. dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightpositive * S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryright))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightnegative. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightnegative + S (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) = S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightnegative. dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryright = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightnegative * S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryright))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue. (((((sto_right_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvaluedecode. (((sto_right_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvaluedecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvaluedecode))) /\ ((dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue = (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryright) + ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryrightvalue))))))))) /\ (((exists dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput. (((H) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)))))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputpositive. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputpositive + S (dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) = S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputpositive. dst_positive_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputpositive * S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_positive_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput))) /\ (((((exists ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputnegative. ff_h_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputnegative + S (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) = S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput)) /\ exists ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputnegative. dst_negative_code_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput = ff_q_pvs_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputnegative * S ((S (sto_index_signed_prefix_sum_pointwise_add_construct_graphentries)) * dst_negative_scale_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue. (((((sto_output_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvaluedecode. (((sto_output_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvaluedecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvaluedecode))) /\ ((dst_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + ge_balance_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue = (dst_negative_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutput) + ge_balance_positive_signed_prefix_sum_pointwise_add_construct_graphentriesentryoutputvalue))))))))) /\ (exists dsa_ap_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation dsa_an_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation dsa_bp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation dsa_bn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation dsa_cp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation dsa_cn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation. (((((sto_left_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * (dsa_ap_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) /\ (dsa_an_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationleft. (((sto_left_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationleft + 1 /\ (dsa_ap_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = 0) /\ (dsa_an_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationleft))) /\ ((((((sto_right_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * (dsa_bp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) /\ (dsa_bn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationright. (((sto_right_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationright + 1 /\ (dsa_bp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = 0) /\ (dsa_bn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationright))) /\ ((((((sto_output_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * (dsa_cp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) /\ (dsa_cn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationoutput. (((sto_output_signed_prefix_sum_pointwise_add_construct_graphentries) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationoutput + 1 /\ (dsa_cp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = 0) /\ (dsa_cn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) = S ge_signed_half_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperationoutput))) /\ ((dsa_ap_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation + dsa_bp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) + dsa_cn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation = (dsa_an_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation + dsa_bn_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation) + dsa_cp_signed_prefix_sum_pointwise_add_construct_graphentriesentryoperation))))))))))))))))))) -> exists a b c. ((exists dst_positive_code_signed_prefix_sum_pointwise_add_result0 dst_positive_scale_signed_prefix_sum_pointwise_add_result0 dst_negative_code_signed_prefix_sum_pointwise_add_result0 dst_negative_scale_signed_prefix_sum_pointwise_add_result0 dst_positive_sum_signed_prefix_sum_pointwise_add_result0 dst_negative_sum_signed_prefix_sum_pointwise_add_result0. (((F) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_result0) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result0)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_result0) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result0)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_result0) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result0))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_result0) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result0)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_result0) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result0)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_result0) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result0))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result0) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result0)))))) /\ (((exists fs_u_dst_signed_prefix_sum_pointwise_add_result0positive fs_v_dst_signed_prefix_sum_pointwise_add_result0positive. