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 slice, row table, column table and signed sum is an actual beta-coded witness. Entries are F((o+s*i)+t*j), for i<m and j<n. Zero dimensions and zero strides are allowed; separately certified endpoints are unused. Table uniqueness concerns values, not codes. No infinite-sum assertion is made.
Exact theorem in conservative defined notation
∀ F. ∀ G. ∀ H. ∀ o. ∀ s. ∀ l. ∀ a. ArithSlice(F,G,o,s,l) → ArithTable(l,H) → ArithTableEqual(G,H,l) → ArithAt(F,o + s · l,a) → ArithAt(H,l,a) → ArithSlice(F,H,o,s,S l)
Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall F G H o s l a. (((exists dst_positive_code_extend_prefixsource_table dst_positive_scale_extend_prefixsource_table dst_negative_code_extend_prefixsource_table dst_negative_scale_extend_prefixsource_table. (((F) = (((((dst_positive_code_extend_prefixsource_table) + (dst_positive_scale_extend_prefixsource_table)) * S ((dst_positive_code_extend_prefixsource_table) + (dst_positive_scale_extend_prefixsource_table)) + ((dst_positive_scale_extend_prefixsource_table) + (dst_positive_scale_extend_prefixsource_table))) + (((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) * S ((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) + ((dst_negative_scale_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)))) * S ((((dst_positive_code_extend_prefixsource_table) + (dst_positive_scale_extend_prefixsource_table)) * S ((dst_positive_code_extend_prefixsource_table) + (dst_positive_scale_extend_prefixsource_table)) + ((dst_positive_scale_extend_prefixsource_table) + (dst_positive_scale_extend_prefixsource_table))) + (((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) * S ((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) + ((dst_negative_scale_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)))) + ((((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) * S ((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) + ((dst_negative_scale_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table))) + (((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) * S ((dst_negative_code_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)) + ((dst_negative_scale_extend_prefixsource_table) + (dst_negative_scale_extend_prefixsource_table)))))) /\ (forall dst_index_extend_prefixsource_table. (exists pvs_le_gap_extend_prefixsource_tabledomain. pvs_le_gap_extend_prefixsource_tabledomain + (dst_index_extend_prefixsource_table) = (0)) -> exists dst_positive_extend_prefixsource_table dst_negative_extend_prefixsource_table dst_value_extend_prefixsource_table. ((((exists ff_h_pvs_extend_prefixsource_tableentrypositive. ff_h_pvs_extend_prefixsource_tableentrypositive + S (dst_positive_extend_prefixsource_table) = S ((S (dst_index_extend_prefixsource_table)) * dst_positive_scale_extend_prefixsource_table)) /\ exists ff_q_pvs_extend_prefixsource_tableentrypositive. dst_positive_code_extend_prefixsource_table = ff_q_pvs_extend_prefixsource_tableentrypositive * S ((S (dst_index_extend_prefixsource_table)) * dst_positive_scale_extend_prefixsource_table) + (dst_positive_extend_prefixsource_table))) /\ (((((exists ff_h_pvs_extend_prefixsource_tableentrynegative. ff_h_pvs_extend_prefixsource_tableentrynegative + S (dst_negative_extend_prefixsource_table) = S ((S (dst_index_extend_prefixsource_table)) * dst_negative_scale_extend_prefixsource_table)) /\ exists ff_q_pvs_extend_prefixsource_tableentrynegative. dst_negative_code_extend_prefixsource_table = ff_q_pvs_extend_prefixsource_tableentrynegative * S ((S (dst_index_extend_prefixsource_table)) * dst_negative_scale_extend_prefixsource_table) + (dst_negative_extend_prefixsource_table))) /\ (exists ge_balance_positive_extend_prefixsource_tableentryvalue ge_balance_negative_extend_prefixsource_tableentryvalue. (((((dst_value_extend_prefixsource_table) = 2 * (ge_balance_positive_extend_prefixsource_tableentryvalue) /\ (ge_balance_negative_extend_prefixsource_tableentryvalue) = 0) \/ exists ge_signed_half_extend_prefixsource_tableentryvaluedecode. (((dst_value_extend_prefixsource_table) = 2 * ge_signed_half_extend_prefixsource_tableentryvaluedecode + 1 /\ (ge_balance_positive_extend_prefixsource_tableentryvalue) = 0) /\ (ge_balance_negative_extend_prefixsource_tableentryvalue) = S ge_signed_half_extend_prefixsource_tableentryvaluedecode))) /\ ((dst_positive_extend_prefixsource_table) + ge_balance_negative_extend_prefixsource_tableentryvalue = (dst_negative_extend_prefixsource_table) + ge_balance_positive_extend_prefixsource_tableentryvalue))))))))) /\ (((exists dst_positive_code_extend_prefixoutput_table dst_positive_scale_extend_prefixoutput_table dst_negative_code_extend_prefixoutput_table dst_negative_scale_extend_prefixoutput_table. (((G) = (((((dst_positive_code_extend_prefixoutput_table) + (dst_positive_scale_extend_prefixoutput_table)) * S ((dst_positive_code_extend_prefixoutput_table) + (dst_positive_scale_extend_prefixoutput_table)) + ((dst_positive_scale_extend_prefixoutput_table) + (dst_positive_scale_extend_prefixoutput_table))) + (((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) * S ((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) + ((dst_negative_scale_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)))) * S ((((dst_positive_code_extend_prefixoutput_table) + (dst_positive_scale_extend_prefixoutput_table)) * S ((dst_positive_code_extend_prefixoutput_table) + (dst_positive_scale_extend_prefixoutput_table)) + ((dst_positive_scale_extend_prefixoutput_table) + (dst_positive_scale_extend_prefixoutput_table))) + (((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) * S ((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) + ((dst_negative_scale_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)))) + ((((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) * S ((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) + ((dst_negative_scale_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table))) + (((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) * S ((dst_negative_code_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)) + ((dst_negative_scale_extend_prefixoutput_table) + (dst_negative_scale_extend_prefixoutput_table)))))) /\ (forall dst_index_extend_prefixoutput_table. (exists pvs_le_gap_extend_prefixoutput_tabledomain. pvs_le_gap_extend_prefixoutput_tabledomain + (dst_index_extend_prefixoutput_table) = (l)) -> exists dst_positive_extend_prefixoutput_table dst_negative_extend_prefixoutput_table dst_value_extend_prefixoutput_table. ((((exists ff_h_pvs_extend_prefixoutput_tableentrypositive. ff_h_pvs_extend_prefixoutput_tableentrypositive + S (dst_positive_extend_prefixoutput_table) = S ((S (dst_index_extend_prefixoutput_table)) * dst_positive_scale_extend_prefixoutput_table)) /\ exists ff_q_pvs_extend_prefixoutput_tableentrypositive. dst_positive_code_extend_prefixoutput_table = ff_q_pvs_extend_prefixoutput_tableentrypositive * S ((S (dst_index_extend_prefixoutput_table)) * dst_positive_scale_extend_prefixoutput_table) + (dst_positive_extend_prefixoutput_table))) /\ (((((exists ff_h_pvs_extend_prefixoutput_tableentrynegative. ff_h_pvs_extend_prefixoutput_tableentrynegative + S (dst_negative_extend_prefixoutput_table) = S ((S (dst_index_extend_prefixoutput_table)) * dst_negative_scale_extend_prefixoutput_table)) /\ exists ff_q_pvs_extend_prefixoutput_tableentrynegative. dst_negative_code_extend_prefixoutput_table = ff_q_pvs_extend_prefixoutput_tableentrynegative * S ((S (dst_index_extend_prefixoutput_table)) * dst_negative_scale_extend_prefixoutput_table) + (dst_negative_extend_prefixoutput_table))) /\ (exists ge_balance_positive_extend_prefixoutput_tableentryvalue ge_balance_negative_extend_prefixoutput_tableentryvalue. (((((dst_value_extend_prefixoutput_table) = 2 * (ge_balance_positive_extend_prefixoutput_tableentryvalue) /\ (ge_balance_negative_extend_prefixoutput_tableentryvalue) = 0) \/ exists ge_signed_half_extend_prefixoutput_tableentryvaluedecode. (((dst_value_extend_prefixoutput_table) = 2 * ge_signed_half_extend_prefixoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_extend_prefixoutput_tableentryvalue) = 0) /\ (ge_balance_negative_extend_prefixoutput_tableentryvalue) = S ge_signed_half_extend_prefixoutput_tableentryvaluedecode))) /\ ((dst_positive_extend_prefixoutput_table) + ge_balance_negative_extend_prefixoutput_tableentryvalue = (dst_negative_extend_prefixoutput_table) + ge_balance_positive_extend_prefixoutput_tableentryvalue))))))))) /\ (forall srs_index_extend_prefix. (exists pvs_gap_extend_prefixbound. pvs_gap_extend_prefixbound + S (srs_index_extend_prefix) = (l)) -> exists srs_value_extend_prefix. (((exists dst_positive_code_extend_prefixentrysource dst_positive_scale_extend_prefixentrysource dst_negative_code_extend_prefixentrysource dst_negative_scale_extend_prefixentrysource dst_positive_extend_prefixentrysource dst_negative_extend_prefixentrysource. (((F) = (((((dst_positive_code_extend_prefixentrysource) + (dst_positive_scale_extend_prefixentrysource)) * S ((dst_positive_code_extend_prefixentrysource) + (dst_positive_scale_extend_prefixentrysource)) + ((dst_positive_scale_extend_prefixentrysource) + (dst_positive_scale_extend_prefixentrysource))) + (((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) * S ((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) + ((dst_negative_scale_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)))) * S ((((dst_positive_code_extend_prefixentrysource) + (dst_positive_scale_extend_prefixentrysource)) * S ((dst_positive_code_extend_prefixentrysource) + (dst_positive_scale_extend_prefixentrysource)) + ((dst_positive_scale_extend_prefixentrysource) + (dst_positive_scale_extend_prefixentrysource))) + (((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) * S ((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) + ((dst_negative_scale_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)))) + ((((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) * S ((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) + ((dst_negative_scale_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource))) + (((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) * S ((dst_negative_code_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)) + ((dst_negative_scale_extend_prefixentrysource) + (dst_negative_scale_extend_prefixentrysource)))))) /\ (((((exists ff_h_pvs_extend_prefixentrysourcepositive. ff_h_pvs_extend_prefixentrysourcepositive + S (dst_positive_extend_prefixentrysource) = S ((S (((o) + ((s) * (srs_index_extend_prefix))))) * dst_positive_scale_extend_prefixentrysource)) /\ exists ff_q_pvs_extend_prefixentrysourcepositive. dst_positive_code_extend_prefixentrysource = ff_q_pvs_extend_prefixentrysourcepositive * S ((S (((o) + ((s) * (srs_index_extend_prefix))))) * dst_positive_scale_extend_prefixentrysource) + (dst_positive_extend_prefixentrysource))) /\ (((((exists ff_h_pvs_extend_prefixentrysourcenegative. ff_h_pvs_extend_prefixentrysourcenegative + S (dst_negative_extend_prefixentrysource) = S ((S (((o) + ((s) * (srs_index_extend_prefix))))) * dst_negative_scale_extend_prefixentrysource)) /\ exists ff_q_pvs_extend_prefixentrysourcenegative. dst_negative_code_extend_prefixentrysource = ff_q_pvs_extend_prefixentrysourcenegative * S ((S (((o) + ((s) * (srs_index_extend_prefix))))) * dst_negative_scale_extend_prefixentrysource) + (dst_negative_extend_prefixentrysource))) /\ (exists ge_balance_positive_extend_prefixentrysourcevalue ge_balance_negative_extend_prefixentrysourcevalue. (((((srs_value_extend_prefix) = 2 * (ge_balance_positive_extend_prefixentrysourcevalue) /\ (ge_balance_negative_extend_prefixentrysourcevalue) = 0) \/ exists ge_signed_half_extend_prefixentrysourcevaluedecode. (((srs_value_extend_prefix) = 2 * ge_signed_half_extend_prefixentrysourcevaluedecode + 1 /\ (ge_balance_positive_extend_prefixentrysourcevalue) = 0) /\ (ge_balance_negative_extend_prefixentrysourcevalue) = S ge_signed_half_extend_prefixentrysourcevaluedecode))) /\ ((dst_positive_extend_prefixentrysource) + ge_balance_negative_extend_prefixentrysourcevalue = (dst_negative_extend_prefixentrysource) + ge_balance_positive_extend_prefixentrysourcevalue))))))))) /\ (exists dst_positive_code_extend_prefixentryoutput dst_positive_scale_extend_prefixentryoutput dst_negative_code_extend_prefixentryoutput dst_negative_scale_extend_prefixentryoutput dst_positive_extend_prefixentryoutput dst_negative_extend_prefixentryoutput. (((G) = (((((dst_positive_code_extend_prefixentryoutput) + (dst_positive_scale_extend_prefixentryoutput)) * S ((dst_positive_code_extend_prefixentryoutput) + (dst_positive_scale_extend_prefixentryoutput)) + ((dst_positive_scale_extend_prefixentryoutput) + (dst_positive_scale_extend_prefixentryoutput))) + (((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) * S ((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) + ((dst_negative_scale_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)))) * S ((((dst_positive_code_extend_prefixentryoutput) + (dst_positive_scale_extend_prefixentryoutput)) * S ((dst_positive_code_extend_prefixentryoutput) + (dst_positive_scale_extend_prefixentryoutput)) + ((dst_positive_scale_extend_prefixentryoutput) + (dst_positive_scale_extend_prefixentryoutput))) + (((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) * S ((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) + ((dst_negative_scale_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)))) + ((((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) * S ((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) + ((dst_negative_scale_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput))) + (((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) * S ((dst_negative_code_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)) + ((dst_negative_scale_extend_prefixentryoutput) + (dst_negative_scale_extend_prefixentryoutput)))))) /\ (((((exists ff_h_pvs_extend_prefixentryoutputpositive. ff_h_pvs_extend_prefixentryoutputpositive + S (dst_positive_extend_prefixentryoutput) = S ((S (srs_index_extend_prefix)) * dst_positive_scale_extend_prefixentryoutput)) /\ exists ff_q_pvs_extend_prefixentryoutputpositive. dst_positive_code_extend_prefixentryoutput = ff_q_pvs_extend_prefixentryoutputpositive * S ((S (srs_index_extend_prefix)) * dst_positive_scale_extend_prefixentryoutput) + (dst_positive_extend_prefixentryoutput))) /\ (((((exists ff_h_pvs_extend_prefixentryoutputnegative. ff_h_pvs_extend_prefixentryoutputnegative + S (dst_negative_extend_prefixentryoutput) = S ((S (srs_index_extend_prefix)) * dst_negative_scale_extend_prefixentryoutput)) /\ exists ff_q_pvs_extend_prefixentryoutputnegative. dst_negative_code_extend_prefixentryoutput = ff_q_pvs_extend_prefixentryoutputnegative * S ((S (srs_index_extend_prefix)) * dst_negative_scale_extend_prefixentryoutput) + (dst_negative_extend_prefixentryoutput))) /\ (exists ge_balance_positive_extend_prefixentryoutputvalue ge_balance_negative_extend_prefixentryoutputvalue. (((((srs_value_extend_prefix) = 2 * (ge_balance_positive_extend_prefixentryoutputvalue) /\ (ge_balance_negative_extend_prefixentryoutputvalue) = 0) \/ exists ge_signed_half_extend_prefixentryoutputvaluedecode. (((srs_value_extend_prefix) = 2 * ge_signed_half_extend_prefixentryoutputvaluedecode + 1 /\ (ge_balance_positive_extend_prefixentryoutputvalue) = 0) /\ (ge_balance_negative_extend_prefixentryoutputvalue) = S ge_signed_half_extend_prefixentryoutputvaluedecode))) /\ ((dst_positive_extend_prefixentryoutput) + ge_balance_negative_extend_prefixentryoutputvalue = (dst_negative_extend_prefixentryoutput) + ge_balance_positive_extend_prefixentryoutputvalue)))))))))))))))) -> (exists dst_positive_code_extend_table dst_positive_scale_extend_table dst_negative_code_extend_table dst_negative_scale_extend_table. (((H) = (((((dst_positive_code_extend_table) + (dst_positive_scale_extend_table)) * S ((dst_positive_code_extend_table) + (dst_positive_scale_extend_table)) + ((dst_positive_scale_extend_table) + (dst_positive_scale_extend_table))) + (((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) * S ((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) + ((dst_negative_scale_extend_table) + (dst_negative_scale_extend_table)))) * S ((((dst_positive_code_extend_table) + (dst_positive_scale_extend_table)) * S ((dst_positive_code_extend_table) + (dst_positive_scale_extend_table)) + ((dst_positive_scale_extend_table) + (dst_positive_scale_extend_table))) + (((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) * S ((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) + ((dst_negative_scale_extend_table) + (dst_negative_scale_extend_table)))) + ((((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) * S ((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) + ((dst_negative_scale_extend_table) + (dst_negative_scale_extend_table))) + (((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) * S ((dst_negative_code_extend_table) + (dst_negative_scale_extend_table)) + ((dst_negative_scale_extend_table) + (dst_negative_scale_extend_table)))))) /\ (forall dst_index_extend_table. (exists pvs_le_gap_extend_tabledomain. pvs_le_gap_extend_tabledomain + (dst_index_extend_table) = (l)) -> exists dst_positive_extend_table dst_negative_extend_table dst_value_extend_table. ((((exists ff_h_pvs_extend_tableentrypositive. ff_h_pvs_extend_tableentrypositive + S (dst_positive_extend_table) = S ((S (dst_index_extend_table)) * dst_positive_scale_extend_table)) /\ exists ff_q_pvs_extend_tableentrypositive. dst_positive_code_extend_table = ff_q_pvs_extend_tableentrypositive * S ((S (dst_index_extend_table)) * dst_positive_scale_extend_table) + (dst_positive_extend_table))) /\ (((((exists ff_h_pvs_extend_tableentrynegative. ff_h_pvs_extend_tableentrynegative + S (dst_negative_extend_table) = S ((S (dst_index_extend_table)) * dst_negative_scale_extend_table)) /\ exists ff_q_pvs_extend_tableentrynegative. dst_negative_code_extend_table = ff_q_pvs_extend_tableentrynegative * S ((S (dst_index_extend_table)) * dst_negative_scale_extend_table) + (dst_negative_extend_table))) /\ (exists ge_balance_positive_extend_tableentryvalue ge_balance_negative_extend_tableentryvalue. (((((dst_value_extend_table) = 2 * (ge_balance_positive_extend_tableentryvalue) /\ (ge_balance_negative_extend_tableentryvalue) = 0) \/ exists ge_signed_half_extend_tableentryvaluedecode. (((dst_value_extend_table) = 2 * ge_signed_half_extend_tableentryvaluedecode + 1 /\ (ge_balance_positive_extend_tableentryvalue) = 0) /\ (ge_balance_negative_extend_tableentryvalue) = S ge_signed_half_extend_tableentryvaluedecode))) /\ ((dst_positive_extend_table) + ge_balance_negative_extend_tableentryvalue = (dst_negative_extend_table) + ge_balance_positive_extend_tableentryvalue))))))))) -> (forall dst_index_extend_preserved dst_first_extend_preserved dst_second_extend_preserved. (exists pvs_gap_extend_preservedbound. pvs_gap_extend_preservedbound + S (dst_index_extend_preserved) = (l)) -> (exists dst_positive_code_extend_preservedfirst