DF000A

dirichlet_grid_flat_prefix_zero

A genuinely constructed singleton supplies the first flat cell; no finite table or choice axiom is used.

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.

Every grid, slice, row sum and intermediate table is constructed. Retained cells have witnessed n=(a*e)*c and value F(a)*(H(e)*G(c)). The flat endpoint is unused. Table associativity includes N=0 and compares only positive values, not encodings. Full G009 remains broader.

Exact theorem in conservative defined notation

∀ F. ∀ G. ∀ H. ∀ n. ∀ T. ∀ z. ArithTable(0,T)ArithAt(T,0,z)DirichletFlatEntry(F,G,H,n,0,z)DirichletFlatPrefix(F,G,H,n,0,T)

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

Definition DAG

Actual proof prerequisites

Original expanded first-order statement
forall F G H n T z. (exists dst_positive_code_base_table dst_positive_scale_base_table dst_negative_code_base_table dst_negative_scale_base_table. (((T) = (((((dst_positive_code_base_table) + (dst_positive_scale_base_table)) * S ((dst_positive_code_base_table) + (dst_positive_scale_base_table)) + ((dst_positive_scale_base_table) + (dst_positive_scale_base_table))) + (((dst_negative_code_base_table) + (dst_negative_scale_base_table)) * S ((dst_negative_code_base_table) + (dst_negative_scale_base_table)) + ((dst_negative_scale_base_table) + (dst_negative_scale_base_table)))) * S ((((dst_positive_code_base_table) + (dst_positive_scale_base_table)) * S ((dst_positive_code_base_table) + (dst_positive_scale_base_table)) + ((dst_positive_scale_base_table) + (dst_positive_scale_base_table))) + (((dst_negative_code_base_table) + (dst_negative_scale_base_table)) * S ((dst_negative_code_base_table) + (dst_negative_scale_base_table)) + ((dst_negative_scale_base_table) + (dst_negative_scale_base_table)))) + ((((dst_negative_code_base_table) + (dst_negative_scale_base_table)) * S ((dst_negative_code_base_table) + (dst_negative_scale_base_table)) + ((dst_negative_scale_base_table) + (dst_negative_scale_base_table))) + (((dst_negative_code_base_table) + (dst_negative_scale_base_table)) * S ((dst_negative_code_base_table) + (dst_negative_scale_base_table)) + ((dst_negative_scale_base_table) + (dst_negative_scale_base_table)))))) /\ (forall dst_index_base_table. (exists pvs_le_gap_base_tabledomain. pvs_le_gap_base_tabledomain + (dst_index_base_table) = (0)) -> exists dst_positive_base_table dst_negative_base_table dst_value_base_table. ((((exists ff_h_pvs_base_tableentrypositive. ff_h_pvs_base_tableentrypositive + S (dst_positive_base_table) = S ((S (dst_index_base_table)) * dst_positive_scale_base_table)) /\ exists ff_q_pvs_base_tableentrypositive. dst_positive_code_base_table = ff_q_pvs_base_tableentrypositive * S ((S (dst_index_base_table)) * dst_positive_scale_base_table) + (dst_positive_base_table))) /\ (((((exists ff_h_pvs_base_tableentrynegative. ff_h_pvs_base_tableentrynegative + S (dst_negative_base_table) = S ((S (dst_index_base_table)) * dst_negative_scale_base_table)) /\ exists ff_q_pvs_base_tableentrynegative. dst_negative_code_base_table = ff_q_pvs_base_tableentrynegative * S ((S (dst_index_base_table)) * dst_negative_scale_base_table) + (dst_negative_base_table))) /\ (exists ge_balance_positive_base_tableentryvalue ge_balance_negative_base_tableentryvalue. (((((dst_value_base_table) = 2 * (ge_balance_positive_base_tableentryvalue) /\ (ge_balance_negative_base_tableentryvalue) = 0) \/ exists ge_signed_half_base_tableentryvaluedecode. (((dst_value_base_table) = 2 * ge_signed_half_base_tableentryvaluedecode + 1 /\ (ge_balance_positive_base_tableentryvalue) = 0) /\ (ge_balance_negative_base_tableentryvalue) = S ge_signed_half_base_tableentryvaluedecode))) /\ ((dst_positive_base_table) + ge_balance_negative_base_tableentryvalue = (dst_negative_base_table) + ge_balance_positive_base_tableentryvalue))))))))) -> (exists dst_positive_code_base_entry dst_positive_scale_base_entry dst_negative_code_base_entry dst_negative_scale_base_entry dst_positive_base_entry dst_negative_base_entry. (((T) = (((((dst_positive_code_base_entry) + (dst_positive_scale_base_entry)) * S ((dst_positive_code_base_entry) + (dst_positive_scale_base_entry)) + ((dst_positive_scale_base_entry) + (dst_positive_scale_base_entry))) + (((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) * S ((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) + ((dst_negative_scale_base_entry) + (dst_negative_scale_base_entry)))) * S ((((dst_positive_code_base_entry) + (dst_positive_scale_base_entry)) * S ((dst_positive_code_base_entry) + (dst_positive_scale_base_entry)) + ((dst_positive_scale_base_entry) + (dst_positive_scale_base_entry))) + (((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) * S ((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) + ((dst_negative_scale_base_entry) + (dst_negative_scale_base_entry)))) + ((((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) * S ((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) + ((dst_negative_scale_base_entry) + (dst_negative_scale_base_entry))) + (((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) * S ((dst_negative_code_base_entry) + (dst_negative_scale_base_entry)) + ((dst_negative_scale_base_entry) + (dst_negative_scale_base_entry)))))) /\ (((((exists ff_h_pvs_base_entrypositive. ff_h_pvs_base_entrypositive + S (dst_positive_base_entry) = S ((S (0)) * dst_positive_scale_base_entry)) /\ exists ff_q_pvs_base_entrypositive. dst_positive_code_base_entry = ff_q_pvs_base_entrypositive * S ((S (0)) * dst_positive_scale_base_entry) + (dst_positive_base_entry))) /\ (((((exists ff_h_pvs_base_entrynegative. ff_h_pvs_base_entrynegative + S (dst_negative_base_entry) = S ((S (0)) * dst_negative_scale_base_entry)) /\ exists ff_q_pvs_base_entrynegative. dst_negative_code_base_entry = ff_q_pvs_base_entrynegative * S ((S (0)) * dst_negative_scale_base_entry) + (dst_negative_base_entry))) /\ (exists ge_balance_positive_base_entryvalue ge_balance_negative_base_entryvalue. (((((z) = 2 * (ge_balance_positive_base_entryvalue) /\ (ge_balance_negative_base_entryvalue) = 0) \/ exists ge_signed_half_base_entryvaluedecode. (((z) = 2 * ge_signed_half_base_entryvaluedecode + 