Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Definition in prerequisite notation
ArithTable(l,T) ∧ (∀ x. ∀ y. Le(x,l) → ArithAt(T,x,y) → SignedIncidenceFlatEntry(A,r,s,M,x,y))
Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
((exists dst_positive_code_g009_definitiontable dst_positive_scale_g009_definitiontable dst_negative_code_g009_definitiontable dst_negative_scale_g009_definitiontable. ((((T)) = (((((dst_positive_code_g009_definitiontable) + (dst_positive_scale_g009_definitiontable)) * S ((dst_positive_code_g009_definitiontable) + (dst_positive_scale_g009_definitiontable)) + ((dst_positive_scale_g009_definitiontable) + (dst_positive_scale_g009_definitiontable))) + (((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) * S ((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) + ((dst_negative_scale_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)))) * S ((((dst_positive_code_g009_definitiontable) + (dst_positive_scale_g009_definitiontable)) * S ((dst_positive_code_g009_definitiontable) + (dst_positive_scale_g009_definitiontable)) + ((dst_positive_scale_g009_definitiontable) + (dst_positive_scale_g009_definitiontable))) + (((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) * S ((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) + ((dst_negative_scale_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)))) + ((((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) * S ((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) + ((dst_negative_scale_g009_definitiontable) + (dst_negative_scale_g009_definitiontable))) + (((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) * S ((dst_negative_code_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)) + ((dst_negative_scale_g009_definitiontable) + (dst_negative_scale_g009_definitiontable)))))) /\ (forall dst_index_g009_definitiontable. (exists pvs_le_gap_g009_definitiontabledomain. pvs_le_gap_g009_definitiontabledomain + (dst_index_g009_definitiontable) = ((l))) -> exists dst_positive_g009_definitiontable dst_negative_g009_definitiontable dst_value_g009_definitiontable. ((((exists ff_h_pvs_g009_definitiontableentrypositive. ff_h_pvs_g009_definitiontableentrypositive + S (dst_positive_g009_definitiontable) = S ((S (dst_index_g009_definitiontable)) * dst_positive_scale_g009_definitiontable)) /\ exists ff_q_pvs_g009_definitiontableentrypositive. dst_positive_code_g009_definitiontable = ff_q_pvs_g009_definitiontableentrypositive * S ((S (dst_index_g009_definitiontable)) * dst_positive_scale_g009_definitiontable) + (dst_positive_g009_definitiontable))) /\ (((((exists ff_h_pvs_g009_definitiontableentrynegative. ff_h_pvs_g009_definitiontableentrynegative + S (dst_negative_g009_definitiontable) = S ((S (dst_index_g009_definitiontable)) * dst_negative_scale_g009_definitiontable)) /\ exists ff_q_pvs_g009_definitiontableentrynegative. dst_negative_code_g009_definitiontable = ff_q_pvs_g009_definitiontableentrynegative * S ((S (dst_index_g009_definitiontable)) * dst_negative_scale_g009_definitiontable) + (dst_negative_g009_definitiontable))) /\ (exists ge_balance_positive_g009_definitiontableentryvalue ge_balance_negative_g009_definitiontableentryvalue. (((((dst_value_g009_definitiontable) = 2 * (ge_balance_positive_g009_definitiontableentryvalue) /\ (ge_balance_negative_g009_definitiontableentryvalue) = 0) \/ exists ge_signed_half_g009_definitiontableentryvaluedecode. (((dst_value_g009_definitiontable) = 2 * ge_signed_half_g009_definitiontableentryvaluedecode + 1 /\ (ge_balance_positive_g009_definitiontableentryvalue) = 0) /\ (ge_balance_negative_g009_definitiontableentryvalue) = S ge_signed_half_g009_definitiontableentryvaluedecode))) /\ ((dst_positive_g009_definitiontable) + ge_balance_negative_g009_definitiontableentryvalue = (dst_negative_g009_definitiontable) + ge_balance_positive_g009_definitiontableentryvalue))))))))) /\ (forall ssr_prefix_index_g009_definition ssr_prefix_value_g009_definition. (exists pvs_le_gap_g009_definitionbound. pvs_le_gap_g009_definitionbound + (ssr_prefix_index_g009_definition) = ((l))) -> (exists dst_positive_code_g009_definitionlookup dst_positive_scale_g009_definitionlookup dst_negative_code_g009_definitionlookup