dst_positive_scale_extend_preservedfirst dst_negative_code_extend_preservedfirst dst_negative_scale_extend_preservedfirst dst_positive_extend_preservedfirst dst_negative_extend_preservedfirst. (((G) = (((((dst_positive_code_extend_preservedfirst) + (dst_positive_scale_extend_preservedfirst)) * S ((dst_positive_code_extend_preservedfirst) + (dst_positive_scale_extend_preservedfirst)) + ((dst_positive_scale_extend_preservedfirst) + (dst_positive_scale_extend_preservedfirst))) + (((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) * S ((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) + ((dst_negative_scale_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)))) * S ((((dst_positive_code_extend_preservedfirst) + (dst_positive_scale_extend_preservedfirst)) * S ((dst_positive_code_extend_preservedfirst) + (dst_positive_scale_extend_preservedfirst)) + ((dst_positive_scale_extend_preservedfirst) + (dst_positive_scale_extend_preservedfirst))) + (((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) * S ((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) + ((dst_negative_scale_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)))) + ((((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) * S ((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) + ((dst_negative_scale_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst))) + (((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) * S ((dst_negative_code_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)) + ((dst_negative_scale_extend_preservedfirst) + (dst_negative_scale_extend_preservedfirst)))))) /\ (((((exists ff_h_pvs_extend_preservedfirstpositive. ff_h_pvs_extend_preservedfirstpositive + S (dst_positive_extend_preservedfirst) = S ((S (dst_index_extend_preserved)) * dst_positive_scale_extend_preservedfirst)) /\ exists ff_q_pvs_extend_preservedfirstpositive. dst_positive_code_extend_preservedfirst = ff_q_pvs_extend_preservedfirstpositive * S ((S (dst_index_extend_preserved)) * dst_positive_scale_extend_preservedfirst) + (dst_positive_extend_preservedfirst))) /\ (((((exists ff_h_pvs_extend_preservedfirstnegative. ff_h_pvs_extend_preservedfirstnegative + S (dst_negative_extend_preservedfirst) = S ((S (dst_index_extend_preserved)) * dst_negative_scale_extend_preservedfirst)) /\ exists ff_q_pvs_extend_preservedfirstnegative. dst_negative_code_extend_preservedfirst = ff_q_pvs_extend_preservedfirstnegative * S ((S (dst_index_extend_preserved)) * dst_negative_scale_extend_preservedfirst) + (dst_negative_extend_preservedfirst))) /\ (exists ge_balance_positive_extend_preservedfirstvalue ge_balance_negative_extend_preservedfirstvalue. (((((dst_first_extend_preserved) = 2 * (ge_balance_positive_extend_preservedfirstvalue) /\ (ge_balance_negative_extend_preservedfirstvalue) = 0) \/ exists ge_signed_half_extend_preservedfirstvaluedecode. (((dst_first_extend_preserved) = 2 * ge_signed_half_extend_preservedfirstvaluedecode + 1 /\ (ge_balance_positive_extend_preservedfirstvalue) = 0) /\ (ge_balance_negative_extend_preservedfirstvalue) = S ge_signed_half_extend_preservedfirstvaluedecode))) /\ ((dst_positive_extend_preservedfirst) + ge_balance_negative_extend_preservedfirstvalue = (dst_negative_extend_preservedfirst) + ge_balance_positive_extend_preservedfirstvalue))))))))) -> (exists dst_positive_code_extend_preservedsecond dst_positive_scale_extend_preservedsecond dst_negative_code_extend_preservedsecond dst_negative_scale_extend_preservedsecond dst_positive_extend_preservedsecond dst_negative_extend_preservedsecond. (((H) = (((((dst_positive_code_extend_preservedsecond) + (dst_positive_scale_extend_preservedsecond)) * S ((dst_positive_code_extend_preservedsecond) + (dst_positive_scale_extend_preservedsecond)) + ((dst_positive_scale_extend_preservedsecond) + (dst_positive_scale_extend_preservedsecond))) + (((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) * S ((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) + ((dst_negative_scale_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)))) * S ((((dst_positive_code_extend_preservedsecond) + (dst_positive_scale_extend_preservedsecond)) * S ((dst_positive_code_extend_preservedsecond) + (dst_positive_scale_extend_preservedsecond)) + ((dst_positive_scale_extend_preservedsecond) + (dst_positive_scale_extend_preservedsecond))) + (((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) * S ((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) + ((dst_negative_scale_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)))) + ((((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) * S ((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) + ((dst_negative_scale_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond))) + (((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) * S ((dst_negative_code_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)) + ((dst_negative_scale_extend_preservedsecond) + (dst_negative_scale_extend_preservedsecond)))))) /\ (((((exists ff_h_pvs_extend_preservedsecondpositive. ff_h_pvs_extend_preservedsecondpositive + S (dst_positive_extend_preservedsecond) = S ((S (dst_index_extend_preserved)) * dst_positive_scale_extend_preservedsecond)) /\ exists ff_q_pvs_extend_preservedsecondpositive. dst_positive_code_extend_preservedsecond = ff_q_pvs_extend_preservedsecondpositive * S ((S (dst_index_extend_preserved)) * dst_positive_scale_extend_preservedsecond) + (dst_positive_extend_preservedsecond))) /\ (((((exists ff_h_pvs_extend_preservedsecondnegative. ff_h_pvs_extend_preservedsecondnegative + S (dst_negative_extend_preservedsecond) = S ((S (dst_index_extend_preserved)) * dst_negative_scale_extend_preservedsecond)) /\ exists ff_q_pvs_extend_preservedsecondnegative. dst_negative_code_extend_preservedsecond = ff_q_pvs_extend_preservedsecondnegative * S ((S (dst_index_extend_preserved)) * dst_negative_scale_extend_preservedsecond) + (dst_negative_extend_preservedsecond))) /\ (exists ge_balance_positive_extend_preservedsecondvalue ge_balance_negative_extend_preservedsecondvalue. (((((dst_second_extend_preserved) = 2 * (ge_balance_positive_extend_preservedsecondvalue) /\ (ge_balance_negative_extend_preservedsecondvalue) = 0) \/ exists ge_signed_half_extend_preservedsecondvaluedecode. (((dst_second_extend_preserved) = 2 * ge_signed_half_extend_preservedsecondvaluedecode + 1 /\ (ge_balance_positive_extend_preservedsecondvalue) = 