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
∃ dst_positive_code_bottomlayer. ∃ dst_positive_scale_bottomlayer. ∃ dst_negative_code_bottomlayer. ∃ dst_negative_scale_bottomlayer. MatrixMinorFourCode(F,dst_positive_code_bottomlayer,dst_positive_scale_bottomlayer,dst_negative_code_bottomlayer,dst_negative_scale_bottomlayer) ∧ (∀ x. Le(x,N) → ∃ y. ∃ z. ∃ n. BetaAt(dst_positive_code_bottomlayer,dst_positive_scale_bottomlayer,x,y) ∧ (BetaAt(dst_negative_code_bottomlayer,dst_negative_scale_bottomlayer,x,z) ∧ SignedBalance(n,y,z)))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists dst_positive_code_bottomlayer dst_positive_scale_bottomlayer dst_negative_code_bottomlayer dst_negative_scale_bottomlayer. ((((F)) = (((((dst_positive_code_bottomlayer) + (dst_positive_scale_bottomlayer)) * S ((dst_positive_code_bottomlayer) + (dst_positive_scale_bottomlayer)) + ((dst_positive_scale_bottomlayer) + (dst_positive_scale_bottomlayer))) + (((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) * S ((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) + ((dst_negative_scale_bottomlayer) + (dst_negative_scale_bottomlayer)))) * S ((((dst_positive_code_bottomlayer) + (dst_positive_scale_bottomlayer)) * S ((dst_positive_code_bottomlayer) + (dst_positive_scale_bottomlayer)) + ((dst_positive_scale_bottomlayer) + (dst_positive_scale_bottomlayer))) + (((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) * S ((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) + ((dst_negative_scale_bottomlayer) + (dst_negative_scale_bottomlayer)))) + ((((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) * S ((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) + ((dst_negative_scale_bottomlayer) + (dst_negative_scale_bottomlayer))) + (((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) * S ((dst_negative_code_bottomlayer) + (dst_negative_scale_bottomlayer)) + ((dst_negative_scale_bottomlayer) + (dst_negative_scale_bottomlayer)))))) /\ (forall dst_index_bottomlayer. (exists pvs_le_gap_bottomlayerdomain. pvs_le_gap_bottomlayerdomain + (dst_index_bottomlayer) = ((N))) -> exists dst_positive_bottomlayer dst_negative_bottomlayer dst_value_bottomlayer. ((((exists ff_h_pvs_bottomlayerentrypositive. ff_h_pvs_bottomlayerentrypositive + S (dst_positive_bottomlayer) = S ((S (dst_index_bottomlayer)) * dst_positive_scale_bottomlayer)) /\ exists ff_q_pvs_bottomlayerentrypositive. dst_positive_code_bottomlayer = ff_q_pvs_bottomlayerentrypositive * S ((S (dst_index_bottomlayer)) * dst_positive_scale_bottomlayer) + (dst_positive_bottomlayer))) /\ (((((exists ff_h_pvs_bottomlayerentrynegative. ff_h_pvs_bottomlayerentrynegative + S (dst_negative_bottomlayer) = S ((S (dst_index_bottomlayer)) * dst_negative_scale_bottomlayer)) /\ exists ff_q_pvs_bottomlayerentrynegative. dst_negative_code_bottomlayer = ff_q_pvs_bottomlayerentrynegative * S ((S (dst_index_bottomlayer)) * dst_negative_scale_bottomlayer) + (dst_negative_bottomlayer))) /\ (exists ge_balance_positive_bottomlayerentryvalue ge_balance_negative_bottomlayerentryvalue. (((((dst_value_bottomlayer) = 2 * (ge_balance_positive_bottomlayerentryvalue) /\ (ge_balance_negative_bottomlayerentryvalue) = 0) \/ exists ge_signed_half_bottomlayerentryvaluedecode. (((dst_value_bottomlayer) = 2 * ge_signed_half_bottomlayerentryvaluedecode + 1 /\ (ge_balance_positive_bottomlayerentryvalue) = 0) /\ (ge_balance_negative_bottomlayerentryvalue) = S ge_signed_half_bottomlayerentryvaluedecode))) /\ ((dst_positive_bottomlayer) + ge_balance_negative_bottomlayerentryvalue = (dst_negative_bottomlayer) + ge_balance_positive_bottomlayerentryvalue))))))))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
Checked theorems using this definition
DV0002 · arithmetic_signed_table_equal_entry_transportDV0003 · arithmetic_signed_table_extend_atDV0004 · arithmetic_signed_table_appendDV0005 · arithmetic_signed_table_singletonDV0006 · arithmetic_signed_sum_existsDV0007 · arithmetic_signed_sum_append_transportDV0008 · mobius_table_zero_constructorDV000A · mobius_table_existsDV0013 · divisor_mask_entry_existsDV0017 · divisor_mask_prefix_zero_constructorDV0019 · divisor_mask_prefix_existsDV0020 · signed_divisor_sum_existsDV0022 · signed_divisor_sum_exists_uniqueDV0024 · signed_divisor_sum_one