dst_negative_scale_g009_definitionlookup dst_positive_g009_definitionlookup dst_negative_g009_definitionlookup. ((((T)) = (((((dst_positive_code_g009_definitionlookup) + (dst_positive_scale_g009_definitionlookup)) * S ((dst_positive_code_g009_definitionlookup) + (dst_positive_scale_g009_definitionlookup)) + ((dst_positive_scale_g009_definitionlookup) + (dst_positive_scale_g009_definitionlookup))) + (((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) * S ((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) + ((dst_negative_scale_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)))) * S ((((dst_positive_code_g009_definitionlookup) + (dst_positive_scale_g009_definitionlookup)) * S ((dst_positive_code_g009_definitionlookup) + (dst_positive_scale_g009_definitionlookup)) + ((dst_positive_scale_g009_definitionlookup) + (dst_positive_scale_g009_definitionlookup))) + (((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) * S ((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) + ((dst_negative_scale_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)))) + ((((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) * S ((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) + ((dst_negative_scale_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup))) + (((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) * S ((dst_negative_code_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)) + ((dst_negative_scale_g009_definitionlookup) + (dst_negative_scale_g009_definitionlookup)))))) /\ (((((exists ff_h_pvs_g009_definitionlookuppositive. ff_h_pvs_g009_definitionlookuppositive + S (dst_positive_g009_definitionlookup) = S ((S (ssr_prefix_index_g009_definition)) * dst_positive_scale_g009_definitionlookup)) /\ exists ff_q_pvs_g009_definitionlookuppositive. dst_positive_code_g009_definitionlookup = ff_q_pvs_g009_definitionlookuppositive * S ((S (ssr_prefix_index_g009_definition)) * dst_positive_scale_g009_definitionlookup) + (dst_positive_g009_definitionlookup))) /\ (((((exists ff_h_pvs_g009_definitionlookupnegative. ff_h_pvs_g009_definitionlookupnegative + S (dst_negative_g009_definitionlookup) = S ((S (ssr_prefix_index_g009_definition)) * dst_negative_scale_g009_definitionlookup)) /\ exists ff_q_pvs_g009_definitionlookupnegative. dst_negative_code_g009_definitionlookup = ff_q_pvs_g009_definitionlookupnegative * S ((S (ssr_prefix_index_g009_definition)) * dst_negative_scale_g009_definitionlookup) + (dst_negative_g009_definitionlookup))) /\ (exists ge_balance_positive_g009_definitionlookupvalue ge_balance_negative_g009_definitionlookupvalue. (((((ssr_prefix_value_g009_definition) = 2 * (ge_balance_positive_g009_definitionlookupvalue) /\ (ge_balance_negative_g009_definitionlookupvalue) = 0) \/ exists ge_signed_half_g009_definitionlookupvaluedecode. (((ssr_prefix_value_g009_definition) = 2 * ge_signed_half_g009_definitionlookupvaluedecode + 1 /\ (ge_balance_positive_g009_definitionlookupvalue) = 0) /\ (ge_balance_negative_g009_definitionlookupvalue) = S ge_signed_half_g009_definitionlookupvaluedecode))) /\ ((dst_positive_g009_definitionlookup) + ge_balance_negative_g009_definitionlookupvalue = (dst_negative_g009_definitionlookup) + ge_balance_positive_g009_definitionlookupvalue))))))))) -> (exists ssr_flat_row_g009_definitionentry ssr_flat_column_g009_definitionentry. (((ssr_prefix_index_g009_definition)=(((S ((M)))*(ssr_flat_row_g009_definitionentry)+(ssr_flat_column_g009_definitionentry)))) /\ (((exists pvs_gap_g009_definitionentryremainder. pvs_gap_g009_definitionentryremainder + S (ssr_flat_column_g009_definitionentry) = (S ((M)))) /\ (exists ssr_entry_value_g009_definitionentryentry ssr_entry_image_g009_definitionentryentry. ((exists dst_positive_code_g009_definitionentryentrysource dst_positive_scale_g009_definitionentryentrysource dst_negative_code_g009_definitionentryentrysource dst_negative_scale_g009_definitionentryentrysource dst_positive_g009_definitionentryentrysource dst_negative_g009_definitionentryentrysource. ((((A)) = (((((dst_positive_code_g009_definitionentryentrysource) + (dst_positive_scale_g009_definitionentryentrysource)) * S ((dst_positive_code_g009_definitionentryentrysource) + (dst_positive_scale_g009_definitionentryentrysource)) + ((dst_positive_scale_g009_definitionentryentrysource) + (dst_positive_scale_g009_definitionentryentrysource))) + (((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) * S ((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) + ((dst_negative_scale_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)))) * S ((((dst_positive_code_g009_definitionentryentrysource) + (dst_positive_scale_g009_definitionentryentrysource)) * S ((dst_positive_code_g009_definitionentryentrysource) + (dst_positive_scale_g009_definitionentryentrysource)) + ((dst_positive_scale_g009_definitionentryentrysource) + (dst_positive_scale_g009_definitionentryentrysource))) + (((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) * S ((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) + ((dst_negative_scale_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)))) + ((((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) * S ((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) + ((dst_negative_scale_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource))) + (((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) * S ((dst_negative_code_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)) + ((dst_negative_scale_g009_definitionentryentrysource) + (dst_negative_scale_g009_definitionentryentrysource)))))) /\ (((((exists ff_h_pvs_g009_definitionentryentrysourcepositive. ff_h_pvs_g009_definitionentryentrysourcepositive + S (dst_positive_g009_definitionentryentrysource) = S ((S (ssr_flat_row_g009_definitionentry)) * dst_positive_scale_g009_definitionentryentrysource)) /\ exists ff_q_pvs_g009_definitionentryentrysourcepositive. dst_positive_code_g009_definitionentryentrysource = ff_q_pvs_g009_definitionentryentrysourcepositive * S ((S (ssr_flat_row_g009_definitionentry)) * dst_positive_scale_g009_definitionentryentrysource) + (dst_positive_g009_definitionentryentrysource))) /\ (((((exists ff_h_pvs_g009_definitionentryentrysourcenegative. ff_h_pvs_g009_definitionentryentrysourcenegative + S (dst_negative_g009_definitionentryentrysource) = S ((S (ssr_flat_row_g009_definitionentry)) * dst_negative_scale_g009_definitionentryentrysource)) /\ exists ff_q_pvs_g009_definitionentryentrysourcenegative. dst_negative_code_g009_definitionentryentrysource = ff_q_pvs_g009_definitionentryentrysourcenegative * S ((S (ssr_flat_row_g009_definitionentry)) * dst_negative_scale_g009_definitionentryentrysource) + (dst_negative_g009_definitionentryentrysource))) /\ (exists ge_balance_positive_g009_definitionentryentrysourcevalue ge_balance_negative_g009_definitionentryentrysourcevalue. (((((ssr_entry_value_g009_definitionentryentry) = 2 * (ge_balance_positive_g009_definitionentryentrysourcevalue) /\ (ge_balance_negative_g009_definitionentryentrysourcevalue) = 0) \/ exists ge_signed_half_g009_definitionentryentrysourcevaluedecode. (((ssr_entry_value_g009_definitionentryentry) = 2 * ge_signed_half_g009_definitionentryentrysourcevaluedecode + 1 /\ (ge_balance_positive_g009_definitionentryentrysourcevalue) = 0) /\ (ge_balance_negative_g009_definitionentryentrysourcevalue) = S ge_signed_half_g009_definitionentryentrysourcevaluedecode))) /\ ((dst_positive_g009_definitionentryentrysource) + ge_balance_negative_g009_definitionentryentrysourcevalue = (dst_negative_g009_definitionentryentrysource) + ge_balance_positive_g009_definitionentryentrysourcevalue))))))))) /\ (((((exists ff_h_pvs_g009_definitionentryentrymap. ff_h_pvs_g009_definitionentryentrymap + S (ssr_entry_image_g009_definitionentryentry) = S ((S (ssr_flat_row_g009_definitionentry)) * (s))) /\ exists ff_q_pvs_g009_definitionentryentrymap. (r) = ff_q_pvs_g009_definitionentryentrymap * S ((S (ssr_flat_row_g009_definitionentry)) * (s)) + (ssr_entry_image_g009_definitionentryentry))) /\ (((((ssr_flat_column_g009_definitionentry)=(ssr_entry_image_g009_definitionentryentry)) /\ ((ssr_prefix_value_g009_definition)=(ssr_entry_value_g009_definitionentryentry)))) \/ (((~((ssr_flat_column_g009_definitionentry)=(ssr_entry_image_g009_definitionentryentry))) /\ ((ssr_prefix_value_g009_definition)=0))))))))))))))
The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
none