EU0012

euler_unit_product_prefix_empty

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

The empty weighted-factor prefix is valid for any beta codes.

Exact expanded first-order arithmetic statement

forall m b c. (forall eu_factor_index_factors_empty. (exists eut_gap_eu_factors_empty_index. eut_gap_eu_factors_empty_index + S (eu_factor_index_factors_empty) = (0)) -> exists eu_factor_value_factors_empty. (((exists fs_h_eu_factors_empty_at. fs_h_eu_factors_empty_at + S (eu_factor_value_factors_empty) = S ((S (eu_factor_index_factors_empty)) * c)) /\ exists fs_q_eu_factors_empty_at. b = fs_q_eu_factors_empty_at * S ((S (eu_factor_index_factors_empty)) * c) + (eu_factor_value_factors_empty))) /\ ((((forall eut_divisor_eu_factors_empty_choice_coprime. (exists eut_left_eu_factors_empty_choice_coprime. (eu_factor_index_factors_empty) = eut_divisor_eu_factors_empty_choice_coprime * eut_left_eu_factors_empty_choice_coprime) -> (exists eut_right_eu_factors_empty_choice_coprime. (m) = eut_divisor_eu_factors_empty_choice_coprime * eut_right_eu_factors_empty_choice_coprime) -> eut_divisor_eu_factors_empty_choice_coprime = 1) /\ (eu_factor_value_factors_empty)=(eu_factor_index_factors_empty)) \/ (~(forall eut_divisor_eu_factors_empty_choice_coprime. (exists eut_left_eu_factors_empty_choice_coprime. (eu_factor_index_factors_empty) = eut_divisor_eu_factors_empty_choice_coprime * eut_left_eu_factors_empty_choice_coprime) -> (exists eut_right_eu_factors_empty_choice_coprime. (m) = eut_divisor_eu_factors_empty_choice_coprime * eut_right_eu_factors_empty_choice_coprime) -> eut_divisor_eu_factors_empty_choice_coprime = 1) /\ (eu_factor_value_factors_empty)=1))))

Constructive proof overview

Generated structural guide

The empty weighted-factor prefix is valid for any beta codes.

The unchanged tactic script uses 2 declared prerequisites and contains 12 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

lt_not_le Alpha theorem; checked-use authorized zero_le Alpha theorem; checked-use authorized

Direct dependents

EU0014 euler_unit_product_prefix_exists

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

12 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 assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro i
  5. L5
    intro hi
02Separate the logical casesL6–6

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

  1. L6
    exfalso
03Use earlier factsL7–12

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

  1. L7
    specialize lt_not_le (i)
  2. L8
    specialize lt_not_le (0)
  3. L9
    apply lt_not_le
  4. L10
    exact hi
  5. L11
    specialize zero_le (i)
  6. L12
    apply zero_le

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro m
  2. 0002intro b
  3. 0003intro c
  4. 0004intro i
  5. 0005intro hi
  6. 0006exfalso
  7. 0007specialize lt_not_le (i)
  8. 0008specialize lt_not_le (0)
  9. 0009apply lt_not_le
  10. 0010exact hi
  11. 0011specialize zero_le (i)
  12. 0012apply zero_le