1 /\ (ge_balance_positive_base_entryvalue) = 0) /\ (ge_balance_negative_base_entryvalue) = S ge_signed_half_base_entryvaluedecode))) /\ ((dst_positive_base_entry) + ge_balance_negative_base_entryvalue = (dst_negative_base_entry) + ge_balance_positive_base_entryvalue))))))))) -> (exists dfg_flat_row_base_value dfg_flat_column_base_value. (((0)=((S (n))*(dfg_flat_row_base_value)+(dfg_flat_column_base_value))) /\ (((exists pvs_gap_base_valueremainder. pvs_gap_base_valueremainder + S (dfg_flat_column_base_value) = (S (n))) /\ ((((~((dfg_flat_row_base_value)=0)) /\ (((~((dfg_flat_column_base_value)=0)) /\ (exists dfg_middle_base_valuecell dfg_first_base_valuecell dfg_last_base_valuecell dfg_value_base_valuecell. (((n)=((dfg_flat_row_base_value)*(dfg_flat_column_base_value))*dfg_middle_base_valuecell) /\ (((exists dst_positive_code_base_valuecellfirst dst_positive_scale_base_valuecellfirst dst_negative_code_base_valuecellfirst dst_negative_scale_base_valuecellfirst dst_positive_base_valuecellfirst dst_negative_base_valuecellfirst. (((F) = (((((dst_positive_code_base_valuecellfirst) + (dst_positive_scale_base_valuecellfirst)) * S ((dst_positive_code_base_valuecellfirst) + (dst_positive_scale_base_valuecellfirst)) + ((dst_positive_scale_base_valuecellfirst) + (dst_positive_scale_base_valuecellfirst))) + (((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) * S ((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) + ((dst_negative_scale_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)))) * S ((((dst_positive_code_base_valuecellfirst) + (dst_positive_scale_base_valuecellfirst)) * S ((dst_positive_code_base_valuecellfirst) + (dst_positive_scale_base_valuecellfirst)) + ((dst_positive_scale_base_valuecellfirst) + (dst_positive_scale_base_valuecellfirst))) + (((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) * S ((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) + ((dst_negative_scale_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)))) + ((((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) * S ((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) + ((dst_negative_scale_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst))) + (((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) * S ((dst_negative_code_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)) + ((dst_negative_scale_base_valuecellfirst) + (dst_negative_scale_base_valuecellfirst)))))) /\ (((((exists ff_h_pvs_base_valuecellfirstpositive. ff_h_pvs_base_valuecellfirstpositive + S (dst_positive_base_valuecellfirst) = S ((S (dfg_flat_row_base_value)) * dst_positive_scale_base_valuecellfirst)) /\ exists ff_q_pvs_base_valuecellfirstpositive. dst_positive_code_base_valuecellfirst = ff_q_pvs_base_valuecellfirstpositive * S ((S (dfg_flat_row_base_value)) * dst_positive_scale_base_valuecellfirst) + (dst_positive_base_valuecellfirst))) /\ (((((exists ff_h_pvs_base_valuecellfirstnegative. ff_h_pvs_base_valuecellfirstnegative + S (dst_negative_base_valuecellfirst) = S ((S (dfg_flat_row_base_value)) * dst_negative_scale_base_valuecellfirst)) /\ exists ff_q_pvs_base_valuecellfirstnegative. dst_negative_code_base_valuecellfirst = ff_q_pvs_base_valuecellfirstnegative * S ((S (dfg_flat_row_base_value)) * dst_negative_scale_base_valuecellfirst) + (dst_negative_base_valuecellfirst))) /\ (exists ge_balance_positive_base_valuecellfirstvalue ge_balance_negative_base_valuecellfirstvalue. (((((dfg_first_base_valuecell) = 2 * (ge_balance_positive_base_valuecellfirstvalue) /\ (ge_balance_negative_base_valuecellfirstvalue) = 0) \/ exists ge_signed_half_base_valuecellfirstvaluedecode. (((dfg_first_base_valuecell) = 2 * ge_signed_half_base_valuecellfirstvaluedecode + 1 /\ (ge_balance_positive_base_valuecellfirstvalue) = 0) /\ (ge_balance_negative_base_valuecellfirstvalue) = S ge_signed_half_base_valuecellfirstvaluedecode))) /\ ((dst_positive_base_valuecellfirst) + ge_balance_negative_base_valuecellfirstvalue = (dst_negative_base_valuecellfirst) + ge_balance_positive_base_valuecellfirstvalue))))))))) /\ (((exists dst_positive_code_base_valuecelllast dst_positive_scale_base_valuecelllast dst_negative_code_base_valuecelllast dst_negative_scale_base_valuecelllast dst_positive_base_valuecelllast dst_negative_base_valuecelllast. (((H) = (((((dst_positive_code_base_valuecelllast) + (dst_positive_scale_base_valuecelllast)) * S ((dst_positive_code_base_valuecelllast) + (dst_positive_scale_base_valuecelllast)) + ((dst_positive_scale_base_valuecelllast) + (dst_positive_scale_base_valuecelllast))) + (((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) * S ((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) + ((dst_negative_scale_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)))) * S ((((dst_positive_code_base_valuecelllast) + (dst_positive_scale_base_valuecelllast)) * S ((dst_positive_code_base_valuecelllast) + (dst_positive_scale_base_valuecelllast)) + ((dst_positive_scale_base_valuecelllast) + (dst_positive_scale_base_valuecelllast))) + (((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) * S ((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) + ((dst_negative_scale_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)))) + ((((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) * S ((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) + ((dst_negative_scale_base_valuecelllast) + (dst_negative_scale_base_valuecelllast))) + (((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) * S ((dst_negative_code_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)) + ((dst_negative_scale_base_valuecelllast) + (dst_negative_scale_base_valuecelllast)))))) /\ (((((exists ff_h_pvs_base_valuecelllastpositive. ff_h_pvs_base_valuecelllastpositive + S (dst_positive_base_valuecelllast) = S ((S (dfg_flat_column_base_value)) * dst_positive_scale_base_valuecelllast)) /\ exists ff_q_pvs_base_valuecelllastpositive. dst_positive_code_base_valuecelllast = ff_q_pvs_base_valuecelllastpositive * S ((S (dfg_flat_column_base_value)) * dst_positive_scale_base_valuecelllast) + (dst_positive_base_valuecelllast))) /\ (((((exists ff_h_pvs_base_valuecelllastnegative. ff_h_pvs_base_valuecelllastnegative + S (dst_negative_base_valuecelllast) = S ((S (dfg_flat_column_base_value)) * dst_negative_scale_base_valuecelllast)) /\ exists ff_q_pvs_base_valuecelllastnegative. dst_negative_code_base_valuecelllast = ff_q_pvs_base_valuecelllastnegative * S ((S (dfg_flat_column_base_value)) * dst_negative_scale_base_valuecelllast) + (dst_negative_base_valuecelllast))) /\ (exists ge_balance_positive_base_valuecelllastvalue ge_balance_negative_base_valuecelllastvalue. (((((dfg_last_base_valuecell) = 2 * (ge_balance_positive_base_valuecelllastvalue) /\ (ge_balance_negative_base_valuecelllastvalue) = 0) \/ exists ge_signed_half_base_valuecelllastvaluedecode. (((dfg_last_base_valuecell) = 2 * ge_signed_half_base_valuecelllastvaluedecode + 1 /\ (ge_balance_positive_base_valuecelllastvalue) = 0) /\ (ge_balance_negative_base_valuecelllastvalue) = S ge_signed_half_base_valuecelllastvaluedecode))) /\ ((dst_positive_base_valuecelllast) + ge_balance_negative_base_valuecelllastvalue = (dst_negative_base_valuecelllast) + ge_balance_positive_base_valuecelllastvalue))))))))) /\ (((exists dst_positive_code_base_valuecellmiddle dst_positive_scale_base_valuecellmiddle dst_negative_code_base_valuecellmiddle dst_negative_scale_base_valuecellmiddle dst_positive_base_valuecellmiddle dst_negative_base_valuecellmiddle. (((G) = (((((dst_positive_code_base_valuecellmiddle) + (dst_positive_scale_base_valuecellmiddle)) * S ((dst_positive_code_base_valuecellmiddle) + (dst_positive_scale_base_valuecellmiddle)) + ((dst_positive_scale_base_valuecellmiddle) + (dst_positive_scale_base_valuecellmiddle))) + (((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) * S ((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) + ((dst_negative_scale_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)))) * S ((((dst_positive_code_base_valuecellmiddle) + (dst_positive_scale_base_valuecellmiddle)) * S ((dst_positive_code_base_valuecellmiddle) + (dst_positive_scale_base_valuecellmiddle)) + ((dst_positive_scale_base_valuecellmiddle) + (dst_positive_scale_base_valuecellmiddle))) + (((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) * S ((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) + ((dst_negative_scale_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)))) + ((((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) * S ((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) + ((dst_negative_scale_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle))) + (((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) * S ((dst_negative_code_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)) + ((dst_negative_scale_base_valuecellmiddle) + (dst_negative_scale_base_valuecellmiddle)))))) /\ (((((exists ff_h_pvs_base_valuecellmiddlepositive. ff_h_pvs_base_valuecellmiddlepositive + S (dst_positive_base_valuecellmiddle) = S ((S (dfg_middle_base_valuecell)) * dst_positive_scale_base_valuecellmiddle)) /\ exists ff_q_pvs_base_valuecellmiddlepositive. dst_positive_code_base_valuecellmiddle = ff_q_pvs_base_valuecellmiddlepositive * S ((S (dfg_middle_base_valuecell)) * dst_positive_scale_base_valuecellmiddle) + (dst_positive_base_valuecellmiddle))) /\ (((((exists ff_h_pvs_base_valuecellmiddlenegative. ff_h_pvs_base_valuecellmiddlenegative + S (dst_negative_base_valuecellmiddle) = S ((S (dfg_middle_base_valuecell)) * dst_negative_scale_base_valuecellmiddle)) /\ exists ff_q_pvs_base_valuecellmiddlenegative. dst_negative_code_base_valuecellmiddle = ff_q_pvs_base_valuecellmiddlenegative * S ((S (dfg_middle_base_valuecell)) * dst_negative_scale_base_valuecellmiddle) + (dst_negative_base_valuecellmiddle))) /\ (exists ge_balance_positive_base_valuecellmiddlevalue ge_balance_negative_base_valuecellmiddlevalue. (((((dfg_value_base_valuecell) = 2 * (ge_balance_positive_base_valuecellmiddlevalue) /\ (ge_balance_negative_base_valuecellmiddlevalue) = 0) \/ exists ge_signed_half_base_valuecellmiddlevaluedecode. (((dfg_value_base_valuecell) = 2 * ge_signed_half_base_valuecellmiddlevaluedecode + 1 /\ (ge_balance_positive_base_valuecellmiddlevalue) = 0) /\ (ge_balance_negative_base_valuecellmiddlevalue) = S ge_signed_half_base_valuecellmiddlevaluedecode))) /\ ((dst_positive_base_valuecellmiddle) + ge_balance_negative_base_valuecellmiddlevalue = (dst_negative_base_valuecellmiddle) + ge_balance_positive_base_valuecellmiddlevalue))))))))) /\ (exists dfg_inner_base_valuecellproduct. ((exists sto_ap_base_valuecellproductinner sto_an_base_valuecellproductinner sto_bp_base_valuecellproductinner sto_bn_base_valuecellproductinner sto_cp_base_valuecellproductinner sto_cn_base_valuecellproductinner. (((((dfg_last_base_valuecell) = 2 * (sto_ap_base_valuecellproductinner) /\ (sto_an_base_valuecellproductinner) = 0) \/ exists ge_signed_half_base_valuecellproductinnerleft. (((dfg_last_base_valuecell) = 2 * ge_signed_half_base_valuecellproductinnerleft + 1 /\ (sto_ap_base_valuecellproductinner) = 0) /\ (sto_an_base_valuecellproductinner) = S ge_signed_half_base_valuecellproductinnerleft))) /\ ((((((dfg_value_base_valuecell) = 2 * (sto_bp_base_valuecellproductinner) /\ (sto_bn_base_valuecellproductinner) = 0) \/ exists ge_signed_half_base_valuecellproductinnerright. (((dfg_value_base_valuecell) = 2 * ge_signed_half_base_valuecellproductinnerright + 1 /\ (sto_bp_base_valuecellproductinner) = 0) /\ (sto_bn_base_valuecellproductinner) = S ge_signed_half_base_valuecellproductinnerright))) /\ ((((((dfg_inner_base_valuecellproduct) = 2 * (sto_cp_base_valuecellproductinner) /\ (sto_cn_base_valuecellproductinner) = 0) \/ exists ge_signed_half_base_valuecellproductinneroutput. (((dfg_inner_base_valuecellproduct) = 2 * ge_signed_half_base_valuecellproductinneroutput + 1 /\ (sto_cp_base_valuecellproductinner) = 0) /\ (sto_cn_base_valuecellproductinner) = S ge_signed_half_base_valuecellproductinneroutput))) /\ ((sto_ap_base_valuecellproductinner * sto_bp_base_valuecellproductinner + sto_an_base_valuecellproductinner * sto_bn_base_valuecellproductinner) + sto_cn_base_valuecellproductinner = (sto_ap_base_valuecellproductinner * sto_bn_base_valuecellproductinner + sto_an_base_valuecellproductinner * sto_bp_base_valuecellproductinner) + sto_cp_base_valuecellproductinner))))))) /\ (exists sto_ap_base_valuecellproductouter sto_an_base_valuecellproductouter sto_bp_base_valuecellproductouter sto_bn_base_valuecellproductouter sto_cp_base_valuecellproductouter sto_cn_base_valuecellproductouter. (((((dfg_first_base_valuecell) = 2 * (sto_ap_base_valuecellproductouter) /\ (sto_an_base_valuecellproductouter) = 0) \/ exists ge_signed_half_base_valuecellproductouterleft. (((dfg_first_base_valuecell) = 2 * ge_signed_half_base_valuecellproductouterleft + 1 /\ (sto_ap_base_valuecellproductouter) = 0) /\ (sto_an_base_valuecellproductouter) = S ge_signed_half_base_valuecellproductouterleft))) /\ ((((((dfg_inner_base_valuecellproduct) = 2 * (sto_bp_base_valuecellproductouter) /\ (sto_bn_base_valuecellproductouter) = 0) \/ exists ge_signed_half_base_valuecellproductouterright. (((dfg_inner_base_valuecellproduct) = 2 * ge_signed_half_base_valuecellproductouterright + 1 /\ (sto_bp_base_valuecellproductouter) = 0) /\ (sto_bn_base_valuecellproductouter) = S ge_signed_half_base_valuecellproductouterright))) /\ ((((((z) = 2 * (sto_cp_base_valuecellproductouter) /\ (sto_cn_base_valuecellproductouter) = 0) \/ exists ge_signed_half_base_valuecellproductouteroutput. (((z) = 2 * ge_signed_half_base_valuecellproductouteroutput + 1 /\ (sto_cp_base_valuecellproductouter) = 0) /\ (sto_cn_base_valuecellproductouter) = S ge_signed_half_base_valuecellproductouteroutput))) /\ ((sto_ap_base_valuecellproductouter * sto_bp_base_valuecellproductouter + sto_an_base_valuecellproductouter * sto_bn_base_valuecellproductouter) + sto_cn_base_valuecellproductouter = (sto_ap_base_valuecellproductouter * sto_bn_base_valuecellproductouter + sto_an_base_valuecellproductouter * sto_bp_base_valuecellproductouter) + sto_cp_base_valuecellproductouter))))))))))))))))))))) \/ ((((dfg_flat_row_base_value)=0 \/ ((dfg_flat_column_base_value)=0 \/ ~(exists pvs_factor_base_valuecellomittednondivisor. (n) = ((dfg_flat_row_base_value)*(dfg_flat_column_base_value)) * pvs_factor_base_valuecellomittednondivisor))) /\ ((z)=0)))))))) -> (((exists dst_positive_code_base_prefixtable dst_positive_scale_base_prefixtable dst_negative_code_base_prefixtable dst_negative_scale_base_prefixtable. (((T) = (((((dst_positive_code_base_prefixtable) + (dst_positive_scale_base_prefixtable)) * S ((dst_positive_code_base_prefixtable) + (dst_positive_scale_base_prefixtable)) + ((dst_positive_scale_base_prefixtable) + (dst_positive_scale_base_prefixtable))) + (((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) * S ((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) + ((dst_negative_scale_base_prefixtable) + (dst_negative_scale_base_prefixtable)))) * S ((((dst_positive_code_base_prefixtable) + (dst_positive_scale_base_prefixtable)) * S ((dst_positive_code_base_prefixtable) + (dst_positive_scale_base_prefixtable)) + ((dst_positive_scale_base_prefixtable) + (dst_positive_scale_base_prefixtable))) + (((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) * S ((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) + ((dst_negative_scale_base_prefixtable) + (dst_negative_scale_base_prefixtable)))) + ((((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) * S ((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) + ((dst_negative_scale_base_prefixtable) + (dst_negative_scale_base_prefixtable))) + (((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) * S ((dst_negative_code_base_prefixtable) + (dst_negative_scale_base_prefixtable)) + ((dst_negative_scale_base_prefixtable) + (dst_negative_scale_base_prefixtable)))))) /\ (forall dst_index_base_prefixtable. (exists pvs_le_gap_base_prefixtabledomain. pvs_le_gap_base_prefixtabledomain + (dst_index_base_prefixtable) = (0)) -> exists dst_positive_base_prefixtable dst_negative_base_prefixtable dst_value_base_prefixtable. ((((exists ff_h_pvs_base_prefixtableentrypositive. ff_h_pvs_base_prefixtableentrypositive + S (dst_positive_base_prefixtable) = S ((S (dst_index_base_prefixtable)) * dst_positive_scale_base_prefixtable)) /\ exists ff_q_pvs_base_prefixtableentrypositive. dst_positive_code_base_prefixtable = ff_q_pvs_base_prefixtableentrypositive * S ((S (dst_index_base_prefixtable)) * dst_positive_scale_base_prefixtable) + (dst_positive_base_prefixtable))) /\ (((((exists ff_h_pvs_base_prefixtableentrynegative. ff_h_pvs_base_prefixtableentrynegative + S (dst_negative_base_prefixtable) = S ((S (dst_index_base_prefixtable)) * dst_negative_scale_base_prefixtable)) /\ exists ff_q_pvs_base_prefixtableentrynegative. dst_negative_code_base_prefixtable = ff_q_pvs_base_prefixtableentrynegative * S ((S (dst_index_base_prefixtable)) * dst_negative_scale_base_prefixtable) + (dst_negative_base_prefixtable))) /\ (exists ge_balance_positive_base_prefixtableentryvalue ge_balance_negative_base_prefixtableentryvalue. (((((dst_value_base_prefixtable) = 2 * (ge_balance_positive_base_prefixtableentryvalue) /\ (ge_balance_negative_base_prefixtableentryvalue) = 0) \/ exists ge_signed_half_base_prefixtableentryvaluedecode. (((dst_value_base_prefixtable) = 2 * ge_signed_half_base_prefixtableentryvaluedecode + 1 /\ (ge_balance_positive_base_prefixtableentryvalue) = 0) /\ (ge_balance_negative_base_prefixtableentryvalue) = S ge_signed_half_base_prefixtableentryvaluedecode))) /\ ((dst_positive_base_prefixtable) + ge_balance_negative_base_prefixtableentryvalue = (dst_negative_base_prefixtable) + ge_balance_positive_base_prefixtableentryvalue))))))))) /\ (forall dfg_flat_index_base_prefix dfg_flat_value_base_prefix. (exists pvs_le_gap_base_prefixbound. pvs_le_gap_base_prefixbound + (dfg_flat_index_base_prefix) = (0)) -> (exists dst_positive_code_base_prefixlookup dst_positive_scale_base_prefixlookup dst_negative_code_base_prefixlookup dst_negative_scale_base_prefixlookup dst_positive_base_prefixlookup dst_negative_base_prefixlookup. (((T) = (((((dst_positive_code_base_prefixlookup) + (dst_positive_scale_base_prefixlookup)) * S ((dst_positive_code_base_prefixlookup) + (dst_positive_scale_base_prefixlookup)) + ((dst_positive_scale_base_prefixlookup) + (dst_positive_scale_base_prefixlookup))) + (((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) * S ((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) + ((dst_negative_scale_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)))) * S ((((dst_positive_code_base_prefixlookup) + (dst_positive_scale_base_prefixlookup)) * S ((dst_positive_code_base_prefixlookup) + (dst_positive_scale_base_prefixlookup)) + ((dst_positive_scale_base_prefixlookup) + (dst_positive_scale_base_prefixlookup))) + (((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) * S ((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) + ((dst_negative_scale_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)))) + ((((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) * S ((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) + ((dst_negative_scale_base_prefixlookup) + (dst_negative_scale_base_prefixlookup))) + (((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) * S ((dst_negative_code_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)) + ((dst_negative_scale_base_prefixlookup) + (dst_negative_scale_base_prefixlookup)))))) /\ (((((exists ff_h_pvs_base_prefixlookuppositive. ff_h_pvs_base_prefixlookuppositive + S (dst_positive_base_prefixlookup) = S ((S (dfg_flat_index_base_prefix)) * dst_positive_scale_base_prefixlookup)) /\ exists ff_q_pvs_base_prefixlookuppositive. dst_positive_code_base_prefixlookup = ff_q_pvs_base_prefixlookuppositive * S ((S (dfg_flat_index_base_prefix)) * dst_positive_scale_base_prefixlookup) + (dst_positive_base_prefixlookup))) /\ (((((exists ff_h_pvs_base_prefixlookupnegative. ff_h_pvs_base_prefixlookupnegative + S (dst_negative_base_prefixlookup) = S ((S (dfg_flat_index_base_prefix)) * dst_negative_scale_base_prefixlookup)) /\ exists ff_q_pvs_base_prefixlookupnegative. dst_negative_code_base_prefixlookup = ff_q_pvs_base_prefixlookupnegative * S ((S (dfg_flat_index_base_prefix)) * dst_negative_scale_base_prefixlookup) + (dst_negative_base_prefixlookup))) /\ (exists ge_balance_positive_base_prefixlookupvalue ge_balance_negative_base_prefixlookupvalue. (((((dfg_flat_value_base_prefix) = 2 * (ge_balance_positive_base_prefixlookupvalue) /\ (ge_balance_negative_base_prefixlookupvalue) = 0) \/ exists ge_signed_half_base_prefixlookupvaluedecode. (((dfg_flat_value_base_prefix) = 2 * ge_signed_half_base_prefixlookupvaluedecode + 1 /\ (ge_balance_positive_base_prefixlookupvalue) = 0) /\ (ge_balance_negative_base_prefixlookupvalue) = S ge_signed_half_base_prefixlookupvaluedecode))) /\ ((dst_positive_base_prefixlookup) + ge_balance_negative_base_prefixlookupvalue = (dst_negative_base_prefixlookup) + ge_balance_positive_base_prefixlookupvalue))))))))) -> (exists dfg_flat_row_base_prefixentry dfg_flat_column_base_prefixentry. (((dfg_flat_index_base_prefix)=((S (n))*(dfg_flat_row_base_prefixentry)+(dfg_flat_column_base_prefixentry))) /\ (((exists pvs_gap_base_prefixentryremainder. pvs_gap_base_prefixentryremainder + S (dfg_flat_column_base_prefixentry) = (S (n))) /\ ((((~((dfg_flat_row_base_prefixentry)=0)) /\ (((~((dfg_flat_column_base_prefixentry)=0)) /\ (exists dfg_middle_base_prefixentrycell dfg_first_base_prefixentrycell dfg_last_base_prefixentrycell dfg_value_base_prefixentrycell. (((n)=((dfg_flat_row_base_prefixentry)*(dfg_flat_column_base_prefixentry))*dfg_middle_base_prefixentrycell) /\ (((exists dst_positive_code_base_prefixentrycellfirst dst_positive_scale_base_prefixentrycellfirst dst_negative_code_base_prefixentrycellfirst dst_negative_scale_base_prefixentrycellfirst dst_positive_base_prefixentrycellfirst dst_negative_base_prefixentrycellfirst. (((F) = (((((dst_positive_code_base_prefixentrycellfirst) + (dst_positive_scale_base_prefixentrycellfirst)) * S ((dst_positive_code_base_prefixentrycellfirst) + (dst_positive_scale_base_prefixentrycellfirst)) + ((dst_positive_scale_base_prefixentrycellfirst) + (dst_positive_scale_base_prefixentrycellfirst))) + (((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) * S ((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) + ((dst_negative_scale_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)))) * S ((((dst_positive_code_base_prefixentrycellfirst) + (dst_positive_scale_base_prefixentrycellfirst)) * S ((dst_positive_code_base_prefixentrycellfirst) + (dst_positive_scale_base_prefixentrycellfirst)) + ((dst_positive_scale_base_prefixentrycellfirst) + (dst_positive_scale_base_prefixentrycellfirst))) + (((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) * S ((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) + ((dst_negative_scale_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)))) + ((((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) * S ((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) + ((dst_negative_scale_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst))) + (((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) * S ((dst_negative_code_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)) + ((dst_negative_scale_base_prefixentrycellfirst) + (dst_negative_scale_base_prefixentrycellfirst)))))) /\ (((((exists ff_h_pvs_base_prefixentrycellfirstpositive. ff_h_pvs_base_prefixentrycellfirstpositive + S (dst_positive_base_prefixentrycellfirst) = S ((S (dfg_flat_row_base_prefixentry)) * dst_positive_scale_base_prefixentrycellfirst)) /\ exists ff_q_pvs_base_prefixentrycellfirstpositive. dst_positive_code_base_prefixentrycellfirst = ff_q_pvs_base_prefixentrycellfirstpositive * S ((S (dfg_flat_row_base_prefixentry)) * dst_positive_scale_base_prefixentrycellfirst) + (dst_positive_base_prefixentrycellfirst))) /\ (((((exists ff_h_pvs_base_prefixentrycellfirstnegative. ff_h_pvs_base_prefixentrycellfirstnegative + S (dst_negative_base_prefixentrycellfirst) = S ((S (dfg_flat_row_base_prefixentry)) * dst_negative_scale_base_prefixentrycellfirst)) /\ exists ff_q_pvs_base_prefixentrycellfirstnegative. dst_negative_code_base_prefixentrycellfirst = ff_q_pvs_base_prefixentrycellfirstnegative * S ((S (dfg_flat_row_base_prefixentry)) * dst_negative_scale_base_prefixentrycellfirst) + (dst_negative_base_prefixentrycellfirst))) /\ (exists ge_balance_positive_base_prefixentrycellfirstvalue ge_balance_negative_base_prefixentrycellfirstvalue. (((((dfg_first_base_prefixentrycell) = 