EU0012

euler_unit_product_prefix_empty

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

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. UnitProductPrefix(m,b,c,0)

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

Definition DAG

UnitProductPrefix(m,b,c,l) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order 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))))

Complete tactic proof in conservative notation

All 12 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

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.

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 assumptionsL1–5

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 defined 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