ND0237

Unit(a,m)

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.

Exactly m>1 with a witnessed inverse b<m and a*b congruent to one. Unlike the old UnitResidue range, this expresses genuine invertibility at composite moduli.

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

Definition in prerequisite notation

Lt(1,m) ∧ (∃ x. Lt(x,m) ∧ ModEq(m,a · x,1))

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

Hygienic expanded first-order definition 
(exists eut_gap_eu_bottomlayer_domain. eut_gap_eu_bottomlayer_domain + S (1) = ((m))) /\ exists eu_inverse_bottomlayer. (exists eut_gap_eu_bottomlayer_bound. eut_gap_eu_bottomlayer_bound + S (eu_inverse_bottomlayer) = ((m))) /\ (exists eu_mod_left_bottomlayer_inverse eu_mod_right_bottomlayer_inverse. (((a))*eu_inverse_bottomlayer) + ((m)) * eu_mod_left_bottomlayer_inverse = (1) + ((m)) * eu_mod_right_bottomlayer_inverse)

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)ModEq(m,a,b)

Definitions depending on this notation

none

Checked theorems using this definition

EU0004 · euler_modular_unit_coprimeEU0005 · euler_coprime_modular_unitEU0021 · euler_modular_unit_totient_powerEU0022 · euler_theorem_for_units