PC0002

prime_bit_prefix_empty

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

The empty primality bit prefix is valid.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall b c. forall pc_index_mask_empty. (exists pc_lt_mask_empty_bound. pc_lt_mask_empty_bound + S (pc_index_mask_empty) = (0)) -> exists pc_bit_mask_empty. (((exists fs_h_pc_mask_empty_entry. fs_h_pc_mask_empty_entry + S (pc_bit_mask_empty) = S ((S (pc_index_mask_empty)) * c)) /\ exists fs_q_pc_mask_empty_entry. b = fs_q_pc_mask_empty_entry * S ((S (pc_index_mask_empty)) * c) + (pc_bit_mask_empty))) /\ (((((~(S (pc_index_mask_empty) = 1) /\ forall bpr_left_pc_mask_empty_choice_prime bpr_right_pc_mask_empty_choice_prime. S (pc_index_mask_empty) = bpr_left_pc_mask_empty_choice_prime * bpr_right_pc_mask_empty_choice_prime -> bpr_left_pc_mask_empty_choice_prime = 1 \/ bpr_right_pc_mask_empty_choice_prime = 1)) /\ pc_bit_mask_empty = 1) \/ (~((~(S (pc_index_mask_empty) = 1) /\ forall bpr_left_pc_mask_empty_choice_prime bpr_right_pc_mask_empty_choice_prime. S (pc_index_mask_empty) = bpr_left_pc_mask_empty_choice_prime * bpr_right_pc_mask_empty_choice_prime -> bpr_left_pc_mask_empty_choice_prime = 1 \/ bpr_right_pc_mask_empty_choice_prime = 1)) /\ pc_bit_mask_empty = 0)))

Constructive proof overview

Generated structural guide

The empty primality bit prefix is valid.

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · 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

PC0006 prime_bit_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

11 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 assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro i
  4. L4
    intro hi
02Separate the logical casesL5–5

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

  1. L5
    exfalso
03Use earlier factsL6–11

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

  1. L6
    specialize lt_not_le i
  2. L7
    specialize lt_not_le 0
  3. L8
    apply lt_not_le
  4. L9
    exact hi
  5. L10
    specialize zero_le i
  6. L11
    apply zero_le

Library-wide reading audit

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