EU0016

euler_unit_product_prefix_entry

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

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

∀ m. ∀ b. ∀ c. ∀ l. ∀ i. ∀ v. UnitProductPrefix(m,b,c,l) → Lt(i,l) → BetaAt(b,c,i,v) → UnitProductFactor(m,i,v)

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

Definition DAG

UnitProductFactor(m,i,v) · 2UnitProductPrefix(m,b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order 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)))

Complete tactic proof in conservative notation

All 27 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

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.

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 assumptionsL1–9

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: BetaAt(b,c,i,w)UnitProductFactor(m,i,w)Original native command in the exact edition
  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 defined 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 : ∃ w. BetaAt(b,c,i,w) ∧ UnitProductFactor(m,i,w)
  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