PD0013

BetaAt(b,c,i,x)

x is the bounded beta-decoded value at index i.

Conservative notation; not a theorem, primitive, or axiom.

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

S x ≤ S (S i · c) ∧ (∃ y. b = y · S (S i · c) + x)

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition 
((exists ff_h_defined_beta_at. ff_h_defined_beta_at + S (x) = S ((S (i)) * c)) /\ exists ff_q_defined_beta_at. b = ff_q_defined_beta_at * S ((S (i)) * c) + (x))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

none — first-order arithmetic only

Definitions depending on this notation

ArithAt(F,i,z)ArithTable(N,F)Sum(b,c,l,z)DivisorPairIndexMap(V,L,r,s)SignedSupportReindex(A,B,r,s,L,M)SignedIncidenceEntry(A,r,s,i,j,z)

Checked theorems using this definition

MX0014 · divisor_pair_index_map_appendMX0015 · divisor_pair_index_map_existsMX0016 · divisor_pair_index_map_lookupMX0017 · divisor_pair_index_map_valueMX0036 · signed_support_incidence_entry_hitMX0037 · signed_support_incidence_entry_missMX0038 · signed_support_incidence_entry_decodeMX003A · signed_support_incidence_entry_existsMX003C · signed_support_incidence_nonzero_source_imageMX0046 · signed_support_incidence_row_sum_valueMX0047 · signed_support_incidence_column_sum_valueMX0052 · dirichlet_coprime_grid_support_preservingMX0053 · dirichlet_coprime_grid_support_injectiveMX0054 · dirichlet_coprime_grid_support_covering