EU0016

euler_unit_product_prefix_entry

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 weighted-factor choice holds for every actual decoded entry, independently of beta encoding.

Exact expanded first-order arithmetic statement

forall m b c l i v. (forall eu_factor_index_factors_entry. (exists eut_gap_eu_factors_entry_index. eut_gap_eu_factors_entry_index + S (eu_factor_index_factors_entry) = (l)) -> exists eu_factor_value_factors_entry. (((exists fs_h_eu_factors_entry_at. fs_h_eu_factors_entry_at + S (eu_factor_value_factors_entry) = S ((S (eu_factor_index_factors_entry)) * c)) /\ exists fs_q_eu_factors_entry_at. b = fs_q_eu_factors_entry_at * S ((S (eu_factor_index_factors_entry)) * c) + (eu_factor_value_factors_entry))) /\ ((((forall eut_divisor_eu_factors_entry_choice_coprime. (exists eut_left_eu_factors_entry_choice_coprime. (eu_factor_index_factors_entry) = eut_divisor_eu_factors_entry_choice_coprime * eut_left_eu_factors_entry_choice_coprime) -> (exists eut_right_eu_factors_entry_choice_coprime. (m) = eut_divisor_eu_factors_entry_choice_coprime * eut_right_eu_factors_entry_choice_coprime) -> eut_divisor_eu_factors_entry_choice_coprime = 1) /\ (eu_factor_value_factors_entry)=(eu_factor_index_factors_entry)) \/ (~(forall eut_divisor_eu_factors_entry_choice_coprime. (exists eut_left_eu_factors_entry_choice_coprime. (eu_factor_index_factors_entry) = eut_divisor_eu_factors_entry_choice_coprime * eut_left_eu_factors_entry_choice_coprime) -> (exists eut_right_eu_factors_entry_choice_coprime. (m) = eut_divisor_eu_factors_entry_choice_coprime * eut_right_eu_factors_entry_choice_coprime) -> eut_divisor_eu_factors_entry_choice_coprime = 1) /\ (eu_factor_value_factors_entry)=1)))) -> (exists eut_gap_eu_factor_entry_bound. eut_gap_eu_factor_entry_bound + S (i) = (l)) -> (((exists fs_h_eu_factor_entry_given. fs_h_eu_factor_entry_given + S (v) = S ((S (i)) * c)) /\ exists fs_q_eu_factor_entry_given. b = fs_q_eu_factor_entry_given * S ((S (i)) * c) + (v))) -> ((((forall eut_divisor_eu_factor_entry_coprime. (exists eut_left_eu_factor_entry_coprime. (i) = eut_divisor_eu_factor_entry_coprime * eut_left_eu_factor_entry_coprime) -> (exists eut_right_eu_factor_entry_coprime. (m) = eut_divisor_eu_factor_entry_coprime * eut_right_eu_factor_entry_coprime) -> eut_divisor_eu_factor_entry_coprime = 1) /\ (v)=(i)) \/ (~(forall eut_divisor_eu_factor_entry_coprime. (exists eut_left_eu_factor_entry_coprime. (i) = eut_divisor_eu_factor_entry_coprime * eut_left_eu_factor_entry_coprime) -> (exists eut_right_eu_factor_entry_coprime. (m) = eut_divisor_eu_factor_entry_coprime * eut_right_eu_factor_entry_coprime) -> eut_divisor_eu_factor_entry_coprime = 1) /\ (v)=1)))

Constructive proof overview

Generated structural guide

The weighted-factor choice holds for every actual decoded entry, independently of beta encoding.

The unchanged tactic script uses 1 declared prerequisite and contains 27 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

beta_at_unique Alpha theorem; checked-use authorized

Direct dependents

EU0017 euler_unit_product_coprime EU001B euler_unit_product_reindex_scale

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

27 script commands · 6 reading checkpoints · 2 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–9

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 l
  5. L5
    intro i
  6. L6
    intro v
  7. L7
    intro h
  8. L8
    intro hi
  9. L9
    intro hv
02Establish hpL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply h.

  1. L10
    have hp : ∃ w. BetaAt(b,c,i,w) ∧ UnitProductFactor(m,i,w)Definitions: UnitProductFactorBetaAt
  2. L11
    specialize h (i)
  3. L12
    apply h
  4. L13
    exact hi
03Separate the logical casesL14–15

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

  1. L14
    cases hp
  2. L15
    cases hp_witness
04Establish heL16–25

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L16
    have he : x=v
  2. L17
    specialize beta_at_unique (b)
  3. L18
    specialize beta_at_unique (c)
  4. L19
    specialize beta_at_unique (i)
  5. L20
    specialize beta_at_unique (x)
  6. L21
    specialize beta_at_unique (v)
  7. L22
    apply beta_at_unique
  8. L23
    exact hp_witness_left
  9. L24
    exact hv
  10. L25
    rewrite he at hp_witness_right
05Calculate and transport equalitiesL26–26

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L26
    rewrite he at hp_witness_right
06Use earlier factsL27–27

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

  1. L27
    exact hp_witness_right

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro m
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro i
  6. 0006intro v
  7. 0007intro h
  8. 0008intro hi
  9. 0009intro hv
  10. 0010have hp : exists w. (((exists fs_h_eu_factor_entry_chosen. fs_h_eu_factor_entry_chosen + S (w) = S ((S (i)) * c)) /\ exists fs_q_eu_factor_entry_chosen. b = fs_q_eu_factor_entry_chosen * S ((S (i)) * c) + (w))) /\ ((((forall eut_divisor_eu_factor_entry_chosen_coprime. (exists eut_left_eu_factor_entry_chosen_coprime. (i) = eut_divisor_eu_factor_entry_chosen_coprime * eut_left_eu_factor_entry_chosen_coprime) -> (exists eut_right_eu_factor_entry_chosen_coprime. (m) = eut_divisor_eu_factor_entry_chosen_coprime * eut_right_eu_factor_entry_chosen_coprime) -> eut_divisor_eu_factor_entry_chosen_coprime = 1) /\ (w)=(i)) \/ (~(forall eut_divisor_eu_factor_entry_chosen_coprime. (exists eut_left_eu_factor_entry_chosen_coprime. (i) = eut_divisor_eu_factor_entry_chosen_coprime * eut_left_eu_factor_entry_chosen_coprime) -> (exists eut_right_eu_factor_entry_chosen_coprime. (m) = eut_divisor_eu_factor_entry_chosen_coprime * eut_right_eu_factor_entry_chosen_coprime) -> eut_divisor_eu_factor_entry_chosen_coprime = 1) /\ (w)=1)))
  11. 0011specialize h (i)
  12. 0012apply h
  13. 0013exact hi
  14. 0014cases hp
  15. 0015cases hp_witness
  16. 0016have he : x=v
  17. 0017specialize beta_at_unique (b)
  18. 0018specialize beta_at_unique (c)
  19. 0019specialize beta_at_unique (i)
  20. 0020specialize beta_at_unique (x)
  21. 0021specialize beta_at_unique (v)
  22. 0022apply beta_at_unique
  23. 0023exact hp_witness_left
  24. 0024exact hv
  25. 0025rewrite he at hp_witness_right
  26. 0026rewrite he at hp_witness_right
  27. 0027exact hp_witness_right