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_start. fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_start + S (0) = S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_start. fs_u_dst_signed_prefix_sum_pointwise_add_result0positive = fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_start * S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive) + (0))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_terminal. fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_terminal + S (dst_positive_sum_signed_prefix_sum_pointwise_add_result0) = S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_terminal. fs_u_dst_signed_prefix_sum_pointwise_add_result0positive = fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_terminal * S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive) + (dst_positive_sum_signed_prefix_sum_pointwise_add_result0))) /\ forall fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps. (exists fs_lt_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_bound. fs_lt_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_bound + S fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps = l) -> exists fs_a_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps fs_r_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps fs_s_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_summand. fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_summand + S (fs_a_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)) * dst_positive_scale_signed_prefix_sum_pointwise_add_result0)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_summand. dst_positive_code_signed_prefix_sum_pointwise_add_result0 = fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_summand * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)) * dst_positive_scale_signed_prefix_sum_pointwise_add_result0) + (fs_a_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_partial. fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_partial + S (fs_r_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_partial. fs_u_dst_signed_prefix_sum_pointwise_add_result0positive = fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_partial * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive) + (fs_r_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_successor. fs_h_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_successor + S (fs_s_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps) = S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_successor. fs_u_dst_signed_prefix_sum_pointwise_add_result0positive = fs_q_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps_successor * S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0positive) + (fs_s_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps))) /\ fs_s_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps = fs_r_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps + fs_a_dst_signed_prefix_sum_pointwise_add_result0positive_body_steps)))))) /\ (((exists fs_u_dst_signed_prefix_sum_pointwise_add_result0negative fs_v_dst_signed_prefix_sum_pointwise_add_result0negative. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_start. fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_start + S (0) = S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_start. fs_u_dst_signed_prefix_sum_pointwise_add_result0negative = fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_start * S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative) + (0))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_terminal. fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_terminal + S (dst_negative_sum_signed_prefix_sum_pointwise_add_result0) = S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_terminal. fs_u_dst_signed_prefix_sum_pointwise_add_result0negative = fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_terminal * S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative) + (dst_negative_sum_signed_prefix_sum_pointwise_add_result0))) /\ forall fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps. (exists fs_lt_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_bound. fs_lt_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_bound + S fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps = l) -> exists fs_a_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps fs_r_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps fs_s_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_summand. fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_summand + S (fs_a_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)) * dst_negative_scale_signed_prefix_sum_pointwise_add_result0)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_summand. dst_negative_code_signed_prefix_sum_pointwise_add_result0 = fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_summand * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)) * dst_negative_scale_signed_prefix_sum_pointwise_add_result0) + (fs_a_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_partial. fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_partial + S (fs_r_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_partial. fs_u_dst_signed_prefix_sum_pointwise_add_result0negative = fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_partial * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative) + (fs_r_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_successor. fs_h_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_successor + S (fs_s_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps) = S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_successor. fs_u_dst_signed_prefix_sum_pointwise_add_result0negative = fs_q_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps_successor * S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result0negative) + (fs_s_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps))) /\ fs_s_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps = fs_r_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps + fs_a_dst_signed_prefix_sum_pointwise_add_result0negative_body_steps)))))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_result0result ge_balance_negative_signed_prefix_sum_pointwise_add_result0result. (((((a) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_result0result) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_result0result) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_result0resultdecode. (((a) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_result0resultdecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_result0result) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_result0result) = S ge_signed_half_signed_prefix_sum_pointwise_add_result0resultdecode))) /\ ((dst_positive_sum_signed_prefix_sum_pointwise_add_result0) + ge_balance_negative_signed_prefix_sum_pointwise_add_result0result = (dst_negative_sum_signed_prefix_sum_pointwise_add_result0) + ge_balance_positive_signed_prefix_sum_pointwise_add_result0result))))))))) /\ (((exists dst_positive_code_signed_prefix_sum_pointwise_add_result1 dst_positive_scale_signed_prefix_sum_pointwise_add_result1 dst_negative_code_signed_prefix_sum_pointwise_add_result1 dst_negative_scale_signed_prefix_sum_pointwise_add_result1 dst_positive_sum_signed_prefix_sum_pointwise_add_result1 dst_negative_sum_signed_prefix_sum_pointwise_add_result1. (((G) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_result1) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result1)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_result1) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result1)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_result1) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result1))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_result1) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result1)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_result1) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result1)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_result1) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result1))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result1) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result1)))))) /\ (((exists fs_u_dst_signed_prefix_sum_pointwise_add_result1positive fs_v_dst_signed_prefix_sum_pointwise_add_result1positive. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_start. fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_start + S (0) = S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_start. fs_u_dst_signed_prefix_sum_pointwise_add_result1positive = fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_start * S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive) + (0))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_terminal. fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_terminal + S (dst_positive_sum_signed_prefix_sum_pointwise_add_result1) = S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_terminal. fs_u_dst_signed_prefix_sum_pointwise_add_result1positive = fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_terminal * S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive) + (dst_positive_sum_signed_prefix_sum_pointwise_add_result1))) /\ forall fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps. (exists fs_lt_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_bound. fs_lt_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_bound + S fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps = l) -> exists fs_a_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps fs_r_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps fs_s_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_summand. fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_summand + S (fs_a_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)) * dst_positive_scale_signed_prefix_sum_pointwise_add_result1)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_summand. dst_positive_code_signed_prefix_sum_pointwise_add_result1 = fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_summand * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)) * dst_positive_scale_signed_prefix_sum_pointwise_add_result1) + (fs_a_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_partial. fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_partial + S (fs_r_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_partial. fs_u_dst_signed_prefix_sum_pointwise_add_result1positive = fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_partial * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive) + (fs_r_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_successor. fs_h_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_successor + S (fs_s_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps) = S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_successor. fs_u_dst_signed_prefix_sum_pointwise_add_result1positive = fs_q_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps_successor * S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1positive) + (fs_s_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps))) /\ fs_s_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps = fs_r_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps + fs_a_dst_signed_prefix_sum_pointwise_add_result1positive_body_steps)))))) /\ (((exists fs_u_dst_signed_prefix_sum_pointwise_add_result1negative fs_v_dst_signed_prefix_sum_pointwise_add_result1negative. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_start. fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_start + S (0) = S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_start. fs_u_dst_signed_prefix_sum_pointwise_add_result1negative = fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_start * S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative) + (0))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_terminal. fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_terminal + S (dst_negative_sum_signed_prefix_sum_pointwise_add_result1) = S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_terminal. fs_u_dst_signed_prefix_sum_pointwise_add_result1negative = fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_terminal * S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative) + (dst_negative_sum_signed_prefix_sum_pointwise_add_result1))) /\ forall fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps. (exists fs_lt_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_bound. fs_lt_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_bound + S fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps = l) -> exists fs_a_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps fs_r_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps fs_s_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_summand. fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_summand + S (fs_a_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)) * dst_negative_scale_signed_prefix_sum_pointwise_add_result1)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_summand. dst_negative_code_signed_prefix_sum_pointwise_add_result1 = fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_summand * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)) * dst_negative_scale_signed_prefix_sum_pointwise_add_result1) + (fs_a_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_partial. fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_partial + S (fs_r_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_partial. fs_u_dst_signed_prefix_sum_pointwise_add_result1negative = fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_partial * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative) + (fs_r_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_successor. fs_h_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_successor + S (fs_s_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps) = S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_successor. fs_u_dst_signed_prefix_sum_pointwise_add_result1negative = fs_q_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps_successor * S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result1negative) + (fs_s_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps))) /\ fs_s_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps = fs_r_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps + fs_a_dst_signed_prefix_sum_pointwise_add_result1negative_body_steps)))))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_result1result ge_balance_negative_signed_prefix_sum_pointwise_add_result1result. (((((b) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_result1result) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_result1result) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_result1resultdecode. (((b) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_result1resultdecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_result1result) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_result1result) = S ge_signed_half_signed_prefix_sum_pointwise_add_result1resultdecode))) /\ ((dst_positive_sum_signed_prefix_sum_pointwise_add_result1) + ge_balance_negative_signed_prefix_sum_pointwise_add_result1result = (dst_negative_sum_signed_prefix_sum_pointwise_add_result1) + ge_balance_positive_signed_prefix_sum_pointwise_add_result1result))))))))) /\ (((exists dst_positive_code_signed_prefix_sum_pointwise_add_result2 dst_positive_scale_signed_prefix_sum_pointwise_add_result2 dst_negative_code_signed_prefix_sum_pointwise_add_result2 dst_negative_scale_signed_prefix_sum_pointwise_add_result2 dst_positive_sum_signed_prefix_sum_pointwise_add_result2 dst_negative_sum_signed_prefix_sum_pointwise_add_result2. (((H) = (((((dst_positive_code_signed_prefix_sum_pointwise_add_result2) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result2)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_result2) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result2)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_result2) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result2))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)))) * S ((((dst_positive_code_signed_prefix_sum_pointwise_add_result2) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result2)) * S ((dst_positive_code_signed_prefix_sum_pointwise_add_result2) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result2)) + ((dst_positive_scale_signed_prefix_sum_pointwise_add_result2) + (dst_positive_scale_signed_prefix_sum_pointwise_add_result2))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)))) + ((((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2))) + (((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) * S ((dst_negative_code_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) + ((dst_negative_scale_signed_prefix_sum_pointwise_add_result2) + (dst_negative_scale_signed_prefix_sum_pointwise_add_result2)))))) /\ (((exists fs_u_dst_signed_prefix_sum_pointwise_add_result2positive fs_v_dst_signed_prefix_sum_pointwise_add_result2positive. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_start. fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_start + S (0) = S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_start. fs_u_dst_signed_prefix_sum_pointwise_add_result2positive = fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_start * S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive) + (0))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_terminal. fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_terminal + S (dst_positive_sum_signed_prefix_sum_pointwise_add_result2) = S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_terminal. fs_u_dst_signed_prefix_sum_pointwise_add_result2positive = fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_terminal * S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive) + (dst_positive_sum_signed_prefix_sum_pointwise_add_result2))) /\ forall fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps. (exists fs_lt_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_bound. fs_lt_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_bound + S fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps = l) -> exists fs_a_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps fs_r_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps fs_s_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_summand. fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_summand + S (fs_a_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)) * dst_positive_scale_signed_prefix_sum_pointwise_add_result2)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_summand. dst_positive_code_signed_prefix_sum_pointwise_add_result2 = fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_summand * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)) * dst_positive_scale_signed_prefix_sum_pointwise_add_result2) + (fs_a_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_partial. fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_partial + S (fs_r_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_partial. fs_u_dst_signed_prefix_sum_pointwise_add_result2positive = fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_partial * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive) + (fs_r_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_successor. fs_h_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_successor + S (fs_s_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps) = S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_successor. fs_u_dst_signed_prefix_sum_pointwise_add_result2positive = fs_q_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps_successor * S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2positive) + (fs_s_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps))) /\ fs_s_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps = fs_r_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps + fs_a_dst_signed_prefix_sum_pointwise_add_result2positive_body_steps)))))) /\ (((exists fs_u_dst_signed_prefix_sum_pointwise_add_result2negative fs_v_dst_signed_prefix_sum_pointwise_add_result2negative. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_start. fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_start + S (0) = S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_start. fs_u_dst_signed_prefix_sum_pointwise_add_result2negative = fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_start * S ((S (0)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative) + (0))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_terminal. fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_terminal + S (dst_negative_sum_signed_prefix_sum_pointwise_add_result2) = S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_terminal. fs_u_dst_signed_prefix_sum_pointwise_add_result2negative = fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_terminal * S ((S (l)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative) + (dst_negative_sum_signed_prefix_sum_pointwise_add_result2))) /\ forall fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps. (exists fs_lt_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_bound. fs_lt_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_bound + S fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps = l) -> exists fs_a_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps fs_r_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps fs_s_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps. ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_summand. fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_summand + S (fs_a_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)) * dst_negative_scale_signed_prefix_sum_pointwise_add_result2)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_summand. dst_negative_code_signed_prefix_sum_pointwise_add_result2 = fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_summand * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)) * dst_negative_scale_signed_prefix_sum_pointwise_add_result2) + (fs_a_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_partial. fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_partial + S (fs_r_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps) = S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_partial. fs_u_dst_signed_prefix_sum_pointwise_add_result2negative = fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_partial * S ((S (fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative) + (fs_r_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps))) /\ ((((exists fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_successor. fs_h_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_successor + S (fs_s_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps) = S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative)) /\ exists fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_successor. fs_u_dst_signed_prefix_sum_pointwise_add_result2negative = fs_q_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps_successor * S ((S (S fs_i_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)) * fs_v_dst_signed_prefix_sum_pointwise_add_result2negative) + (fs_s_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps))) /\ fs_s_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps = fs_r_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps + fs_a_dst_signed_prefix_sum_pointwise_add_result2negative_body_steps)))))) /\ (exists ge_balance_positive_signed_prefix_sum_pointwise_add_result2result ge_balance_negative_signed_prefix_sum_pointwise_add_result2result. (((((c) = 2 * (ge_balance_positive_signed_prefix_sum_pointwise_add_result2result) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_result2result) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_result2resultdecode. (((c) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_result2resultdecode + 1 /\ (ge_balance_positive_signed_prefix_sum_pointwise_add_result2result) = 0) /\ (ge_balance_negative_signed_prefix_sum_pointwise_add_result2result) = S ge_signed_half_signed_prefix_sum_pointwise_add_result2resultdecode))) /\ ((dst_positive_sum_signed_prefix_sum_pointwise_add_result2) + ge_balance_negative_signed_prefix_sum_pointwise_add_result2result = (dst_negative_sum_signed_prefix_sum_pointwise_add_result2) + ge_balance_positive_signed_prefix_sum_pointwise_add_result2result))))))))) /\ (exists dsa_ap_signed_prefix_sum_pointwise_add_result_operation dsa_an_signed_prefix_sum_pointwise_add_result_operation dsa_bp_signed_prefix_sum_pointwise_add_result_operation dsa_bn_signed_prefix_sum_pointwise_add_result_operation dsa_cp_signed_prefix_sum_pointwise_add_result_operation dsa_cn_signed_prefix_sum_pointwise_add_result_operation. (((((a) = 2 * (dsa_ap_signed_prefix_sum_pointwise_add_result_operation) /\ (dsa_an_signed_prefix_sum_pointwise_add_result_operation) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_result_operationleft. (((a) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_result_operationleft + 1 /\ (dsa_ap_signed_prefix_sum_pointwise_add_result_operation) = 0) /\ (dsa_an_signed_prefix_sum_pointwise_add_result_operation) = S ge_signed_half_signed_prefix_sum_pointwise_add_result_operationleft))) /\ ((((((b) = 2 * (dsa_bp_signed_prefix_sum_pointwise_add_result_operation) /\ (dsa_bn_signed_prefix_sum_pointwise_add_result_operation) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_result_operationright. (((b) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_result_operationright + 1 /\ (dsa_bp_signed_prefix_sum_pointwise_add_result_operation) = 0) /\ (dsa_bn_signed_prefix_sum_pointwise_add_result_operation) = S ge_signed_half_signed_prefix_sum_pointwise_add_result_operationright))) /\ ((((((c) = 2 * (dsa_cp_signed_prefix_sum_pointwise_add_result_operation) /\ (dsa_cn_signed_prefix_sum_pointwise_add_result_operation) = 0) \/ exists ge_signed_half_signed_prefix_sum_pointwise_add_result_operationoutput. (((c) = 2 * ge_signed_half_signed_prefix_sum_pointwise_add_result_operationoutput + 1 /\ (dsa_cp_signed_prefix_sum_pointwise_add_result_operation) = 0) /\ (dsa_cn_signed_prefix_sum_pointwise_add_result_operation) = S ge_signed_half_signed_prefix_sum_pointwise_add_result_operationoutput))) /\ ((dsa_ap_signed_prefix_sum_pointwise_add_result_operation + dsa_bp_signed_prefix_sum_pointwise_add_result_operation) + dsa_cn_signed_prefix_sum_pointwise_add_result_operation = (dsa_an_signed_prefix_sum_pointwise_add_result_operation + dsa_bn_signed_prefix_sum_pointwise_add_result_operation) + dsa_cp_signed_prefix_sum_pointwise_add_result_operation))))))))))))Complete tactic proof in conservative notation
All 56 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
56 script commands · 21 reading checkpoints · 3 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 (1)
01Fix variables and assumptionsL1–5
02Establish hs0L6–10
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply arithmetic signed sum exists.
- L6
have hs0 : ∃ z. SignedPrefixSum(F,l,z)Definitions: SignedPrefixSum(F,l,z)Original native command in the exact edition - L7
specialize arithmetic_signed_sum_exists (l) - L8
specialize arithmetic_signed_sum_exists (F) - L9
specialize arithmetic_signed_sum_exists (l) - L10
apply arithmetic_signed_sum_exists
03Separate the logical casesL11–13
04Use earlier factsL14–14
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L14
exact hpoint_left
05Separate the logical casesL15–15
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L15
cases hs0
06Establish hs1L16–20
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply arithmetic signed sum exists.
- L16
have hs1 : ∃ z. SignedPrefixSum(G,l,z)Definitions: SignedPrefixSum(G,l,z)Original native command in the exact edition - L17
specialize arithmetic_signed_sum_exists (l) - L18
specialize arithmetic_signed_sum_exists (G) - L19
specialize arithmetic_signed_sum_exists (l) - L20
apply arithmetic_signed_sum_exists
07Separate the logical casesL21–23
08Use earlier factsL24–24
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L24
exact hpoint_right_left
09Separate the logical casesL25–25
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L25
cases hs1
10Establish hs2L26–30
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply arithmetic signed sum exists.