2 * (ge_balance_positive_base_prefixentrycellfirstvalue) /\ (ge_balance_negative_base_prefixentrycellfirstvalue) = 0) \/ exists ge_signed_half_base_prefixentrycellfirstvaluedecode. (((dfg_first_base_prefixentrycell) = 2 * ge_signed_half_base_prefixentrycellfirstvaluedecode + 1 /\ (ge_balance_positive_base_prefixentrycellfirstvalue) = 0) /\ (ge_balance_negative_base_prefixentrycellfirstvalue) = S ge_signed_half_base_prefixentrycellfirstvaluedecode))) /\ ((dst_positive_base_prefixentrycellfirst) + ge_balance_negative_base_prefixentrycellfirstvalue = (dst_negative_base_prefixentrycellfirst) + ge_balance_positive_base_prefixentrycellfirstvalue))))))))) /\ (((exists dst_positive_code_base_prefixentrycelllast dst_positive_scale_base_prefixentrycelllast dst_negative_code_base_prefixentrycelllast dst_negative_scale_base_prefixentrycelllast dst_positive_base_prefixentrycelllast dst_negative_base_prefixentrycelllast. (((H) = (((((dst_positive_code_base_prefixentrycelllast) + (dst_positive_scale_base_prefixentrycelllast)) * S ((dst_positive_code_base_prefixentrycelllast) + (dst_positive_scale_base_prefixentrycelllast)) + ((dst_positive_scale_base_prefixentrycelllast) + (dst_positive_scale_base_prefixentrycelllast))) + (((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) * S ((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) + ((dst_negative_scale_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)))) * S ((((dst_positive_code_base_prefixentrycelllast) + (dst_positive_scale_base_prefixentrycelllast)) * S ((dst_positive_code_base_prefixentrycelllast) + (dst_positive_scale_base_prefixentrycelllast)) + ((dst_positive_scale_base_prefixentrycelllast) + (dst_positive_scale_base_prefixentrycelllast))) + (((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) * S ((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) + ((dst_negative_scale_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)))) + ((((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) * S ((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) + ((dst_negative_scale_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast))) + (((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) * S ((dst_negative_code_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)) + ((dst_negative_scale_base_prefixentrycelllast) + (dst_negative_scale_base_prefixentrycelllast)))))) /\ (((((exists ff_h_pvs_base_prefixentrycelllastpositive. ff_h_pvs_base_prefixentrycelllastpositive + S (dst_positive_base_prefixentrycelllast) = S ((S (dfg_flat_column_base_prefixentry)) * dst_positive_scale_base_prefixentrycelllast)) /\ exists ff_q_pvs_base_prefixentrycelllastpositive. dst_positive_code_base_prefixentrycelllast = ff_q_pvs_base_prefixentrycelllastpositive * S ((S (dfg_flat_column_base_prefixentry)) * dst_positive_scale_base_prefixentrycelllast) + (dst_positive_base_prefixentrycelllast))) /\ (((((exists ff_h_pvs_base_prefixentrycelllastnegative. ff_h_pvs_base_prefixentrycelllastnegative + S (dst_negative_base_prefixentrycelllast) = S ((S (dfg_flat_column_base_prefixentry)) * dst_negative_scale_base_prefixentrycelllast)) /\ exists ff_q_pvs_base_prefixentrycelllastnegative. dst_negative_code_base_prefixentrycelllast = ff_q_pvs_base_prefixentrycelllastnegative * S ((S (dfg_flat_column_base_prefixentry)) * dst_negative_scale_base_prefixentrycelllast) + (dst_negative_base_prefixentrycelllast))) /\ (exists ge_balance_positive_base_prefixentrycelllastvalue ge_balance_negative_base_prefixentrycelllastvalue. (((((dfg_last_base_prefixentrycell) = 2 * (ge_balance_positive_base_prefixentrycelllastvalue) /\ (ge_balance_negative_base_prefixentrycelllastvalue) = 0) \/ exists ge_signed_half_base_prefixentrycelllastvaluedecode. (((dfg_last_base_prefixentrycell) = 2 * ge_signed_half_base_prefixentrycelllastvaluedecode + 1 /\ (ge_balance_positive_base_prefixentrycelllastvalue) = 0) /\ (ge_balance_negative_base_prefixentrycelllastvalue) = S ge_signed_half_base_prefixentrycelllastvaluedecode))) /\ ((dst_positive_base_prefixentrycelllast) + ge_balance_negative_base_prefixentrycelllastvalue = (dst_negative_base_prefixentrycelllast) + ge_balance_positive_base_prefixentrycelllastvalue))))))))) /\ (((exists dst_positive_code_base_prefixentrycellmiddle dst_positive_scale_base_prefixentrycellmiddle dst_negative_code_base_prefixentrycellmiddle dst_negative_scale_base_prefixentrycellmiddle dst_positive_base_prefixentrycellmiddle dst_negative_base_prefixentrycellmiddle. (((G) = (((((dst_positive_code_base_prefixentrycellmiddle) + (dst_positive_scale_base_prefixentrycellmiddle)) * S ((dst_positive_code_base_prefixentrycellmiddle) + (dst_positive_scale_base_prefixentrycellmiddle)) + ((dst_positive_scale_base_prefixentrycellmiddle) + (dst_positive_scale_base_prefixentrycellmiddle))) + (((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) * S ((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) + ((dst_negative_scale_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)))) * S ((((dst_positive_code_base_prefixentrycellmiddle) + (dst_positive_scale_base_prefixentrycellmiddle)) * S ((dst_positive_code_base_prefixentrycellmiddle) + (dst_positive_scale_base_prefixentrycellmiddle)) + ((dst_positive_scale_base_prefixentrycellmiddle) + (dst_positive_scale_base_prefixentrycellmiddle))) + (((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) * S ((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) + ((dst_negative_scale_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)))) + ((((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) * S ((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) + ((dst_negative_scale_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle))) + (((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) * S ((dst_negative_code_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)) + ((dst_negative_scale_base_prefixentrycellmiddle) + (dst_negative_scale_base_prefixentrycellmiddle)))))) /\ (((((exists ff_h_pvs_base_prefixentrycellmiddlepositive. ff_h_pvs_base_prefixentrycellmiddlepositive + S (dst_positive_base_prefixentrycellmiddle) = S ((S (dfg_middle_base_prefixentrycell)) * dst_positive_scale_base_prefixentrycellmiddle)) /\ exists ff_q_pvs_base_prefixentrycellmiddlepositive. dst_positive_code_base_prefixentrycellmiddle = ff_q_pvs_base_prefixentrycellmiddlepositive * S ((S (dfg_middle_base_prefixentrycell)) * dst_positive_scale_base_prefixentrycellmiddle) + (dst_positive_base_prefixentrycellmiddle))) /\ (((((exists ff_h_pvs_base_prefixentrycellmiddlenegative. ff_h_pvs_base_prefixentrycellmiddlenegative + S (dst_negative_base_prefixentrycellmiddle) = S ((S (dfg_middle_base_prefixentrycell)) * dst_negative_scale_base_prefixentrycellmiddle)) /\ exists ff_q_pvs_base_prefixentrycellmiddlenegative. dst_negative_code_base_prefixentrycellmiddle = ff_q_pvs_base_prefixentrycellmiddlenegative * S ((S (dfg_middle_base_prefixentrycell)) * dst_negative_scale_base_prefixentrycellmiddle) + (dst_negative_base_prefixentrycellmiddle))) /\ (exists ge_balance_positive_base_prefixentrycellmiddlevalue ge_balance_negative_base_prefixentrycellmiddlevalue. (((((dfg_value_base_prefixentrycell) = 2 * (ge_balance_positive_base_prefixentrycellmiddlevalue) /\ (ge_balance_negative_base_prefixentrycellmiddlevalue) = 0) \/ exists ge_signed_half_base_prefixentrycellmiddlevaluedecode. (((dfg_value_base_prefixentrycell) = 2 * ge_signed_half_base_prefixentrycellmiddlevaluedecode + 1 /\ (ge_balance_positive_base_prefixentrycellmiddlevalue) = 0) /\ (ge_balance_negative_base_prefixentrycellmiddlevalue) = S ge_signed_half_base_prefixentrycellmiddlevaluedecode))) /\ ((dst_positive_base_prefixentrycellmiddle) + ge_balance_negative_base_prefixentrycellmiddlevalue = (dst_negative_base_prefixentrycellmiddle) + ge_balance_positive_base_prefixentrycellmiddlevalue))))))))) /\ (exists dfg_inner_base_prefixentrycellproduct. ((exists sto_ap_base_prefixentrycellproductinner sto_an_base_prefixentrycellproductinner sto_bp_base_prefixentrycellproductinner sto_bn_base_prefixentrycellproductinner sto_cp_base_prefixentrycellproductinner sto_cn_base_prefixentrycellproductinner. (((((dfg_last_base_prefixentrycell) = 2 * (sto_ap_base_prefixentrycellproductinner) /\ (sto_an_base_prefixentrycellproductinner) = 0) \/ exists ge_signed_half_base_prefixentrycellproductinnerleft. (((dfg_last_base_prefixentrycell) = 2 * ge_signed_half_base_prefixentrycellproductinnerleft + 1 /\ (sto_ap_base_prefixentrycellproductinner) = 0) /\ (sto_an_base_prefixentrycellproductinner) = S ge_signed_half_base_prefixentrycellproductinnerleft))) /\ ((((((dfg_value_base_prefixentrycell) = 2 * (sto_bp_base_prefixentrycellproductinner) /\ (sto_bn_base_prefixentrycellproductinner) = 0) \/ exists ge_signed_half_base_prefixentrycellproductinnerright. (((dfg_value_base_prefixentrycell) = 2 * ge_signed_half_base_prefixentrycellproductinnerright + 1 /\ (sto_bp_base_prefixentrycellproductinner) = 0) /\ (sto_bn_base_prefixentrycellproductinner) = S ge_signed_half_base_prefixentrycellproductinnerright))) /\ ((((((dfg_inner_base_prefixentrycellproduct) = 2 * (sto_cp_base_prefixentrycellproductinner) /\ (sto_cn_base_prefixentrycellproductinner) = 0) \/ exists ge_signed_half_base_prefixentrycellproductinneroutput. (((dfg_inner_base_prefixentrycellproduct) = 2 * ge_signed_half_base_prefixentrycellproductinneroutput + 1 /\ (sto_cp_base_prefixentrycellproductinner) = 0) /\ (sto_cn_base_prefixentrycellproductinner) = S ge_signed_half_base_prefixentrycellproductinneroutput))) /\ ((sto_ap_base_prefixentrycellproductinner * sto_bp_base_prefixentrycellproductinner + sto_an_base_prefixentrycellproductinner * sto_bn_base_prefixentrycellproductinner) + sto_cn_base_prefixentrycellproductinner = (sto_ap_base_prefixentrycellproductinner * sto_bn_base_prefixentrycellproductinner + sto_an_base_prefixentrycellproductinner * sto_bp_base_prefixentrycellproductinner) + sto_cp_base_prefixentrycellproductinner))))))) /\ (exists sto_ap_base_prefixentrycellproductouter sto_an_base_prefixentrycellproductouter sto_bp_base_prefixentrycellproductouter sto_bn_base_prefixentrycellproductouter sto_cp_base_prefixentrycellproductouter sto_cn_base_prefixentrycellproductouter. (((((dfg_first_base_prefixentrycell) = 2 * (sto_ap_base_prefixentrycellproductouter) /\ (sto_an_base_prefixentrycellproductouter) = 0) \/ exists ge_signed_half_base_prefixentrycellproductouterleft. (((dfg_first_base_prefixentrycell) = 2 * ge_signed_half_base_prefixentrycellproductouterleft + 1 /\ (sto_ap_base_prefixentrycellproductouter) = 0) /\ (sto_an_base_prefixentrycellproductouter) = S ge_signed_half_base_prefixentrycellproductouterleft))) /\ ((((((dfg_inner_base_prefixentrycellproduct) = 2 * (sto_bp_base_prefixentrycellproductouter) /\ (sto_bn_base_prefixentrycellproductouter) = 0) \/ exists ge_signed_half_base_prefixentrycellproductouterright. (((dfg_inner_base_prefixentrycellproduct) = 2 * ge_signed_half_base_prefixentrycellproductouterright + 1 /\ (sto_bp_base_prefixentrycellproductouter) = 0) /\ (sto_bn_base_prefixentrycellproductouter) = S ge_signed_half_base_prefixentrycellproductouterright))) /\ ((((((dfg_flat_value_base_prefix) = 2 * (sto_cp_base_prefixentrycellproductouter) /\ (sto_cn_base_prefixentrycellproductouter) = 0) \/ exists ge_signed_half_base_prefixentrycellproductouteroutput. (((dfg_flat_value_base_prefix) = 2 * ge_signed_half_base_prefixentrycellproductouteroutput + 1 /\ (sto_cp_base_prefixentrycellproductouter) = 0) /\ (sto_cn_base_prefixentrycellproductouter) = S ge_signed_half_base_prefixentrycellproductouteroutput))) /\ ((sto_ap_base_prefixentrycellproductouter * sto_bp_base_prefixentrycellproductouter + sto_an_base_prefixentrycellproductouter * sto_bn_base_prefixentrycellproductouter) + sto_cn_base_prefixentrycellproductouter = (sto_ap_base_prefixentrycellproductouter * sto_bn_base_prefixentrycellproductouter + sto_an_base_prefixentrycellproductouter * sto_bp_base_prefixentrycellproductouter) + sto_cp_base_prefixentrycellproductouter))))))))))))))))))))) \/ ((((dfg_flat_row_base_prefixentry)=0 \/ ((dfg_flat_column_base_prefixentry)=0 \/ ~(exists pvs_factor_base_prefixentrycellomittednondivisor. (n) = ((dfg_flat_row_base_prefixentry)*(dfg_flat_column_base_prefixentry)) * pvs_factor_base_prefixentrycellomittednondivisor))) /\ ((dfg_flat_value_base_prefix)=0)))))))))))

