EU0015

euler_unit_product_prefix_drop_last

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.

Restrict an actual weighted-factor prefix to its predecessor interval.

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

The exact G014 theorem is proved in this research checkpoint for m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem also handles m=1 by congruence, not by asserting that one is a canonical remainder. No multiplicative-order or RSA theorem is claimed. The published atlas and Alpha membership are unchanged.

Exact theorem in conservative defined notation

∀ m. ∀ b. ∀ c. ∀ l. UnitProductPrefix(m,b,c,S l) → UnitProductPrefix(m,b,c,l)

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

Definition DAG

UnitProductPrefix(m,b,c,l) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall m b c l. (forall eu_factor_index_factors_drop_old. (exists eut_gap_eu_factors_drop_old_index. eut_gap_eu_factors_drop_old_index + S (eu_factor_index_factors_drop_old) = (S l)) -> exists eu_factor_value_factors_drop_old. (((exists fs_h_eu_factors_drop_old_at. fs_h_eu_factors_drop_old_at + S (eu_factor_value_factors_drop_old) = S ((S (eu_factor_index_factors_drop_old)) * c)) /\ exists fs_q_eu_factors_drop_old_at. b = fs_q_eu_factors_drop_old_at * S ((S (eu_factor_index_factors_drop_old)) * c) + (eu_factor_value_factors_drop_old))) /\ ((((forall eut_divisor_eu_factors_drop_old_choice_coprime. (exists eut_left_eu_factors_drop_old_choice_coprime. (eu_factor_index_factors_drop_old) = eut_divisor_eu_factors_drop_old_choice_coprime * eut_left_eu_factors_drop_old_choice_coprime) -> (exists eut_right_eu_factors_drop_old_choice_coprime. (m) = eut_divisor_eu_factors_drop_old_choice_coprime * eut_right_eu_factors_drop_old_choice_coprime) -> eut_divisor_eu_factors_drop_old_choice_coprime = 1) /\ (eu_factor_value_factors_drop_old)=(eu_factor_index_factors_drop_old)) \/ (~(forall eut_divisor_eu_factors_drop_old_choice_coprime. (exists eut_left_eu_factors_drop_old_choice_coprime. (eu_factor_index_factors_drop_old) = eut_divisor_eu_factors_drop_old_choice_coprime * eut_left_eu_factors_drop_old_choice_coprime) -> (exists eut_right_eu_factors_drop_old_choice_coprime. (m) = eut_divisor_eu_factors_drop_old_choice_coprime * eut_right_eu_factors_drop_old_choice_coprime) -> eut_divisor_eu_factors_drop_old_choice_coprime = 1) /\ (eu_factor_value_factors_drop_old)=1)))) -> (forall eu_factor_index_factors_drop_new. (exists eut_gap_eu_factors_drop_new_index. eut_gap_eu_factors_drop_new_index + S (eu_factor_index_factors_drop_new) = (l)) -> exists eu_factor_value_factors_drop_new. (((exists fs_h_eu_factors_drop_new_at. fs_h_eu_factors_drop_new_at + S (eu_factor_value_factors_drop_new) = S ((S (eu_factor_index_factors_drop_new)) * c)) /\ exists fs_q_eu_factors_drop_new_at. b = fs_q_eu_factors_drop_new_at * S ((S (eu_factor_index_factors_drop_new)) * c) + (eu_factor_value_factors_drop_new))) /\ ((((forall eut_divisor_eu_factors_drop_new_choice_coprime. (exists eut_left_eu_factors_drop_new_choice_coprime. (eu_factor_index_factors_drop_new) = eut_divisor_eu_factors_drop_new_choice_coprime * eut_left_eu_factors_drop_new_choice_coprime) -> (exists eut_right_eu_factors_drop_new_choice_coprime. (m) = eut_divisor_eu_factors_drop_new_choice_coprime * eut_right_eu_factors_drop_new_choice_coprime) -> eut_divisor_eu_factors_drop_new_choice_coprime = 1) /\ (eu_factor_value_factors_drop_new)=(eu_factor_index_factors_drop_new)) \/ (~(forall eut_divisor_eu_factors_drop_new_choice_coprime. (exists eut_left_eu_factors_drop_new_choice_coprime. (eu_factor_index_factors_drop_new) = eut_divisor_eu_factors_drop_new_choice_coprime * eut_left_eu_factors_drop_new_choice_coprime) -> (exists eut_right_eu_factors_drop_new_choice_coprime. (m) = eut_divisor_eu_factors_drop_new_choice_coprime * eut_right_eu_factors_drop_new_choice_coprime) -> eut_divisor_eu_factors_drop_new_choice_coprime = 1) /\ (eu_factor_value_factors_drop_new)=1))))

Complete tactic proof in conservative notation

All 13 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

13 script commands · 2 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 assumptionsL1–7

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 h
  6. L6
    intro i
  7. L7
    intro hi
02Use earlier factsL8–13

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

  1. L8
    specialize h (i)
  2. L9
    apply h
  3. L10
    specialize le_succ (S i)
  4. L11
    specialize le_succ (l)
  5. L12
    apply le_succ
  6. L13
    exact hi

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro m
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro h
  6. 0006intro i
  7. 0007intro hi
  8. 0008specialize h (i)
  9. 0009apply h
  10. 0010specialize le_succ (S i)
  11. 0011specialize le_succ (l)
  12. 0012apply le_succ
  13. 0013exact hi