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. Exact original first-admission records.

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

UnitBitPrefix(n,b,c,l)Sum(b,c,l,z)Product(b,c,l,z)Repeat(b,c,a,l)EulerFactorPrefix(pb,pc,eb,ec,fb,fc,l)InjectivePrefix(b,c,l)PrimeExponentEntries(n,pb,pc,eb,ec,vb,vc,l)PrimeDivisorSupport(n,pb,pc,l)AllPrime(b,c,l)

Checked theorems using this definition

TP0006 · totient_unit_prefix_entryTP0007 · totient_unit_prefix_extendTP0009 · totient_unit_prefix_all_bitsTP000A · totient_unit_prefix_equal_entryTP000F · totient_unit_count_succ_decomposeTP003E · totient_coprime_multiplicative_from_prime_listTP0043 · totient_prime_entries_decoded_powerTP0044 · totient_prime_entries_selected_powerTP0045 · totient_prime_support_powers_pairwise_coprimeTP0046 · totient_pairwise_coprime_product_foldTP0049 · totient_euler_factor_prefix_appendTP004A · totient_euler_factor_prefix_existsTP004B · totient_euler_factor_prefix_counts