EU0004

euler_modular_unit_coprime

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

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

The exact G014 theorem covers m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem handles m=1 by congruence, not by asserting that one is a canonical remainder. Multiplicative-order and RSA statements are not claimed.

Exact theorem in conservative defined notation

∀ a. ∀ m. Unit(a,m) → Coprime(a,m)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Unit(a,m) · 1Coprime(a,b) · 1

Actual proof prerequisites

mod_inverse_implies_coprime · checked external prerequisite
Original expanded first-order 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)

Complete tactic proof in conservative notation

All 11 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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