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
∀ a. ∀ F. ∀ G. ∀ l. ArithScale(a,F,G,S l) → ArithScale(a,F,G,l)
Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall a F G l. (((exists dst_positive_code_scalar_restrict_inputinput_table dst_positive_scale_scalar_restrict_inputinput_table dst_negative_code_scalar_restrict_inputinput_table dst_negative_scale_scalar_restrict_inputinput_table. (((F) = (((((dst_positive_code_scalar_restrict_inputinput_table) + (dst_positive_scale_scalar_restrict_inputinput_table)) * S ((dst_positive_code_scalar_restrict_inputinput_table) + (dst_positive_scale_scalar_restrict_inputinput_table)) + ((dst_positive_scale_scalar_restrict_inputinput_table) + (dst_positive_scale_scalar_restrict_inputinput_table))) + (((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) * S ((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) + ((dst_negative_scale_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)))) * S ((((dst_positive_code_scalar_restrict_inputinput_table) + (dst_positive_scale_scalar_restrict_inputinput_table)) * S ((dst_positive_code_scalar_restrict_inputinput_table) + (dst_positive_scale_scalar_restrict_inputinput_table)) + ((dst_positive_scale_scalar_restrict_inputinput_table) + (dst_positive_scale_scalar_restrict_inputinput_table))) + (((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) * S ((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) + ((dst_negative_scale_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)))) + ((((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) * S ((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) + ((dst_negative_scale_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table))) + (((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) * S ((dst_negative_code_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)) + ((dst_negative_scale_scalar_restrict_inputinput_table) + (dst_negative_scale_scalar_restrict_inputinput_table)))))) /\ (forall dst_index_scalar_restrict_inputinput_table. (exists pvs_le_gap_scalar_restrict_inputinput_tabledomain. pvs_le_gap_scalar_restrict_inputinput_tabledomain + (dst_index_scalar_restrict_inputinput_table) = (S l)) -> exists dst_positive_scalar_restrict_inputinput_table dst_negative_scalar_restrict_inputinput_table dst_value_scalar_restrict_inputinput_table. ((((exists ff_h_pvs_scalar_restrict_inputinput_tableentrypositive. ff_h_pvs_scalar_restrict_inputinput_tableentrypositive + S (dst_positive_scalar_restrict_inputinput_table) = S ((S (dst_index_scalar_restrict_inputinput_table)) * dst_positive_scale_scalar_restrict_inputinput_table)) /\ exists ff_q_pvs_scalar_restrict_inputinput_tableentrypositive. dst_positive_code_scalar_restrict_inputinput_table = ff_q_pvs_scalar_restrict_inputinput_tableentrypositive * S ((S (dst_index_scalar_restrict_inputinput_table)) * dst_positive_scale_scalar_restrict_inputinput_table) + (dst_positive_scalar_restrict_inputinput_table))) /\ (((((exists ff_h_pvs_scalar_restrict_inputinput_tableentrynegative. ff_h_pvs_scalar_restrict_inputinput_tableentrynegative + S (dst_negative_scalar_restrict_inputinput_table) = S ((S (dst_index_scalar_restrict_inputinput_table)) * dst_negative_scale_scalar_restrict_inputinput_table)) /\ exists ff_q_pvs_scalar_restrict_inputinput_tableentrynegative. dst_negative_code_scalar_restrict_inputinput_table = ff_q_pvs_scalar_restrict_inputinput_tableentrynegative * S ((S (dst_index_scalar_restrict_inputinput_table)) * dst_negative_scale_scalar_restrict_inputinput_table) + (dst_negative_scalar_restrict_inputinput_table))) /\ (exists ge_balance_positive_scalar_restrict_inputinput_tableentryvalue ge_balance_negative_scalar_restrict_inputinput_tableentryvalue. (((((dst_value_scalar_restrict_inputinput_table) = 2 * (ge_balance_positive_scalar_restrict_inputinput_tableentryvalue) /\ (ge_balance_negative_scalar_restrict_inputinput_tableentryvalue) = 0) \/ exists ge_signed_half_scalar_restrict_inputinput_tableentryvaluedecode. (((dst_value_scalar_restrict_inputinput_table) = 2 * ge_signed_half_scalar_restrict_inputinput_tableentryvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_inputinput_tableentryvalue) = 0) /\ (ge_balance_negative_scalar_restrict_inputinput_tableentryvalue) = S ge_signed_half_scalar_restrict_inputinput_tableentryvaluedecode))) /\ ((dst_positive_scalar_restrict_inputinput_table) + ge_balance_negative_scalar_restrict_inputinput_tableentryvalue = (dst_negative_scalar_restrict_inputinput_table) + ge_balance_positive_scalar_restrict_inputinput_tableentryvalue))))))))) /\ (((exists dst_positive_code_scalar_restrict_inputoutput_table dst_positive_scale_scalar_restrict_inputoutput_table dst_negative_code_scalar_restrict_inputoutput_table dst_negative_scale_scalar_restrict_inputoutput_table. (((G) = (((((dst_positive_code_scalar_restrict_inputoutput_table) + (dst_positive_scale_scalar_restrict_inputoutput_table)) * S ((dst_positive_code_scalar_restrict_inputoutput_table) + (dst_positive_scale_scalar_restrict_inputoutput_table)) + ((dst_positive_scale_scalar_restrict_inputoutput_table) + (dst_positive_scale_scalar_restrict_inputoutput_table))) + (((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) * S ((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) + ((dst_negative_scale_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)))) * S ((((dst_positive_code_scalar_restrict_inputoutput_table) + (dst_positive_scale_scalar_restrict_inputoutput_table)) * S ((dst_positive_code_scalar_restrict_inputoutput_table) + (dst_positive_scale_scalar_restrict_inputoutput_table)) + ((dst_positive_scale_scalar_restrict_inputoutput_table) + (dst_positive_scale_scalar_restrict_inputoutput_table))) + (((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) * S ((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) + ((dst_negative_scale_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)))) + ((((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) * S ((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) + ((dst_negative_scale_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table))) + (((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) * S ((dst_negative_code_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)) + ((dst_negative_scale_scalar_restrict_inputoutput_table) + (dst_negative_scale_scalar_restrict_inputoutput_table)))))) /\ (forall dst_index_scalar_restrict_inputoutput_table. (exists pvs_le_gap_scalar_restrict_inputoutput_tabledomain. pvs_le_gap_scalar_restrict_inputoutput_tabledomain + (dst_index_scalar_restrict_inputoutput_table) = (S l)) -> exists dst_positive_scalar_restrict_inputoutput_table dst_negative_scalar_restrict_inputoutput_table dst_value_scalar_restrict_inputoutput_table. ((((exists ff_h_pvs_scalar_restrict_inputoutput_tableentrypositive. ff_h_pvs_scalar_restrict_inputoutput_tableentrypositive + S (dst_positive_scalar_restrict_inputoutput_table) = S ((S (dst_index_scalar_restrict_inputoutput_table)) * dst_positive_scale_scalar_restrict_inputoutput_table)) /\ exists ff_q_pvs_scalar_restrict_inputoutput_tableentrypositive. dst_positive_code_scalar_restrict_inputoutput_table = ff_q_pvs_scalar_restrict_inputoutput_tableentrypositive * S ((S (dst_index_scalar_restrict_inputoutput_table)) * dst_positive_scale_scalar_restrict_inputoutput_table) + (dst_positive_scalar_restrict_inputoutput_table))) /\ (((((exists ff_h_pvs_scalar_restrict_inputoutput_tableentrynegative. ff_h_pvs_scalar_restrict_inputoutput_tableentrynegative + S (dst_negative_scalar_restrict_inputoutput_table) = S ((S (dst_index_scalar_restrict_inputoutput_table)) * dst_negative_scale_scalar_restrict_inputoutput_table)) /\ exists ff_q_pvs_scalar_restrict_inputoutput_tableentrynegative. dst_negative_code_scalar_restrict_inputoutput_table = ff_q_pvs_scalar_restrict_inputoutput_tableentrynegative * S ((S (dst_index_scalar_restrict_inputoutput_table)) * dst_negative_scale_scalar_restrict_inputoutput_table) + (dst_negative_scalar_restrict_inputoutput_table))) /\ (exists ge_balance_positive_scalar_restrict_inputoutput_tableentryvalue ge_balance_negative_scalar_restrict_inputoutput_tableentryvalue. (((((dst_value_scalar_restrict_inputoutput_table) = 2 * (ge_balance_positive_scalar_restrict_inputoutput_tableentryvalue) /\ (ge_balance_negative_scalar_restrict_inputoutput_tableentryvalue) = 0) \/ exists ge_signed_half_scalar_restrict_inputoutput_tableentryvaluedecode. (((dst_value_scalar_restrict_inputoutput_table) = 2 * ge_signed_half_scalar_restrict_inputoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_inputoutput_tableentryvalue) = 0) /\ (ge_balance_negative_scalar_restrict_inputoutput_tableentryvalue) = S ge_signed_half_scalar_restrict_inputoutput_tableentryvaluedecode))) /\ ((dst_positive_scalar_restrict_inputoutput_table) + ge_balance_negative_scalar_restrict_inputoutput_tableentryvalue = (dst_negative_scalar_restrict_inputoutput_table) + ge_balance_positive_scalar_restrict_inputoutput_tableentryvalue))))))))) /\ (forall sto_index_scalar_restrict_inputentries. (exists pvs_gap_scalar_restrict_inputentriesbound. pvs_gap_scalar_restrict_inputentriesbound + S (sto_index_scalar_restrict_inputentries) = (S l)) -> exists sto_input_scalar_restrict_inputentries sto_output_scalar_restrict_inputentries. ((exists dst_positive_code_scalar_restrict_inputentriesentryinput dst_positive_scale_scalar_restrict_inputentriesentryinput dst_negative_code_scalar_restrict_inputentriesentryinput dst_negative_scale_scalar_restrict_inputentriesentryinput dst_positive_scalar_restrict_inputentriesentryinput dst_negative_scalar_restrict_inputentriesentryinput. (((F) = (((((dst_positive_code_scalar_restrict_inputentriesentryinput) + (dst_positive_scale_scalar_restrict_inputentriesentryinput)) * S ((dst_positive_code_scalar_restrict_inputentriesentryinput) + (dst_positive_scale_scalar_restrict_inputentriesentryinput)) + ((dst_positive_scale_scalar_restrict_inputentriesentryinput) + (dst_positive_scale_scalar_restrict_inputentriesentryinput))) + (((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)))) * S ((((dst_positive_code_scalar_restrict_inputentriesentryinput) + (dst_positive_scale_scalar_restrict_inputentriesentryinput)) * S ((dst_positive_code_scalar_restrict_inputentriesentryinput) + (dst_positive_scale_scalar_restrict_inputentriesentryinput)) + ((dst_positive_scale_scalar_restrict_inputentriesentryinput) + (dst_positive_scale_scalar_restrict_inputentriesentryinput))) + (((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)))) + ((((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput))) + (((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryinput) + (dst_negative_scale_scalar_restrict_inputentriesentryinput)))))) /\ (((((exists ff_h_pvs_scalar_restrict_inputentriesentryinputpositive. ff_h_pvs_scalar_restrict_inputentriesentryinputpositive + S (dst_positive_scalar_restrict_inputentriesentryinput) = S ((S (sto_index_scalar_restrict_inputentries)) * dst_positive_scale_scalar_restrict_inputentriesentryinput)) /\ exists ff_q_pvs_scalar_restrict_inputentriesentryinputpositive. dst_positive_code_scalar_restrict_inputentriesentryinput = ff_q_pvs_scalar_restrict_inputentriesentryinputpositive * S ((S (sto_index_scalar_restrict_inputentries)) * dst_positive_scale_scalar_restrict_inputentriesentryinput) + (dst_positive_scalar_restrict_inputentriesentryinput))) /\ (((((exists ff_h_pvs_scalar_restrict_inputentriesentryinputnegative. ff_h_pvs_scalar_restrict_inputentriesentryinputnegative + S (dst_negative_scalar_restrict_inputentriesentryinput) = S ((S (sto_index_scalar_restrict_inputentries)) * dst_negative_scale_scalar_restrict_inputentriesentryinput)) /\ exists ff_q_pvs_scalar_restrict_inputentriesentryinputnegative. dst_negative_code_scalar_restrict_inputentriesentryinput = ff_q_pvs_scalar_restrict_inputentriesentryinputnegative * S ((S (sto_index_scalar_restrict_inputentries)) * dst_negative_scale_scalar_restrict_inputentriesentryinput) + (dst_negative_scalar_restrict_inputentriesentryinput))) /\ (exists ge_balance_positive_scalar_restrict_inputentriesentryinputvalue ge_balance_negative_scalar_restrict_inputentriesentryinputvalue. (((((sto_input_scalar_restrict_inputentries) = 2 * (ge_balance_positive_scalar_restrict_inputentriesentryinputvalue) /\ (ge_balance_negative_scalar_restrict_inputentriesentryinputvalue) = 0) \/ exists ge_signed_half_scalar_restrict_inputentriesentryinputvaluedecode. (((sto_input_scalar_restrict_inputentries) = 2 * ge_signed_half_scalar_restrict_inputentriesentryinputvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_inputentriesentryinputvalue) = 0) /\ (ge_balance_negative_scalar_restrict_inputentriesentryinputvalue) = S ge_signed_half_scalar_restrict_inputentriesentryinputvaluedecode))) /\ ((dst_positive_scalar_restrict_inputentriesentryinput) + ge_balance_negative_scalar_restrict_inputentriesentryinputvalue = (dst_negative_scalar_restrict_inputentriesentryinput) + ge_balance_positive_scalar_restrict_inputentriesentryinputvalue))))))))) /\ (((exists dst_positive_code_scalar_restrict_inputentriesentryoutput dst_positive_scale_scalar_restrict_inputentriesentryoutput dst_negative_code_scalar_restrict_inputentriesentryoutput dst_negative_scale_scalar_restrict_inputentriesentryoutput dst_positive_scalar_restrict_inputentriesentryoutput dst_negative_scalar_restrict_inputentriesentryoutput. (((G) = (((((dst_positive_code_scalar_restrict_inputentriesentryoutput) + (dst_positive_scale_scalar_restrict_inputentriesentryoutput)) * S ((dst_positive_code_scalar_restrict_inputentriesentryoutput) + (dst_positive_scale_scalar_restrict_inputentriesentryoutput)) + ((dst_positive_scale_scalar_restrict_inputentriesentryoutput) + (dst_positive_scale_scalar_restrict_inputentriesentryoutput))) + (((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)))) * S ((((dst_positive_code_scalar_restrict_inputentriesentryoutput) + (dst_positive_scale_scalar_restrict_inputentriesentryoutput)) * S ((dst_positive_code_scalar_restrict_inputentriesentryoutput) + (dst_positive_scale_scalar_restrict_inputentriesentryoutput)) + ((dst_positive_scale_scalar_restrict_inputentriesentryoutput) + (dst_positive_scale_scalar_restrict_inputentriesentryoutput))) + (((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)))) + ((((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput))) + (((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_inputentriesentryoutput) + (dst_negative_scale_scalar_restrict_inputentriesentryoutput)))))) /\ (((((exists ff_h_pvs_scalar_restrict_inputentriesentryoutputpositive. ff_h_pvs_scalar_restrict_inputentriesentryoutputpositive + S (dst_positive_scalar_restrict_inputentriesentryoutput) = S ((S (sto_index_scalar_restrict_inputentries)) * dst_positive_scale_scalar_restrict_inputentriesentryoutput)) /\ exists ff_q_pvs_scalar_restrict_inputentriesentryoutputpositive. dst_positive_code_scalar_restrict_inputentriesentryoutput = ff_q_pvs_scalar_restrict_inputentriesentryoutputpositive * S ((S (sto_index_scalar_restrict_inputentries)) * dst_positive_scale_scalar_restrict_inputentriesentryoutput) + (dst_positive_scalar_restrict_inputentriesentryoutput))) /\ (((((exists ff_h_pvs_scalar_restrict_inputentriesentryoutputnegative. ff_h_pvs_scalar_restrict_inputentriesentryoutputnegative + S (dst_negative_scalar_restrict_inputentriesentryoutput) = S ((S (sto_index_scalar_restrict_inputentries)) * dst_negative_scale_scalar_restrict_inputentriesentryoutput)) /\ exists ff_q_pvs_scalar_restrict_inputentriesentryoutputnegative. dst_negative_code_scalar_restrict_inputentriesentryoutput = ff_q_pvs_scalar_restrict_inputentriesentryoutputnegative * S ((S (sto_index_scalar_restrict_inputentries)) * dst_negative_scale_scalar_restrict_inputentriesentryoutput) + (dst_negative_scalar_restrict_inputentriesentryoutput))) /\ (exists ge_balance_positive_scalar_restrict_inputentriesentryoutputvalue ge_balance_negative_scalar_restrict_inputentriesentryoutputvalue. (((((sto_output_scalar_restrict_inputentries) = 2 * (ge_balance_positive_scalar_restrict_inputentriesentryoutputvalue) /\ (ge_balance_negative_scalar_restrict_inputentriesentryoutputvalue) = 0) \/ exists ge_signed_half_scalar_restrict_inputentriesentryoutputvaluedecode. (((sto_output_scalar_restrict_inputentries) = 2 * ge_signed_half_scalar_restrict_inputentriesentryoutputvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_inputentriesentryoutputvalue) = 0) /\ (ge_balance_negative_scalar_restrict_inputentriesentryoutputvalue) = S ge_signed_half_scalar_restrict_inputentriesentryoutputvaluedecode))) /\ ((dst_positive_scalar_restrict_inputentriesentryoutput) + ge_balance_negative_scalar_restrict_inputentriesentryoutputvalue = (dst_negative_scalar_restrict_inputentriesentryoutput) + ge_balance_positive_scalar_restrict_inputentriesentryoutputvalue))))))))) /\ (exists sto_ap_scalar_restrict_inputentriesentryoperation sto_an_scalar_restrict_inputentriesentryoperation sto_bp_scalar_restrict_inputentriesentryoperation sto_bn_scalar_restrict_inputentriesentryoperation sto_cp_scalar_restrict_inputentriesentryoperation sto_cn_scalar_restrict_inputentriesentryoperation. (((((a) = 2 * (sto_ap_scalar_restrict_inputentriesentryoperation) /\ (sto_an_scalar_restrict_inputentriesentryoperation) = 0) \/ exists ge_signed_half_scalar_restrict_inputentriesentryoperationleft. (((a) = 2 * ge_signed_half_scalar_restrict_inputentriesentryoperationleft + 1 /\ (sto_ap_scalar_restrict_inputentriesentryoperation) = 0) /\ (sto_an_scalar_restrict_inputentriesentryoperation) = S ge_signed_half_scalar_restrict_inputentriesentryoperationleft))) /\ ((((((sto_input_scalar_restrict_inputentries) = 2 * (sto_bp_scalar_restrict_inputentriesentryoperation) /\ (sto_bn_scalar_restrict_inputentriesentryoperation) = 0) \/ exists ge_signed_half_scalar_restrict_inputentriesentryoperationright. (((sto_input_scalar_restrict_inputentries) = 2 * ge_signed_half_scalar_restrict_inputentriesentryoperationright + 1 /\ (sto_bp_scalar_restrict_inputentriesentryoperation) = 0) /\ (sto_bn_scalar_restrict_inputentriesentryoperation) = S ge_signed_half_scalar_restrict_inputentriesentryoperationright))) /\ ((((((sto_output_scalar_restrict_inputentries) = 2 * (sto_cp_scalar_restrict_inputentriesentryoperation) /\ (sto_cn_scalar_restrict_inputentriesentryoperation) = 0) \/ exists ge_signed_half_scalar_restrict_inputentriesentryoperationoutput. (((sto_output_scalar_restrict_inputentries) = 2 * ge_signed_half_scalar_restrict_inputentriesentryoperationoutput + 1 /\ (sto_cp_scalar_restrict_inputentriesentryoperation) = 0) /\ (sto_cn_scalar_restrict_inputentriesentryoperation) = S ge_signed_half_scalar_restrict_inputentriesentryoperationoutput))) /\ ((sto_ap_scalar_restrict_inputentriesentryoperation * sto_bp_scalar_restrict_inputentriesentryoperation + sto_an_scalar_restrict_inputentriesentryoperation * sto_bn_scalar_restrict_inputentriesentryoperation) + sto_cn_scalar_restrict_inputentriesentryoperation = (sto_ap_scalar_restrict_inputentriesentryoperation * sto_bn_scalar_restrict_inputentriesentryoperation + sto_an_scalar_restrict_inputentriesentryoperation * sto_bp_scalar_restrict_inputentriesentryoperation) + sto_cp_scalar_restrict_inputentriesentryoperation))))))))))))))) -> (((exists dst_positive_code_scalar_restrict_outputinput_table dst_positive_scale_scalar_restrict_outputinput_table dst_negative_code_scalar_restrict_outputinput_table dst_negative_scale_scalar_restrict_outputinput_table. (((F) = (((((dst_positive_code_scalar_restrict_outputinput_table) + (dst_positive_scale_scalar_restrict_outputinput_table)) * S ((dst_positive_code_scalar_restrict_outputinput_table) + (dst_positive_scale_scalar_restrict_outputinput_table)) + ((dst_positive_scale_scalar_restrict_outputinput_table) + (dst_positive_scale_scalar_restrict_outputinput_table))) + (((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) * S ((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) + ((dst_negative_scale_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)))) * S ((((dst_positive_code_scalar_restrict_outputinput_table) + (dst_positive_scale_scalar_restrict_outputinput_table)) * S ((dst_positive_code_scalar_restrict_outputinput_table) + (dst_positive_scale_scalar_restrict_outputinput_table)) + ((dst_positive_scale_scalar_restrict_outputinput_table) + (dst_positive_scale_scalar_restrict_outputinput_table))) + (((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) * S ((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) + ((dst_negative_scale_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)))) + ((((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) * S ((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) + ((dst_negative_scale_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table))) + (((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) * S ((dst_negative_code_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)) + ((dst_negative_scale_scalar_restrict_outputinput_table) + (dst_negative_scale_scalar_restrict_outputinput_table)))))) /\ (forall dst_index_scalar_restrict_outputinput_table. (exists pvs_le_gap_scalar_restrict_outputinput_tabledomain. pvs_le_gap_scalar_restrict_outputinput_tabledomain + (dst_index_scalar_restrict_outputinput_table) = (l)) -> exists dst_positive_scalar_restrict_outputinput_table dst_negative_scalar_restrict_outputinput_table dst_value_scalar_restrict_outputinput_table. ((((exists ff_h_pvs_scalar_restrict_outputinput_tableentrypositive. ff_h_pvs_scalar_restrict_outputinput_tableentrypositive + S (dst_positive_scalar_restrict_outputinput_table) = S ((S (dst_index_scalar_restrict_outputinput_table)) * dst_positive_scale_scalar_restrict_outputinput_table)) /\ exists ff_q_pvs_scalar_restrict_outputinput_tableentrypositive. dst_positive_code_scalar_restrict_outputinput_table = ff_q_pvs_scalar_restrict_outputinput_tableentrypositive * S ((S (dst_index_scalar_restrict_outputinput_table)) * dst_positive_scale_scalar_restrict_outputinput_table) + (dst_positive_scalar_restrict_outputinput_table))) /\ (((((exists ff_h_pvs_scalar_restrict_outputinput_tableentrynegative. ff_h_pvs_scalar_restrict_outputinput_tableentrynegative + S (dst_negative_scalar_restrict_outputinput_table) = S ((S (dst_index_scalar_restrict_outputinput_table)) * dst_negative_scale_scalar_restrict_outputinput_table)) /\ exists ff_q_pvs_scalar_restrict_outputinput_tableentrynegative. dst_negative_code_scalar_restrict_outputinput_table = ff_q_pvs_scalar_restrict_outputinput_tableentrynegative * S ((S (dst_index_scalar_restrict_outputinput_table)) * dst_negative_scale_scalar_restrict_outputinput_table) + (dst_negative_scalar_restrict_outputinput_table))) /\ (exists ge_balance_positive_scalar_restrict_outputinput_tableentryvalue ge_balance_negative_scalar_restrict_outputinput_tableentryvalue. (((((dst_value_scalar_restrict_outputinput_table) = 2 * (ge_balance_positive_scalar_restrict_outputinput_tableentryvalue) /\ (ge_balance_negative_scalar_restrict_outputinput_tableentryvalue) = 0) \/ exists ge_signed_half_scalar_restrict_outputinput_tableentryvaluedecode. (((dst_value_scalar_restrict_outputinput_table) = 2 * ge_signed_half_scalar_restrict_outputinput_tableentryvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_outputinput_tableentryvalue) = 0) /\ (ge_balance_negative_scalar_restrict_outputinput_tableentryvalue) = S ge_signed_half_scalar_restrict_outputinput_tableentryvaluedecode))) /\ ((dst_positive_scalar_restrict_outputinput_table) + ge_balance_negative_scalar_restrict_outputinput_tableentryvalue = (dst_negative_scalar_restrict_outputinput_table) + ge_balance_positive_scalar_restrict_outputinput_tableentryvalue))))))))) /\ (((exists dst_positive_code_scalar_restrict_outputoutput_table dst_positive_scale_scalar_restrict_outputoutput_table dst_negative_code_scalar_restrict_outputoutput_table dst_negative_scale_scalar_restrict_outputoutput_table. (((G) = (((((dst_positive_code_scalar_restrict_outputoutput_table) + (dst_positive_scale_scalar_restrict_outputoutput_table)) * S ((dst_positive_code_scalar_restrict_outputoutput_table) + (dst_positive_scale_scalar_restrict_outputoutput_table)) + ((dst_positive_scale_scalar_restrict_outputoutput_table) + (dst_positive_scale_scalar_restrict_outputoutput_table))) + (((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) * S ((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) + ((dst_negative_scale_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)))) * S ((((dst_positive_code_scalar_restrict_outputoutput_table) + (dst_positive_scale_scalar_restrict_outputoutput_table)) * S ((dst_positive_code_scalar_restrict_outputoutput_table) + (dst_positive_scale_scalar_restrict_outputoutput_table)) + ((dst_positive_scale_scalar_restrict_outputoutput_table) + (dst_positive_scale_scalar_restrict_outputoutput_table))) + (((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) * S ((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) + ((dst_negative_scale_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)))) + ((((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) * S ((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) + ((dst_negative_scale_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table))) + (((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) * S ((dst_negative_code_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)) + ((dst_negative_scale_scalar_restrict_outputoutput_table) + (dst_negative_scale_scalar_restrict_outputoutput_table)))))) /\ (forall dst_index_scalar_restrict_outputoutput_table. (exists pvs_le_gap_scalar_restrict_outputoutput_tabledomain. pvs_le_gap_scalar_restrict_outputoutput_tabledomain + (dst_index_scalar_restrict_outputoutput_table) = (l)) -> exists dst_positive_scalar_restrict_outputoutput_table dst_negative_scalar_restrict_outputoutput_table dst_value_scalar_restrict_outputoutput_table. ((((exists ff_h_pvs_scalar_restrict_outputoutput_tableentrypositive. ff_h_pvs_scalar_restrict_outputoutput_tableentrypositive + S (dst_positive_scalar_restrict_outputoutput_table) = S ((S (dst_index_scalar_restrict_outputoutput_table)) * dst_positive_scale_scalar_restrict_outputoutput_table)) /\ exists ff_q_pvs_scalar_restrict_outputoutput_tableentrypositive. dst_positive_code_scalar_restrict_outputoutput_table = ff_q_pvs_scalar_restrict_outputoutput_tableentrypositive * S ((S (dst_index_scalar_restrict_outputoutput_table)) * dst_positive_scale_scalar_restrict_outputoutput_table) + (dst_positive_scalar_restrict_outputoutput_table))) /\ (((((exists ff_h_pvs_scalar_restrict_outputoutput_tableentrynegative. ff_h_pvs_scalar_restrict_outputoutput_tableentrynegative + S (dst_negative_scalar_restrict_outputoutput_table) = S ((S (dst_index_scalar_restrict_outputoutput_table)) * dst_negative_scale_scalar_restrict_outputoutput_table)) /\ exists ff_q_pvs_scalar_restrict_outputoutput_tableentrynegative. dst_negative_code_scalar_restrict_outputoutput_table = ff_q_pvs_scalar_restrict_outputoutput_tableentrynegative * S ((S (dst_index_scalar_restrict_outputoutput_table)) * dst_negative_scale_scalar_restrict_outputoutput_table) + (dst_negative_scalar_restrict_outputoutput_table))) /\ (exists ge_balance_positive_scalar_restrict_outputoutput_tableentryvalue ge_balance_negative_scalar_restrict_outputoutput_tableentryvalue. (((((dst_value_scalar_restrict_outputoutput_table) = 2 * (ge_balance_positive_scalar_restrict_outputoutput_tableentryvalue) /\ (ge_balance_negative_scalar_restrict_outputoutput_tableentryvalue) = 0) \/ exists ge_signed_half_scalar_restrict_outputoutput_tableentryvaluedecode. (((dst_value_scalar_restrict_outputoutput_table) = 2 * ge_signed_half_scalar_restrict_outputoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_outputoutput_tableentryvalue) = 0) /\ (ge_balance_negative_scalar_restrict_outputoutput_tableentryvalue) = S ge_signed_half_scalar_restrict_outputoutput_tableentryvaluedecode))) /\ ((dst_positive_scalar_restrict_outputoutput_table) + ge_balance_negative_scalar_restrict_outputoutput_tableentryvalue = (dst_negative_scalar_restrict_outputoutput_table) + ge_balance_positive_scalar_restrict_outputoutput_tableentryvalue))))))))) /\ (forall sto_index_scalar_restrict_outputentries. (exists pvs_gap_scalar_restrict_outputentriesbound. pvs_gap_scalar_restrict_outputentriesbound + S (sto_index_scalar_restrict_outputentries) = (l)) -> exists sto_input_scalar_restrict_outputentries sto_output_scalar_restrict_outputentries. ((exists dst_positive_code_scalar_restrict_outputentriesentryinput dst_positive_scale_scalar_restrict_outputentriesentryinput dst_negative_code_scalar_restrict_outputentriesentryinput dst_negative_scale_scalar_restrict_outputentriesentryinput dst_positive_scalar_restrict_outputentriesentryinput dst_negative_scalar_restrict_outputentriesentryinput. (((F) = (((((dst_positive_code_scalar_restrict_outputentriesentryinput) + (dst_positive_scale_scalar_restrict_outputentriesentryinput)) * S ((dst_positive_code_scalar_restrict_outputentriesentryinput) + (dst_positive_scale_scalar_restrict_outputentriesentryinput)) + ((dst_positive_scale_scalar_restrict_outputentriesentryinput) + (dst_positive_scale_scalar_restrict_outputentriesentryinput))) + (((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)))) * S ((((dst_positive_code_scalar_restrict_outputentriesentryinput) + (dst_positive_scale_scalar_restrict_outputentriesentryinput)) * S ((dst_positive_code_scalar_restrict_outputentriesentryinput) + (dst_positive_scale_scalar_restrict_outputentriesentryinput)) + ((dst_positive_scale_scalar_restrict_outputentriesentryinput) + (dst_positive_scale_scalar_restrict_outputentriesentryinput))) + (((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)))) + ((((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput))) + (((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryinput) + (dst_negative_scale_scalar_restrict_outputentriesentryinput)))))) /\ (((((exists