- L26
have hs2 : ∃ z. SignedPrefixSum(H,l,z)Definitions: SignedPrefixSum(H,l,z)Original native command in the exact edition - L27
specialize arithmetic_signed_sum_exists (l) - L28
specialize arithmetic_signed_sum_exists (H) - L29
specialize arithmetic_signed_sum_exists (l) - L30
apply arithmetic_signed_sum_exists
11Separate the logical casesL31–33
12Use earlier factsL34–34
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L34
exact hpoint_right_right_left
13Separate the logical casesL35–35
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L35
cases hs2
14Construct an explicit witnessL36–38
15Separate the logical casesL39–39
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L39
split
16Use earlier factsL40–40
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L40
exact hs0_witness
17Separate the logical casesL41–41
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L41
split
18Use earlier factsL42–42
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L42
exact hs1_witness
19Separate the logical casesL43–43
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L43
split
20Use earlier factsL44–53
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L44
exact hs2_witness - L45
specialize signed_prefix_sum_pointwise_add (l) - L46
specialize signed_prefix_sum_pointwise_add (F) - L47
specialize signed_prefix_sum_pointwise_add (G) - L48
specialize signed_prefix_sum_pointwise_add (H) - L49
specialize signed_prefix_sum_pointwise_add (x) - L50
specialize signed_prefix_sum_pointwise_add (x1) - L51
specialize signed_prefix_sum_pointwise_add (x2) - L52
apply signed_prefix_sum_pointwise_add - L53
exact hpoint
Original defined command ledger · 56 lines
- 0001
intro l - 0002
intro F - 0003
intro G - 0004
intro H - 0005
intro hpoint - 0006
have hs0 : ∃ z. SignedPrefixSum(F,l,z) - 0007
specialize arithmetic_signed_sum_exists (l) - 0008
specialize arithmetic_signed_sum_exists (F) - 0009
specialize arithmetic_signed_sum_exists (l) - 0010
apply arithmetic_signed_sum_exists - 0011
cases hpoint - 0012
cases hpoint_right - 0013
cases hpoint_right_right - 0014
exact hpoint_left - 0015
cases hs0 - 0016
have hs1 : ∃ z. SignedPrefixSum(G,l,z) - 0017
specialize arithmetic_signed_sum_exists (l) - 0018
specialize arithmetic_signed_sum_exists (G) - 0019
specialize arithmetic_signed_sum_exists (l) - 0020
apply arithmetic_signed_sum_exists - 0021
cases hpoint - 0022
cases hpoint_right - 0023
cases hpoint_right_right - 0024
exact hpoint_right_left - 0025
cases hs1 - 0026
have hs2 : ∃ z. SignedPrefixSum(H,l,z) - 0027
specialize arithmetic_signed_sum_exists (l) - 0028
specialize arithmetic_signed_sum_exists (H) - 0029
specialize arithmetic_signed_sum_exists (l) - 0030
apply arithmetic_signed_sum_exists - 0031
cases hpoint - 0032
cases hpoint_right - 0033
cases hpoint_right_right - 0034
exact hpoint_right_right_left - 0035
cases hs2 - 0036
exists x - 0037
exists x1 - 0038
exists x2 - 0039
split - 0040
exact hs0_witness - 0041
split - 0042
exact hs1_witness - 0043
split - 0044
exact hs2_witness - 0045
specialize signed_prefix_sum_pointwise_add (l) - 0046
specialize signed_prefix_sum_pointwise_add (F) - 0047
specialize signed_prefix_sum_pointwise_add (G) - 0048
specialize signed_prefix_sum_pointwise_add (H) - 0049
specialize signed_prefix_sum_pointwise_add (x) - 0050
specialize signed_prefix_sum_pointwise_add (x1) - 0051
specialize signed_prefix_sum_pointwise_add (x2) - 0052
apply signed_prefix_sum_pointwise_add - 0053
exact hpoint - 0054
exact hs0_witness - 0055
exact hs1_witness - 0056
exact hs2_witness