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

InitialPrimeChain(b,c,k)Product(b,c,l,z)Repeat(b,c,a,l)

Checked theorems using this definition

PE0008 · initial_prime_chain_prefix_extendPE000A · initial_prime_chain_terminal_is_primePE000B · initial_prime_chain_bounded_existsPE000C · first_primes_double_exponential_boundPE000D · initial_prime_chain_strict_orderPE000E · initial_prime_chain_exhausts_primesPE0010 · prime_list_every_entry_is_primePE0011 · prime_list_omits_no_smaller_primePE0012 · prime_list_strictly_increasingPE0013 · first_primes_list_exists