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. ∃ dst_positive_bottomlayer. ∃ dst_negative_bottomlayer. MatrixMinorFourCode(F,dst_positive_code_bottomlayer,dst_positive_scale_bottomlayer,dst_negative_code_bottomlayer,dst_negative_scale_bottomlayer) ∧ (BetaAt(dst_positive_code_bottomlayer,dst_positive_scale_bottomlayer,i,dst_positive_bottomlayer) ∧ (BetaAt(dst_negative_code_bottomlayer,dst_negative_scale_bottomlayer,i,dst_negative_bottomlayer) ∧ SignedBalance(z,dst_positive_bottomlayer,dst_negative_bottomlayer)))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 dst_positive_bottomlayer dst_negative_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)))))) /\ (((((exists ff_h_pvs_bottomlayerpositive. ff_h_pvs_bottomlayerpositive + S (dst_positive_bottomlayer) = S ((S ((i))) * dst_positive_scale_bottomlayer)) /\ exists ff_q_pvs_bottomlayerpositive. dst_positive_code_bottomlayer = ff_q_pvs_bottomlayerpositive * S ((S ((i))) * dst_positive_scale_bottomlayer) + (dst_positive_bottomlayer))) /\ (((((exists ff_h_pvs_bottomlayernegative. ff_h_pvs_bottomlayernegative + S (dst_negative_bottomlayer) = S ((S ((i))) * dst_negative_scale_bottomlayer)) /\ exists ff_q_pvs_bottomlayernegative. dst_negative_code_bottomlayer = ff_q_pvs_bottomlayernegative * S ((S ((i))) * dst_negative_scale_bottomlayer) + (dst_negative_bottomlayer))) /\ (exists ge_balance_positive_bottomlayervalue ge_balance_negative_bottomlayervalue. ((((((z)) = 2 * (ge_balance_positive_bottomlayervalue) /\ (ge_balance_negative_bottomlayervalue) = 0) \/ exists ge_signed_half_bottomlayervaluedecode. ((((z)) = 2 * ge_signed_half_bottomlayervaluedecode + 1 /\ (ge_balance_positive_bottomlayervalue) = 0) /\ (ge_balance_negative_bottomlayervalue) = S ge_signed_half_bottomlayervaluedecode))) /\ ((dst_positive_bottomlayer) + ge_balance_negative_bottomlayervalue = (dst_negative_bottomlayer) + ge_balance_positive_bottomlayervalue))))))))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
DirichletEntry(F,G,n,d,z)DirichletPrefix(F,G,n,l,M)DirichletTable(N,F,G,H)SignedZeroWindow(F,k,l)ArithSlice(F,G,o,s,l)ArithRowSums(F,R,o,s,t,m,n)ArithScale(a,F,G,l)ArithPositiveEqual(F,G,N)DirichletGridEntry(F,G,H,n,a,e,z)DirichletFlatPrefix(F,G,H,n,l,T)DirichletGrid(F,G,H,n,T)DirichletFactorRow(F,G,H,n,a,V)
Checked theorems using this definition
DF0002 · dirichlet_grid_entry_from_factorizationDF0004 · dirichlet_grid_entry_factor_productDF0006 · dirichlet_grid_entry_existsDF000A · dirichlet_grid_flat_prefix_zeroDF000B · dirichlet_grid_flat_prefix_appendDF000C · dirichlet_grid_flat_prefix_existsDF000F · dirichlet_grid_table_lookupDF0011 · dirichlet_grid_entry_from_convolution_entryDF0012 · dirichlet_grid_entry_convolution_productDF0014 · dirichlet_factor_row_scalarDF0015 · dirichlet_grid_row_sliceDF0016 · dirichlet_grid_column_sliceDF0019 · dirichlet_factor_row_sum_productDF001A · dirichlet_factor_row_nested_entry