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

Product(b,c,l,z)Repeat(b,c,a,l)BoundedPrefix(b,c,l)InjectivePrefix(b,c,l)UnitBitPrefix(n,b,c,l)Sum(b,c,l,z)SurjectivePrefix(b,c,l)UnitMultiplierPrefix(a,m,b,c,l)UnitProductPrefix(m,b,c,l)UnitScaledPrefix(a,m,b,c,d,e,l)

Checked theorems using this definition

EU0008 · euler_multiplier_prefix_extendEU000A · euler_multiplier_prefix_entryEU000B · euler_multiplier_prefix_bounded_injectiveEU0013 · euler_unit_product_prefix_extendEU0016 · euler_unit_product_prefix_entryEU0017 · euler_unit_product_coprimeEU001B · euler_unit_product_reindex_scaleEU001E · euler_unit_count_product_balanceEU001F · euler_coprime_totient_power_value