EU0015

euler_unit_product_prefix_drop_last

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic 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))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 13 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

le_succ Stable theorem; checked-use authorized

Direct dependents

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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