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

Repeat(b,c,a,l)DivisionPrefix(m,b,c,qb,qc,rb,rc,l)BetaPrefixInto(b,c,l,B)BetaPrefixEqual(b,c,d,e,l)FpCoefficientReduction(p,b,c,d,e,l)FpPolyAdd(p,ab,ac,bb,bc,cb,cc,l)FpPolyScale(p,k,ab,ac,bb,bc,l)FpHornerStep(p,b,c,x,u,v,i)FpHornerTrace(p,b,c,x,l,r,u,v)

Checked theorems using this definition

PP0001 · prime_field_polynomial_normalization_from_divisionPP0003 · prime_field_polynomial_normalization_entryPP0004 · prime_field_polynomial_normalization_boundedPP0005 · prime_field_polynomial_normalization_functionalPP0006 · prime_field_polynomial_normalization_reflexivePP0007 · prime_field_polynomial_normalization_transportPP000C · prime_field_polynomial_add_from_normalizationPP000D · prime_field_polynomial_add_existsPP000E · prime_field_polynomial_add_entryPP000F · prime_field_polynomial_add_boundedPP0010 · prime_field_polynomial_add_functionalPP0011 · prime_field_polynomial_add_transportPP0012 · prime_field_polynomial_add_commutativePP0013 · prime_field_polynomial_add_zero_rightPP0014 · prime_field_polynomial_scale_from_normalizationPP0015 · prime_field_polynomial_scale_existsPP0016 · prime_field_polynomial_scale_entryPP0017 · prime_field_polynomial_scale_boundedPP0018 · prime_field_polynomial_scale_functionalPP0019 · prime_field_polynomial_scale_transportPP001A · prime_field_polynomial_scale_onePP001B · prime_field_polynomial_scale_zeroPP001C · prime_field_polynomial_add_associativePP001D · prime_field_polynomial_scale_associativePP001E · prime_field_polynomial_scale_distributes_over_addPP001F · prime_field_polynomial_scalar_add_distributesPP0021 · prime_field_polynomial_horner_trace_from_normalizationPP0025 · prime_field_polynomial_horner_successor_decomposePP0027 · prime_field_polynomial_horner_normalization_residuePP002C · prime_field_polynomial_horner_successor_constructPP002D · prime_field_polynomial_horner_constant