WS000C

signed_table_scalar_restrict

Restrict the strict pointwise window from S l to l while retaining genuine input and output table certificates.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

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

Exact theorem in conservative defined notation

∀ 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

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

  1. L1
    intro a
  2. L2
    intro F
  3. L3
    intro G
  4. L4
    intro l
  5. L5
    intro h
02Separate the logical casesL6–8

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

  1. L6
    cases h
  2. L7
    cases h_right
  3. L8
    split
03Use earlier factsL9–13

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

  1. L9
    specialize signed_table_domain_resize (S l)
  2. L10
    specialize signed_table_domain_resize (l)
  3. L11
    specialize signed_table_domain_resize (F)
  4. L12
    apply signed_table_domain_resize
  5. L13
    exact h_left
04Separate the logical casesL14–14

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

  1. L14
    split
05Use earlier factsL15–19

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

  1. L15
    specialize signed_table_domain_resize (S l)
  2. L16
    specialize signed_table_domain_resize (l)
  3. L17
    specialize signed_table_domain_resize (G)
  4. L18
    apply signed_table_domain_resize
  5. L19
    exact h_right_left
06Fix variables and assumptionsL20–21

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

  1. L20
    intro i
  2. L21
    intro hi
07Use earlier factsL22–27

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

  1. L22
    specialize h_right_right (i)
  2. L23
    apply h_right_right
  3. L24
    specialize le_succ (S i)
  4. L25
    specialize le_succ (l)
  5. L26
    apply le_succ
  6. L27
    exact hi

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro a
  2. 0002intro F
  3. 0003intro G
  4. 0004intro l
  5. 0005intro h
  6. 0006cases h
  7. 0007cases h_right
  8. 0008split
  9. 0009specialize signed_table_domain_resize (S l)
  10. 0010specialize signed_table_domain_resize (l)
  11. 0011specialize signed_table_domain_resize (F)
  12. 0012apply signed_table_domain_resize
  13. 0013exact h_left
  14. 0014split
  15. 0015specialize signed_table_domain_resize (S l)
  16. 0016specialize signed_table_domain_resize (l)
  17. 0017specialize signed_table_domain_resize (G)
  18. 0018apply signed_table_domain_resize
  19. 0019exact h_right_left
  20. 0020intro i
  21. 0021intro hi
  22. 0022specialize h_right_right (i)
  23. 0023apply h_right_right
  24. 0024specialize le_succ (S i)
  25. 0025specialize le_succ (l)
  26. 0026apply le_succ
  27. 0027exact hi