EU0005

euler_coprime_modular_unit

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.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

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

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 15 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

binary_modulus_nontrivial_nonzero Alpha theorem; checked-use authorized coprime_bounded_mod_inverse Alpha theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 exact 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