EU0007

euler_multiplier_prefix_empty

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

The empty multiplier prefix has no residue obligations.

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 a m b c. forall eu_index_map_empty. (exists eut_gap_eu_map_empty_index. eut_gap_eu_map_empty_index + S (eu_index_map_empty) = (0)) -> exists eu_residue_map_empty. (((exists fs_h_eu_map_empty_at. fs_h_eu_map_empty_at + S (eu_residue_map_empty) = S ((S (eu_index_map_empty)) * c)) /\ exists fs_q_eu_map_empty_at. b = fs_q_eu_map_empty_at * S ((S (eu_index_map_empty)) * c) + (eu_residue_map_empty))) /\ ((exists eut_gap_eu_map_empty_bound. eut_gap_eu_map_empty_bound + S (eu_residue_map_empty) = (m)) /\ (exists eu_mod_left_map_empty_mod eu_mod_right_map_empty_mod. ((a)*eu_index_map_empty) + (m) * eu_mod_left_map_empty_mod = (eu_residue_map_empty) + (m) * eu_mod_right_map_empty_mod))

Constructive proof overview

Generated structural guide

The empty multiplier prefix has no residue obligations.

The unchanged tactic script uses 2 declared prerequisites 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

lt_not_le Stable theorem; checked-use authorized zero_le Stable theorem; checked-use authorized

Direct dependents

EU0009 euler_multiplier_prefix_exists

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 · 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.

01Fix variables and assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro i
  6. L6
    intro hi
02Separate the logical casesL7–7

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

  1. L7
    exfalso
03Use earlier factsL8–13

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

  1. L8
    specialize lt_not_le (i)
  2. L9
    specialize lt_not_le (0)
  3. L10
    apply lt_not_le
  4. L11
    exact hi
  5. L12
    specialize zero_le (i)
  6. L13
    apply zero_le

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro b
  4. 0004intro c
  5. 0005intro i
  6. 0006intro hi
  7. 0007exfalso
  8. 0008specialize lt_not_le (i)
  9. 0009specialize lt_not_le (0)
  10. 0010apply lt_not_le
  11. 0011exact hi
  12. 0012specialize zero_le (i)
  13. 0013apply zero_le