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

FpAddPrefix(p,b,c,l)FpMulPrefix(p,b,c,l)FpNegPrefix(p,b,c,l)FpInvPrefix(p,b,c,l)IdentityMatrixSelector(b,c,l)FpCardinality(p,b,c)FpUnitSteps(p,b,c,n)FpUnitTrace(p,b,c,n,r)

Checked theorems using this definition

FP002F · prime_field_add_prefix_choiceFP0030 · prime_field_multiply_prefix_choiceFP0031 · prime_field_negate_prefix_choiceFP0032 · prime_field_inverse_prefix_choiceFP0039 · prime_field_add_table_lookupFP003A · prime_field_add_table_reflectFP003C · prime_field_multiply_table_lookupFP003D · prime_field_multiply_table_reflectFP003E · prime_field_negate_table_lookupFP003F · prime_field_negate_table_reflectFP0040 · prime_field_inverse_table_lookupFP0041 · prime_field_inverse_table_reflectFP0042 · prime_field_add_table_commutativeFP0043 · prime_field_add_table_associativeFP0044 · prime_field_multiply_table_commutativeFP0045 · prime_field_multiply_table_associativeFP0046 · prime_field_inverse_table_zeroFP0047 · prime_field_inverse_table_nonzeroFP0048 · prime_field_left_table_distributiveFP0049 · prime_field_right_table_distributiveFP004A · prime_field_enumeration_valueFP004D · prime_field_unit_trace_recodeFP004E · prime_field_unit_trace_successorFP004F · prime_field_unit_trace_residue