Complete tactic proof in conservative notation

All 36 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

36 script commands · 8 reading checkpoints · 2 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–9

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

  1. L1
    intro F
  2. L2
    intro G
  3. L3
    intro H
  4. L4
    intro n
  5. L5
    intro T
  6. L6
    intro z
  7. L7
    intro hT
  8. L8
    intro hz
  9. L9
    intro hv
02Separate the logical casesL10–10

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

  1. L10
    split
03Use earlier factsL11–11

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

  1. L11
    exact hT
04Fix variables and assumptionsL12–15

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

  1. L12
    intro i
  2. L13
    intro u
  3. L14
    intro hi
  4. L15
    intro hu
05Establish hi0L16–23

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

  1. L16
    have hi0 : i=0
  2. L17
    specialize le_zero (i)
  3. L18
    apply le_zero
  4. L19
    exact hi
  5. L20
    rewrite hi0 at hu
  6. L21
    rewrite hi0 at hu
  7. L22
    rewrite hi0 at hu
  8. L23
    rewrite hi0 at hu
06Establish hvalueL24–33

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

  1. L24
    have hvalue : z=u
  2. L25
    specialize divisor_signed_table_at_functional (T)
  3. L26
    specialize divisor_signed_table_at_functional (0)
  4. L27
    specialize divisor_signed_table_at_functional (z)
  5. L28
    specialize divisor_signed_table_at_functional (u)
  6. L29
    apply divisor_signed_table_at_functional
  7. L30
    exact hz
  8. L31
    exact hu
  9. L32
    rewrite hvalue at hv
  10. L33
    rewrite hvalue at hv
07Calculate and transport equalitiesL34–35

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L34
    rewrite hvalue at hv
  2. L35
    rewrite hi0
08Use earlier factsL36–36

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

  1. L36
    exact hv

Library-wide reading audit

Original defined command ledger · 36 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro H
  4. 0004intro n
  5. 0005intro T
  6. 0006intro z
  7. 0007intro hT
  8. 0008intro hz
  9. 0009intro hv
  10. 0010split
  11. 0011exact hT
  12. 0012intro i
  13. 0013intro u
  14. 0014intro hi
  15. 0015intro hu
  16. 0016have hi0 : i=0
  17. 0017specialize le_zero (i)
  18. 0018apply le_zero
  19. 0019exact hi
  20. 0020rewrite hi0 at hu
  21. 0021rewrite hi0 at hu
  22. 0022rewrite hi0 at hu
  23. 0023rewrite hi0 at hu
  24. 0024have hvalue : z=u
  25. 0025specialize divisor_signed_table_at_functional (T)
  26. 0026specialize divisor_signed_table_at_functional (0)
  27. 0027specialize divisor_signed_table_at_functional (z)
  28. 0028specialize divisor_signed_table_at_functional (u)
  29. 0029apply divisor_signed_table_at_functional
  30. 0030exact hz
  31. 0031exact hu
  32. 0032rewrite hvalue at hv
  33. 0033rewrite hvalue at hv
  34. 0034rewrite hvalue at hv
  35. 0035rewrite hi0
  36. 0036exact hv