EU0004

euler_modular_unit_coprime

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

An actual bounded inverse implies the frozen common-divisor coprimality graph.

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 expanded first-order arithmetic statement

forall a m. ((exists eut_gap_eu_unit_given_domain. eut_gap_eu_unit_given_domain + S (1) = (m)) /\ exists eu_inverse_unit_given. (exists eut_gap_eu_unit_given_bound. eut_gap_eu_unit_given_bound + S (eu_inverse_unit_given) = (m)) /\ (exists eu_mod_left_unit_given_inverse eu_mod_right_unit_given_inverse. ((a)*eu_inverse_unit_given) + (m) * eu_mod_left_unit_given_inverse = (1) + (m) * eu_mod_right_unit_given_inverse)) -> (forall eut_divisor_eu_unit_coprime. (exists eut_left_eu_unit_coprime. (a) = eut_divisor_eu_unit_coprime * eut_left_eu_unit_coprime) -> (exists eut_right_eu_unit_coprime. (m) = eut_divisor_eu_unit_coprime * eut_right_eu_unit_coprime) -> eut_divisor_eu_unit_coprime = 1)

Constructive proof overview

Generated structural guide

An actual bounded inverse implies the frozen common-divisor coprimality graph.

The unchanged tactic script uses 1 declared prerequisite and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mod_inverse_implies_coprime Stable theorem; checked-use authorized

Direct dependents

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

11 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

01Fix variables and assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro hu
02Separate the logical casesL4–6

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases hu
  2. L5
    cases hu_right
  3. L6
    cases hu_right_witness
03Use earlier factsL7–11

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L7
    specialize mod_inverse_implies_coprime (a)
  2. L8
    specialize mod_inverse_implies_coprime (m)
  3. L9
    specialize mod_inverse_implies_coprime (x)
  4. L10
    apply mod_inverse_implies_coprime
  5. L11
    exact hu_right_witness_right

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro hu
  4. 0004cases hu
  5. 0005cases hu_right
  6. 0006cases hu_right_witness
  7. 0007specialize mod_inverse_implies_coprime (a)
  8. 0008specialize mod_inverse_implies_coprime (m)
  9. 0009specialize mod_inverse_implies_coprime (x)
  10. 0010apply mod_inverse_implies_coprime
  11. 0011exact hu_right_witness_right