0) /\ (ge_balance_negative_extend_preservedsecondvalue) = S ge_signed_half_extend_preservedsecondvaluedecode))) /\ ((dst_positive_extend_preservedsecond) + ge_balance_negative_extend_preservedsecondvalue = (dst_negative_extend_preservedsecond) + ge_balance_positive_extend_preservedsecondvalue))))))))) -> dst_first_extend_preserved = dst_second_extend_preserved) -> (exists dst_positive_code_extend_source dst_positive_scale_extend_source dst_negative_code_extend_source dst_negative_scale_extend_source dst_positive_extend_source dst_negative_extend_source. (((F) = (((((dst_positive_code_extend_source) + (dst_positive_scale_extend_source)) * S ((dst_positive_code_extend_source) + (dst_positive_scale_extend_source)) + ((dst_positive_scale_extend_source) + (dst_positive_scale_extend_source))) + (((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) * S ((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) + ((dst_negative_scale_extend_source) + (dst_negative_scale_extend_source)))) * S ((((dst_positive_code_extend_source) + (dst_positive_scale_extend_source)) * S ((dst_positive_code_extend_source) + (dst_positive_scale_extend_source)) + ((dst_positive_scale_extend_source) + (dst_positive_scale_extend_source))) + (((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) * S ((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) + ((dst_negative_scale_extend_source) + (dst_negative_scale_extend_source)))) + ((((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) * S ((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) + ((dst_negative_scale_extend_source) + (dst_negative_scale_extend_source))) + (((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) * S ((dst_negative_code_extend_source) + (dst_negative_scale_extend_source)) + ((dst_negative_scale_extend_source) + (dst_negative_scale_extend_source)))))) /\ (((((exists ff_h_pvs_extend_sourcepositive. ff_h_pvs_extend_sourcepositive + S (dst_positive_extend_source) = S ((S (((o) + ((s) * (l))))) * dst_positive_scale_extend_source)) /\ exists ff_q_pvs_extend_sourcepositive. dst_positive_code_extend_source = ff_q_pvs_extend_sourcepositive * S ((S (((o) + ((s) * (l))))) * dst_positive_scale_extend_source) + (dst_positive_extend_source))) /\ (((((exists ff_h_pvs_extend_sourcenegative. ff_h_pvs_extend_sourcenegative + S (dst_negative_extend_source) = S ((S (((o) + ((s) * (l))))) * dst_negative_scale_extend_source)) /\ exists ff_q_pvs_extend_sourcenegative. dst_negative_code_extend_source = ff_q_pvs_extend_sourcenegative * S ((S (((o) + ((s) * (l))))) * dst_negative_scale_extend_source) + (dst_negative_extend_source))) /\ (exists ge_balance_positive_extend_sourcevalue ge_balance_negative_extend_sourcevalue. (((((a) = 2 * (ge_balance_positive_extend_sourcevalue) /\ (ge_balance_negative_extend_sourcevalue) = 0) \/ exists ge_signed_half_extend_sourcevaluedecode. (((a) = 2 * ge_signed_half_extend_sourcevaluedecode + 1 /\ (ge_balance_positive_extend_sourcevalue) = 0) /\ (ge_balance_negative_extend_sourcevalue) = S ge_signed_half_extend_sourcevaluedecode))) /\ ((dst_positive_extend_source) + ge_balance_negative_extend_sourcevalue = (dst_negative_extend_source) + ge_balance_positive_extend_sourcevalue))))))))) -> (exists dst_positive_code_extend_value dst_positive_scale_extend_value dst_negative_code_extend_value dst_negative_scale_extend_value dst_positive_extend_value dst_negative_extend_value. (((H) = (((((dst_positive_code_extend_value) + (dst_positive_scale_extend_value)) * S ((dst_positive_code_extend_value) + (dst_positive_scale_extend_value)) + ((dst_positive_scale_extend_value) + (dst_positive_scale_extend_value))) + (((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) * S ((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) + ((dst_negative_scale_extend_value) + (dst_negative_scale_extend_value)))) * S ((((dst_positive_code_extend_value) + (dst_positive_scale_extend_value)) * S ((dst_positive_code_extend_value) + (dst_positive_scale_extend_value)) + ((dst_positive_scale_extend_value) + (dst_positive_scale_extend_value))) + (((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) * S ((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) + ((dst_negative_scale_extend_value) + (dst_negative_scale_extend_value)))) + ((((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) * S ((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) + ((dst_negative_scale_extend_value) + (dst_negative_scale_extend_value))) + (((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) * S ((dst_negative_code_extend_value) + (dst_negative_scale_extend_value)) + ((dst_negative_scale_extend_value) + (dst_negative_scale_extend_value)))))) /\ (((((exists ff_h_pvs_extend_valuepositive. ff_h_pvs_extend_valuepositive + S (dst_positive_extend_value) = S ((S (l)) * dst_positive_scale_extend_value)) /\ exists ff_q_pvs_extend_valuepositive. dst_positive_code_extend_value = ff_q_pvs_extend_valuepositive * S ((S (l)) * dst_positive_scale_extend_value) + (dst_positive_extend_value))) /\ (((((exists ff_h_pvs_extend_valuenegative. ff_h_pvs_extend_valuenegative + S (dst_negative_extend_value) = S ((S (l)) * dst_negative_scale_extend_value)) /\ exists ff_q_pvs_extend_valuenegative. dst_negative_code_extend_value = ff_q_pvs_extend_valuenegative * S ((S (l)) * dst_negative_scale_extend_value) + (dst_negative_extend_value))) /\ (exists ge_balance_positive_extend_valuevalue ge_balance_negative_extend_valuevalue. (((((a) = 2 * (ge_balance_positive_extend_valuevalue) /\ (ge_balance_negative_extend_valuevalue) = 0) \/ exists ge_signed_half_extend_valuevaluedecode. (((a) = 2 * ge_signed_half_extend_valuevaluedecode + 1 /\ (ge_balance_positive_extend_valuevalue) = 0) /\ (ge_balance_negative_extend_valuevalue) = S ge_signed_half_extend_valuevaluedecode))) /\ ((dst_positive_extend_value) + ge_balance_negative_extend_valuevalue = (dst_negative_extend_value) + ge_balance_positive_extend_valuevalue))))))))) -> (((exists dst_positive_code_extend_resultsource_table dst_positive_scale_extend_resultsource_table dst_negative_code_extend_resultsource_table dst_negative_scale_extend_resultsource_table. (((F) = (((((dst_positive_code_extend_resultsource_table) + (dst_positive_scale_extend_resultsource_table)) * S ((dst_positive_code_extend_resultsource_table) + (dst_positive_scale_extend_resultsource_table)) + ((dst_positive_scale_extend_resultsource_table) + (dst_positive_scale_extend_resultsource_table))) + (((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) * S ((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) + ((dst_negative_scale_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)))) * S ((((dst_positive_code_extend_resultsource_table) + (dst_positive_scale_extend_resultsource_table)) * S ((dst_positive_code_extend_resultsource_table) + (dst_positive_scale_extend_resultsource_table)) + ((dst_positive_scale_extend_resultsource_table) + (dst_positive_scale_extend_resultsource_table))) + (((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) * S ((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) + ((dst_negative_scale_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)))) + ((((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) * S ((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) + ((dst_negative_scale_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table))) + (((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) * S ((dst_negative_code_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)) + ((dst_negative_scale_extend_resultsource_table) + (dst_negative_scale_extend_resultsource_table)))))) /\ (forall dst_index_extend_resultsource_table. (exists pvs_le_gap_extend_resultsource_tabledomain. pvs_le_gap_extend_resultsource_tabledomain + (dst_index_extend_resultsource_table) = (0)) -> exists dst_positive_extend_resultsource_table dst_negative_extend_resultsource_table dst_value_extend_resultsource_table. ((((exists ff_h_pvs_extend_resultsource_tableentrypositive. ff_h_pvs_extend_resultsource_tableentrypositive + S (dst_positive_extend_resultsource_table) = S ((S (dst_index_extend_resultsource_table)) * dst_positive_scale_extend_resultsource_table)) /\ exists ff_q_pvs_extend_resultsource_tableentrypositive. dst_positive_code_extend_resultsource_table = ff_q_pvs_extend_resultsource_tableentrypositive * S ((S (dst_index_extend_resultsource_table)) * dst_positive_scale_extend_resultsource_table) + (dst_positive_extend_resultsource_table))) /\ (((((exists ff_h_pvs_extend_resultsource_tableentrynegative. ff_h_pvs_extend_resultsource_tableentrynegative + S (dst_negative_extend_resultsource_table) = S ((S (dst_index_extend_resultsource_table)) * dst_negative_scale_extend_resultsource_table)) /\ exists ff_q_pvs_extend_resultsource_tableentrynegative. dst_negative_code_extend_resultsource_table = ff_q_pvs_extend_resultsource_tableentrynegative * S ((S (dst_index_extend_resultsource_table)) * dst_negative_scale_extend_resultsource_table) + (dst_negative_extend_resultsource_table))) /\ (exists ge_balance_positive_extend_resultsource_tableentryvalue ge_balance_negative_extend_resultsource_tableentryvalue. (((((dst_value_extend_resultsource_table) = 2 * (ge_balance_positive_extend_resultsource_tableentryvalue) /\ (ge_balance_negative_extend_resultsource_tableentryvalue) = 0) \/ exists ge_signed_half_extend_resultsource_tableentryvaluedecode. (((dst_value_extend_resultsource_table) = 2 * ge_signed_half_extend_resultsource_tableentryvaluedecode + 1 /\ (ge_balance_positive_extend_resultsource_tableentryvalue) = 0) /\ (ge_balance_negative_extend_resultsource_tableentryvalue) = S ge_signed_half_extend_resultsource_tableentryvaluedecode))) /\ ((dst_positive_extend_resultsource_table) + ge_balance_negative_extend_resultsource_tableentryvalue = (dst_negative_extend_resultsource_table) + ge_balance_positive_extend_resultsource_tableentryvalue))))))))) /\ (((exists dst_positive_code_extend_resultoutput_table dst_positive_scale_extend_resultoutput_table dst_negative_code_extend_resultoutput_table dst_negative_scale_extend_resultoutput_table. (((H) = (((((dst_positive_code_extend_resultoutput_table) + (dst_positive_scale_extend_resultoutput_table)) * S ((dst_positive_code_extend_resultoutput_table) + (dst_positive_scale_extend_resultoutput_table)) + ((dst_positive_scale_extend_resultoutput_table) + (dst_positive_scale_extend_resultoutput_table))) + (((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) * S ((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) + ((dst_negative_scale_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)))) * S ((((dst_positive_code_extend_resultoutput_table) + (dst_positive_scale_extend_resultoutput_table)) * S ((dst_positive_code_extend_resultoutput_table) + (dst_positive_scale_extend_resultoutput_table)) + ((dst_positive_scale_extend_resultoutput_table) + (dst_positive_scale_extend_resultoutput_table))) + (((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) * S ((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) + ((dst_negative_scale_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)))) + ((((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) * S ((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) + ((dst_negative_scale_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table))) + (((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) * S ((dst_negative_code_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)) + ((dst_negative_scale_extend_resultoutput_table) + (dst_negative_scale_extend_resultoutput_table)))))) /\ (forall dst_index_extend_resultoutput_table. (exists pvs_le_gap_extend_resultoutput_tabledomain. pvs_le_gap_extend_resultoutput_tabledomain + (dst_index_extend_resultoutput_table) = (S l)) -> exists dst_positive_extend_resultoutput_table dst_negative_extend_resultoutput_table dst_value_extend_resultoutput_table. ((((exists ff_h_pvs_extend_resultoutput_tableentrypositive. ff_h_pvs_extend_resultoutput_tableentrypositive + S (dst_positive_extend_resultoutput_table) = S ((S (dst_index_extend_resultoutput_table)) * dst_positive_scale_extend_resultoutput_table)) /\ exists ff_q_pvs_extend_resultoutput_tableentrypositive. dst_positive_code_extend_resultoutput_table = ff_q_pvs_extend_resultoutput_tableentrypositive * S ((S (dst_index_extend_resultoutput_table)) * dst_positive_scale_extend_resultoutput_table) + (dst_positive_extend_resultoutput_table))) /\ (((((exists ff_h_pvs_extend_resultoutput_tableentrynegative. ff_h_pvs_extend_resultoutput_tableentrynegative + S (dst_negative_extend_resultoutput_table) = S ((S (dst_index_extend_resultoutput_table)) * dst_negative_scale_extend_resultoutput_table)) /\ exists ff_q_pvs_extend_resultoutput_tableentrynegative. dst_negative_code_extend_resultoutput_table = ff_q_pvs_extend_resultoutput_tableentrynegative * S ((S (dst_index_extend_resultoutput_table)) * dst_negative_scale_extend_resultoutput_table) + (dst_negative_extend_resultoutput_table))) /\ (exists ge_balance_positive_extend_resultoutput_tableentryvalue ge_balance_negative_extend_resultoutput_tableentryvalue. (((((dst_value_extend_resultoutput_table) = 2 * (ge_balance_positive_extend_resultoutput_tableentryvalue) /\ (ge_balance_negative_extend_resultoutput_tableentryvalue) = 0) \/ exists ge_signed_half_extend_resultoutput_tableentryvaluedecode. (((dst_value_extend_resultoutput_table) = 2 * ge_signed_half_extend_resultoutput_tableentryvaluedecode + 1 /\ (ge_balance_positive_extend_resultoutput_tableentryvalue) = 0) /\ (ge_balance_negative_extend_resultoutput_tableentryvalue) = S ge_signed_half_extend_resultoutput_tableentryvaluedecode))) /\ ((dst_positive_extend_resultoutput_table) + ge_balance_negative_extend_resultoutput_tableentryvalue = (dst_negative_extend_resultoutput_table) + ge_balance_positive_extend_resultoutput_tableentryvalue))))))))) /\ (forall srs_index_extend_result. (exists pvs_gap_extend_resultbound. pvs_gap_extend_resultbound + S (srs_index_extend_result) = (S l)) -> exists srs_value_extend_result. (((exists dst_positive_code_extend_resultentrysource dst_positive_scale_extend_resultentrysource dst_negative_code_extend_resultentrysource dst_negative_scale_extend_resultentrysource dst_positive_extend_resultentrysource dst_negative_extend_resultentrysource. (((F) = (((((dst_positive_code_extend_resultentrysource) + (dst_positive_scale_extend_resultentrysource)) * S ((dst_positive_code_extend_resultentrysource) + (dst_positive_scale_extend_resultentrysource)) + ((dst_positive_scale_extend_resultentrysource) + (dst_positive_scale_extend_resultentrysource))) + (((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) * S ((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) + ((dst_negative_scale_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)))) * S ((((dst_positive_code_extend_resultentrysource) + (dst_positive_scale_extend_resultentrysource)) * S ((dst_positive_code_extend_resultentrysource) + (dst_positive_scale_extend_resultentrysource)) + ((dst_positive_scale_extend_resultentrysource) + (dst_positive_scale_extend_resultentrysource))) + (((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) * S ((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) + ((dst_negative_scale_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)))) + ((((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) * S ((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) + ((dst_negative_scale_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource))) + (((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) * S ((dst_negative_code_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)) + ((dst_negative_scale_extend_resultentrysource) + (dst_negative_scale_extend_resultentrysource)))))) /\ (((((exists ff_h_pvs_extend_resultentrysourcepositive. ff_h_pvs_extend_resultentrysourcepositive + S (dst_positive_extend_resultentrysource) = S ((S (((o) + ((s) * (srs_index_extend_result))))) * dst_positive_scale_extend_resultentrysource)) /\ exists ff_q_pvs_extend_resultentrysourcepositive. dst_positive_code_extend_resultentrysource = ff_q_pvs_extend_resultentrysourcepositive * S ((S (((o) + ((s) * (srs_index_extend_result))))) * dst_positive_scale_extend_resultentrysource) + (dst_positive_extend_resultentrysource))) /\ (((((exists ff_h_pvs_extend_resultentrysourcenegative. ff_h_pvs_extend_resultentrysourcenegative + S (dst_negative_extend_resultentrysource) = S ((S (((o) + ((s) * (srs_index_extend_result))))) * dst_negative_scale_extend_resultentrysource)) /\ exists ff_q_pvs_extend_resultentrysourcenegative. dst_negative_code_extend_resultentrysource = ff_q_pvs_extend_resultentrysourcenegative * S ((S (((o) + ((s) * (srs_index_extend_result))))) * dst_negative_scale_extend_resultentrysource) + (dst_negative_extend_resultentrysource))) /\ (exists ge_balance_positive_extend_resultentrysourcevalue ge_balance_negative_extend_resultentrysourcevalue. (((((srs_value_extend_result) = 2 * (ge_balance_positive_extend_resultentrysourcevalue) /\ (ge_balance_negative_extend_resultentrysourcevalue) = 0) \/ exists ge_signed_half_extend_resultentrysourcevaluedecode. (((srs_value_extend_result) = 2 * ge_signed_half_extend_resultentrysourcevaluedecode + 1 /\ (ge_balance_positive_extend_resultentrysourcevalue) = 0) /\ (ge_balance_negative_extend_resultentrysourcevalue) = S ge_signed_half_extend_resultentrysourcevaluedecode))) /\ ((dst_positive_extend_resultentrysource) + ge_balance_negative_extend_resultentrysourcevalue = (dst_negative_extend_resultentrysource) + ge_balance_positive_extend_resultentrysourcevalue))))))))) /\ (exists dst_positive_code_extend_resultentryoutput dst_positive_scale_extend_resultentryoutput dst_negative_code_extend_resultentryoutput dst_negative_scale_extend_resultentryoutput dst_positive_extend_resultentryoutput dst_negative_extend_resultentryoutput. (((H) = (((((dst_positive_code_extend_resultentryoutput) + (dst_positive_scale_extend_resultentryoutput)) * S ((dst_positive_code_extend_resultentryoutput) + (dst_positive_scale_extend_resultentryoutput)) + ((dst_positive_scale_extend_resultentryoutput) + (dst_positive_scale_extend_resultentryoutput))) + (((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) * S ((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) + ((dst_negative_scale_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)))) * S ((((dst_positive_code_extend_resultentryoutput) + (dst_positive_scale_extend_resultentryoutput)) * S ((dst_positive_code_extend_resultentryoutput) + (dst_positive_scale_extend_resultentryoutput)) + ((dst_positive_scale_extend_resultentryoutput) + (dst_positive_scale_extend_resultentryoutput))) + (((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) * S ((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) + ((dst_negative_scale_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)))) + ((((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) * S ((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) + ((dst_negative_scale_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput))) + (((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) * S ((dst_negative_code_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)) + ((dst_negative_scale_extend_resultentryoutput) + (dst_negative_scale_extend_resultentryoutput)))))) /\ (((((exists ff_h_pvs_extend_resultentryoutputpositive. ff_h_pvs_extend_resultentryoutputpositive + S (dst_positive_extend_resultentryoutput) = S ((S (srs_index_extend_result)) * dst_positive_scale_extend_resultentryoutput)) /\ exists ff_q_pvs_extend_resultentryoutputpositive. dst_positive_code_extend_resultentryoutput = ff_q_pvs_extend_resultentryoutputpositive * S ((S (srs_index_extend_result)) * dst_positive_scale_extend_resultentryoutput) + (dst_positive_extend_resultentryoutput))) /\ (((((exists ff_h_pvs_extend_resultentryoutputnegative. ff_h_pvs_extend_resultentryoutputnegative + S (dst_negative_extend_resultentryoutput) = S ((S (srs_index_extend_result)) * dst_negative_scale_extend_resultentryoutput)) /\ exists ff_q_pvs_extend_resultentryoutputnegative. dst_negative_code_extend_resultentryoutput = ff_q_pvs_extend_resultentryoutputnegative * S ((S (srs_index_extend_result)) * dst_negative_scale_extend_resultentryoutput) + (dst_negative_extend_resultentryoutput))) /\ (exists ge_balance_positive_extend_resultentryoutputvalue ge_balance_negative_extend_resultentryoutputvalue. (((((srs_value_extend_result) = 2 * (ge_balance_positive_extend_resultentryoutputvalue) /\ (ge_balance_negative_extend_resultentryoutputvalue) = 0) \/ exists ge_signed_half_extend_resultentryoutputvaluedecode. (((srs_value_extend_result) = 2 * ge_signed_half_extend_resultentryoutputvaluedecode + 1 /\ (ge_balance_positive_extend_resultentryoutputvalue) = 0) /\ (ge_balance_negative_extend_resultentryoutputvalue) = S ge_signed_half_extend_resultentryoutputvaluedecode))) /\ ((dst_positive_extend_resultentryoutput) + ge_balance_negative_extend_resultentryoutputvalue = (dst_negative_extend_resultentryoutput) + ge_balance_positive_extend_resultentryoutputvalue))))))))))))))))Complete tactic proof in conservative notation
All 68 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
68 script commands · 19 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 assumptionsL1–10
02Fix variables and assumptionsL11–12
03Separate the logical casesL13–15
04Use earlier factsL16–16
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L16
exact hs_left
05Separate the logical casesL17–17
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L17
split
06Use earlier factsL18–22
07Fix variables and assumptionsL23–24
08Establish hcaseL25–29
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite lt succ eq or lt.
09Separate the logical casesL30–30
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L30
cases hcase
10Calculate and transport equalitiesL31–38
11Construct an explicit witnessL39–39
Supply the displayed value, then prove that it has the required property.
- L39
exists a
12Separate the logical casesL40–40
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L40
split
13Use earlier factsL41–42
14Establish holdL43–46
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hs right right.
- L43
have hold : ∃ z. ArithAt(F,o + s · i,z) ∧ ArithAt(G,i,z)Definitions: ArithAt(F,o + s · i,z)ArithAt(G,i,z)Original native command in the exact edition - L44
specialize hs_right_right (i) - L45
apply hs_right_right - L46
exact hcase_right
15Separate the logical casesL47–48
16Construct an explicit witnessL49–49
Supply the displayed value, then prove that it has the required property.
- L49
exists x
17Separate the logical casesL50–50
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L50
split
18Use earlier factsL51–60
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L51
exact hold_witness_left - L52
specialize arithmetic_signed_table_equal_entry_transport (i) - L53
specialize arithmetic_signed_table_equal_entry_transport (G) - L54
specialize arithmetic_signed_table_equal_entry_transport (H) - L55
specialize arithmetic_signed_table_equal_entry_transport (l) - L56
specialize arithmetic_signed_table_equal_entry_transport (i) - L57
specialize arithmetic_signed_table_equal_entry_transport (x) - L58
apply arithmetic_signed_table_equal_entry_transport - L59
specialize signed_table_domain_resize (l) - L60
specialize signed_table_domain_resize (i)
19Use earlier factsL61–68
Original defined command ledger · 68 lines
- 0001
intro F - 0002
intro G - 0003
intro H - 0004
intro o - 0005
intro s - 0006
intro l - 0007
intro a - 0008
intro hs - 0009
intro hH - 0010
intro hequal - 0011
intro hsource - 0012
intro hentry - 0013
cases hs - 0014
cases hs_right - 0015
split - 0016
exact hs_left - 0017
split - 0018
specialize signed_table_domain_resize (l) - 0019
specialize signed_table_domain_resize (S l) - 0020
specialize signed_table_domain_resize (H) - 0021
apply signed_table_domain_resize - 0022
exact hH - 0023
intro i - 0024
intro hi - 0025
have hcase : i = l ∨ Lt(i,l) - 0026
specialize finite_lt_succ_eq_or_lt (l) - 0027
specialize finite_lt_succ_eq_or_lt (i) - 0028
apply finite_lt_succ_eq_or_lt - 0029
exact hi - 0030
cases hcase - 0031
rewrite hcase_left - 0032
rewrite hcase_left - 0033
rewrite hcase_left - 0034
rewrite hcase_left - 0035
rewrite hcase_left - 0036
rewrite hcase_left - 0037
rewrite hcase_left - 0038
rewrite hcase_left - 0039
exists a - 0040
split - 0041
exact hsource - 0042
exact hentry - 0043
have hold : ∃ z. ArithAt(F,o + s · i,z) ∧ ArithAt(G,i,z) - 0044
specialize hs_right_right (i) - 0045
apply hs_right_right - 0046
exact hcase_right - 0047
cases hold - 0048
cases hold_witness - 0049
exists x - 0050
split - 0051
exact hold_witness_left - 0052
specialize arithmetic_signed_table_equal_entry_transport (i) - 0053
specialize arithmetic_signed_table_equal_entry_transport (G) - 0054
specialize arithmetic_signed_table_equal_entry_transport (H) - 0055
specialize arithmetic_signed_table_equal_entry_transport (l) - 0056
specialize arithmetic_signed_table_equal_entry_transport (i) - 0057
specialize arithmetic_signed_table_equal_entry_transport (x) - 0058
apply arithmetic_signed_table_equal_entry_transport - 0059
specialize signed_table_domain_resize (l) - 0060
specialize signed_table_domain_resize (i) - 0061
specialize signed_table_domain_resize (H) - 0062
apply signed_table_domain_resize - 0063
exact hH - 0064
exact hequal - 0065
specialize le_refl (i) - 0066
apply le_refl - 0067
exact hcase_right - 0068
exact hold_witness_right