ND0238

UnitMultiplierPrefix(a,m,b,c,l)

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

At every index i<l, the actual beta entry is the canonical residue of a*i. A bijection is constructed by theorem, not assumed in this graph.

Conservative notation; not a theorem, primitive, or axiom.

Definition in prerequisite notation

∀ eu_index_bottomlayer. Lt(eu_index_bottomlayer,l) → ∃ x. BetaAt(b,c,eu_index_bottomlayer,x) ∧ CanonicalModularResidue(m,a · eu_index_bottomlayer,x)

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition 
forall eu_index_bottomlayer. (exists eut_gap_eu_bottomlayer_index. eut_gap_eu_bottomlayer_index + S (eu_index_bottomlayer) = ((l))) -> exists eu_residue_bottomlayer. (((exists fs_h_eu_bottomlayer_at. fs_h_eu_bottomlayer_at + S (eu_residue_bottomlayer) = S ((S (eu_index_bottomlayer)) * (c))) /\ exists fs_q_eu_bottomlayer_at. (b) = fs_q_eu_bottomlayer_at * S ((S (eu_index_bottomlayer)) * (c)) + (eu_residue_bottomlayer))) /\ ((exists eut_gap_eu_bottomlayer_bound. eut_gap_eu_bottomlayer_bound + S (eu_residue_bottomlayer) = ((m))) /\ (exists eu_mod_left_bottomlayer_mod eu_mod_right_bottomlayer_mod. (((a))*eu_index_bottomlayer) + ((m)) * eu_mod_left_bottomlayer_mod = (eu_residue_bottomlayer) + ((m)) * eu_mod_right_bottomlayer_mod))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Lt(a,b)BetaAt(b,c,i,x)CanonicalModularResidue(m,a,r)

Definitions depending on this notation

none

Checked theorems using this definition

EU0007 · euler_multiplier_prefix_emptyEU0008 · euler_multiplier_prefix_extendEU0009 · euler_multiplier_prefix_existsEU000A · euler_multiplier_prefix_entryEU000B · euler_multiplier_prefix_bounded_injectiveEU000C · euler_multiplier_prefix_permutationEU000D · euler_multiplier_permutation_existsEU001B · euler_unit_product_reindex_scaleEU001F · euler_coprime_totient_power_value