ff_h_pvs_scalar_restrict_outputentriesentryinputpositive. ff_h_pvs_scalar_restrict_outputentriesentryinputpositive + S (dst_positive_scalar_restrict_outputentriesentryinput) = S ((S (sto_index_scalar_restrict_outputentries)) * dst_positive_scale_scalar_restrict_outputentriesentryinput)) /\ exists ff_q_pvs_scalar_restrict_outputentriesentryinputpositive. dst_positive_code_scalar_restrict_outputentriesentryinput = ff_q_pvs_scalar_restrict_outputentriesentryinputpositive * S ((S (sto_index_scalar_restrict_outputentries)) * dst_positive_scale_scalar_restrict_outputentriesentryinput) + (dst_positive_scalar_restrict_outputentriesentryinput))) /\ (((((exists ff_h_pvs_scalar_restrict_outputentriesentryinputnegative. ff_h_pvs_scalar_restrict_outputentriesentryinputnegative + S (dst_negative_scalar_restrict_outputentriesentryinput) = S ((S (sto_index_scalar_restrict_outputentries)) * dst_negative_scale_scalar_restrict_outputentriesentryinput)) /\ exists ff_q_pvs_scalar_restrict_outputentriesentryinputnegative. dst_negative_code_scalar_restrict_outputentriesentryinput = ff_q_pvs_scalar_restrict_outputentriesentryinputnegative * S ((S (sto_index_scalar_restrict_outputentries)) * dst_negative_scale_scalar_restrict_outputentriesentryinput) + (dst_negative_scalar_restrict_outputentriesentryinput))) /\ (exists ge_balance_positive_scalar_restrict_outputentriesentryinputvalue ge_balance_negative_scalar_restrict_outputentriesentryinputvalue. (((((sto_input_scalar_restrict_outputentries) = 2 * (ge_balance_positive_scalar_restrict_outputentriesentryinputvalue) /\ (ge_balance_negative_scalar_restrict_outputentriesentryinputvalue) = 0) \/ exists ge_signed_half_scalar_restrict_outputentriesentryinputvaluedecode. (((sto_input_scalar_restrict_outputentries) = 2 * ge_signed_half_scalar_restrict_outputentriesentryinputvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_outputentriesentryinputvalue) = 0) /\ (ge_balance_negative_scalar_restrict_outputentriesentryinputvalue) = S ge_signed_half_scalar_restrict_outputentriesentryinputvaluedecode))) /\ ((dst_positive_scalar_restrict_outputentriesentryinput) + ge_balance_negative_scalar_restrict_outputentriesentryinputvalue = (dst_negative_scalar_restrict_outputentriesentryinput) + ge_balance_positive_scalar_restrict_outputentriesentryinputvalue))))))))) /\ (((exists dst_positive_code_scalar_restrict_outputentriesentryoutput dst_positive_scale_scalar_restrict_outputentriesentryoutput dst_negative_code_scalar_restrict_outputentriesentryoutput dst_negative_scale_scalar_restrict_outputentriesentryoutput dst_positive_scalar_restrict_outputentriesentryoutput dst_negative_scalar_restrict_outputentriesentryoutput. (((G) = (((((dst_positive_code_scalar_restrict_outputentriesentryoutput) + (dst_positive_scale_scalar_restrict_outputentriesentryoutput)) * S ((dst_positive_code_scalar_restrict_outputentriesentryoutput) + (dst_positive_scale_scalar_restrict_outputentriesentryoutput)) + ((dst_positive_scale_scalar_restrict_outputentriesentryoutput) + (dst_positive_scale_scalar_restrict_outputentriesentryoutput))) + (((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)))) * S ((((dst_positive_code_scalar_restrict_outputentriesentryoutput) + (dst_positive_scale_scalar_restrict_outputentriesentryoutput)) * S ((dst_positive_code_scalar_restrict_outputentriesentryoutput) + (dst_positive_scale_scalar_restrict_outputentriesentryoutput)) + ((dst_positive_scale_scalar_restrict_outputentriesentryoutput) + (dst_positive_scale_scalar_restrict_outputentriesentryoutput))) + (((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)))) + ((((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput))) + (((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) * S ((dst_negative_code_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)) + ((dst_negative_scale_scalar_restrict_outputentriesentryoutput) + (dst_negative_scale_scalar_restrict_outputentriesentryoutput)))))) /\ (((((exists ff_h_pvs_scalar_restrict_outputentriesentryoutputpositive. ff_h_pvs_scalar_restrict_outputentriesentryoutputpositive + S (dst_positive_scalar_restrict_outputentriesentryoutput) = S ((S (sto_index_scalar_restrict_outputentries)) * dst_positive_scale_scalar_restrict_outputentriesentryoutput)) /\ exists ff_q_pvs_scalar_restrict_outputentriesentryoutputpositive. dst_positive_code_scalar_restrict_outputentriesentryoutput = ff_q_pvs_scalar_restrict_outputentriesentryoutputpositive * S ((S (sto_index_scalar_restrict_outputentries)) * dst_positive_scale_scalar_restrict_outputentriesentryoutput) + (dst_positive_scalar_restrict_outputentriesentryoutput))) /\ (((((exists ff_h_pvs_scalar_restrict_outputentriesentryoutputnegative. ff_h_pvs_scalar_restrict_outputentriesentryoutputnegative + S (dst_negative_scalar_restrict_outputentriesentryoutput) = S ((S (sto_index_scalar_restrict_outputentries)) * dst_negative_scale_scalar_restrict_outputentriesentryoutput)) /\ exists ff_q_pvs_scalar_restrict_outputentriesentryoutputnegative. dst_negative_code_scalar_restrict_outputentriesentryoutput = ff_q_pvs_scalar_restrict_outputentriesentryoutputnegative * S ((S (sto_index_scalar_restrict_outputentries)) * dst_negative_scale_scalar_restrict_outputentriesentryoutput) + (dst_negative_scalar_restrict_outputentriesentryoutput))) /\ (exists ge_balance_positive_scalar_restrict_outputentriesentryoutputvalue ge_balance_negative_scalar_restrict_outputentriesentryoutputvalue. (((((sto_output_scalar_restrict_outputentries) = 2 * (ge_balance_positive_scalar_restrict_outputentriesentryoutputvalue) /\ (ge_balance_negative_scalar_restrict_outputentriesentryoutputvalue) = 0) \/ exists ge_signed_half_scalar_restrict_outputentriesentryoutputvaluedecode. (((sto_output_scalar_restrict_outputentries) = 2 * ge_signed_half_scalar_restrict_outputentriesentryoutputvaluedecode + 1 /\ (ge_balance_positive_scalar_restrict_outputentriesentryoutputvalue) = 0) /\ (ge_balance_negative_scalar_restrict_outputentriesentryoutputvalue) = S ge_signed_half_scalar_restrict_outputentriesentryoutputvaluedecode))) /\ ((dst_positive_scalar_restrict_outputentriesentryoutput) + ge_balance_negative_scalar_restrict_outputentriesentryoutputvalue = (dst_negative_scalar_restrict_outputentriesentryoutput) + ge_balance_positive_scalar_restrict_outputentriesentryoutputvalue))))))))) /\ (exists sto_ap_scalar_restrict_outputentriesentryoperation sto_an_scalar_restrict_outputentriesentryoperation sto_bp_scalar_restrict_outputentriesentryoperation sto_bn_scalar_restrict_outputentriesentryoperation sto_cp_scalar_restrict_outputentriesentryoperation sto_cn_scalar_restrict_outputentriesentryoperation. (((((a) = 2 * (sto_ap_scalar_restrict_outputentriesentryoperation) /\ (sto_an_scalar_restrict_outputentriesentryoperation) = 0) \/ exists ge_signed_half_scalar_restrict_outputentriesentryoperationleft. (((a) = 2 * ge_signed_half_scalar_restrict_outputentriesentryoperationleft + 1 /\ (sto_ap_scalar_restrict_outputentriesentryoperation) = 0) /\ (sto_an_scalar_restrict_outputentriesentryoperation) = S ge_signed_half_scalar_restrict_outputentriesentryoperationleft))) /\ ((((((sto_input_scalar_restrict_outputentries) = 2 * (sto_bp_scalar_restrict_outputentriesentryoperation) /\ (sto_bn_scalar_restrict_outputentriesentryoperation) = 0) \/ exists ge_signed_half_scalar_restrict_outputentriesentryoperationright. (((sto_input_scalar_restrict_outputentries) = 2 * ge_signed_half_scalar_restrict_outputentriesentryoperationright + 1 /\ (sto_bp_scalar_restrict_outputentriesentryoperation) = 0) /\ (sto_bn_scalar_restrict_outputentriesentryoperation) = S ge_signed_half_scalar_restrict_outputentriesentryoperationright))) /\ ((((((sto_output_scalar_restrict_outputentries) = 2 * (sto_cp_scalar_restrict_outputentriesentryoperation) /\ (sto_cn_scalar_restrict_outputentriesentryoperation) = 0) \/ exists ge_signed_half_scalar_restrict_outputentriesentryoperationoutput. (((sto_output_scalar_restrict_outputentries) = 2 * ge_signed_half_scalar_restrict_outputentriesentryoperationoutput + 1 /\ (sto_cp_scalar_restrict_outputentriesentryoperation) = 0) /\ (sto_cn_scalar_restrict_outputentriesentryoperation) = S ge_signed_half_scalar_restrict_outputentriesentryoperationoutput))) /\ ((sto_ap_scalar_restrict_outputentriesentryoperation * sto_bp_scalar_restrict_outputentriesentryoperation + sto_an_scalar_restrict_outputentriesentryoperation * sto_bn_scalar_restrict_outputentriesentryoperation) + sto_cn_scalar_restrict_outputentriesentryoperation = (sto_ap_scalar_restrict_outputentriesentryoperation * sto_bn_scalar_restrict_outputentriesentryoperation + sto_an_scalar_restrict_outputentriesentryoperation * sto_bp_scalar_restrict_outputentriesentryoperation) + sto_cp_scalar_restrict_outputentriesentryoperation)))))))))))))))Complete tactic proof in conservative notation
All 27 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
27 script commands · 7 reading checkpoints · 0 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
02Separate the logical casesL6–8
03Use earlier factsL9–13
04Separate the logical casesL14–14
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L14
split
05Use earlier factsL15–19
06Fix variables and assumptionsL20–21
Original defined command ledger · 27 lines
- 0001
intro a - 0002
intro F - 0003
intro G - 0004
intro l - 0005
intro h - 0006
cases h - 0007
cases h_right - 0008
split - 0009
specialize signed_table_domain_resize (S l) - 0010
specialize signed_table_domain_resize (l) - 0011
specialize signed_table_domain_resize (F) - 0012
apply signed_table_domain_resize - 0013
exact h_left - 0014
split - 0015
specialize signed_table_domain_resize (S l) - 0016
specialize signed_table_domain_resize (l) - 0017
specialize signed_table_domain_resize (G) - 0018
apply signed_table_domain_resize - 0019
exact h_right_left - 0020
intro i - 0021
intro hi - 0022
specialize h_right_right (i) - 0023
apply h_right_right - 0024
specialize le_succ (S i) - 0025
specialize le_succ (l) - 0026
apply le_succ - 0027
exact hi