EU0005

euler_coprime_modular_unit

Above modulus one, coprimality constructs the blueprint's actual bounded inverse without assuming it.

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. Lt(1,m) → Coprime(a,m) → Unit(a,m)

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

Definition DAG

Unit(a,m) · 1Lt(a,b) · 1Coprime(a,b) · 1

Actual proof prerequisites

binary_modulus_nontrivial_nonzero · checked external prerequisitecoprime_bounded_mod_inverse · checked external prerequisite
Original expanded first-order statement 
forall a m. (exists eut_gap_eu_unit_domain. eut_gap_eu_unit_domain + S (1) = (m)) -> (forall eut_divisor_eu_unit_coprime_given. (exists eut_left_eu_unit_coprime_given. (a) = eut_divisor_eu_unit_coprime_given * eut_left_eu_unit_coprime_given) -> (exists eut_right_eu_unit_coprime_given. (m) = eut_divisor_eu_unit_coprime_given * eut_right_eu_unit_coprime_given) -> eut_divisor_eu_unit_coprime_given = 1) -> ((exists eut_gap_eu_unit_built_domain. eut_gap_eu_unit_built_domain + S (1) = (m)) /\ exists eu_inverse_unit_built. (exists eut_gap_eu_unit_built_bound. eut_gap_eu_unit_built_bound + S (eu_inverse_unit_built) = (m)) /\ (exists eu_mod_left_unit_built_inverse eu_mod_right_unit_built_inverse. ((a)*eu_inverse_unit_built) + (m) * eu_mod_left_unit_built_inverse = (1) + (m) * eu_mod_right_unit_built_inverse))

Complete tactic proof in conservative notation

All 15 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

15 script commands · 5 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro hm
  4. L4
    intro hc
02Separate the logical casesL5–5

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

  1. L5
    split
03Use earlier factsL6–9

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

  1. L6
    exact hm
  2. L7
    specialize coprime_bounded_mod_inverse (a)
  3. L8
    specialize coprime_bounded_mod_inverse (m)
  4. L9
    apply coprime_bounded_mod_inverse
04Fix variables and assumptionsL10–10

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

  1. L10
    intro hz
05Use earlier factsL11–15

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

  1. L11
    specialize binary_modulus_nontrivial_nonzero (m)
  2. L12
    apply binary_modulus_nontrivial_nonzero
  3. L13
    exact hm
  4. L14
    exact hz
  5. L15
    exact hc

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro hm
  4. 0004intro hc
  5. 0005split
  6. 0006exact hm
  7. 0007specialize coprime_bounded_mod_inverse (a)
  8. 0008specialize coprime_bounded_mod_inverse (m)
  9. 0009apply coprime_bounded_mod_inverse
  10. 0010intro hz
  11. 0011specialize binary_modulus_nontrivial_nonzero (m)
  12. 0012apply binary_modulus_nontrivial_nonzero
  13. 0013exact hm
  14. 0014exact hz
  15. 0015exact hc