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

Product(b,c,l,z)Repeat(b,c,a,l)PrimeExponentPrefixGCD(b,c,l,g)PerfectPowerRootTable(n,g,b,c)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

SK001B · prime_exponent_common_divisor_dropSK001C · prime_exponent_common_divisor_successorSK001D · prime_exponent_common_divisor_factorSK001F · prime_exponent_prefix_gcd_successorSK0020 · prime_exponent_prefix_gcd_existsSK0023 · prime_valuation_support_exponent_gcd_nonzeroSK0024 · prime_exponent_entry_has_prime_valuationSK0025 · prime_support_common_divisor_implies_all_valuationsSK0026 · prime_support_all_valuations_implies_common_divisorSK002A · perfect_power_root_table_prefix_appendSK002C · perfect_power_root_table_prefix_existsSK002D · perfect_power_root_table_existsSK0031 · perfect_power_profile_data_root_lookup