PC0007

prime_bit_prefix_all_bits

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

A primality mask consists of actual zero/one entries.

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 l. (forall pc_index_mask_bits_source. (exists pc_lt_mask_bits_source_bound. pc_lt_mask_bits_source_bound + S (pc_index_mask_bits_source) = (l)) -> exists pc_bit_mask_bits_source. (((exists fs_h_pc_mask_bits_source_entry. fs_h_pc_mask_bits_source_entry + S (pc_bit_mask_bits_source) = S ((S (pc_index_mask_bits_source)) * c)) /\ exists fs_q_pc_mask_bits_source_entry. b = fs_q_pc_mask_bits_source_entry * S ((S (pc_index_mask_bits_source)) * c) + (pc_bit_mask_bits_source))) /\ (((((~(S (pc_index_mask_bits_source) = 1) /\ forall bpr_left_pc_mask_bits_source_choice_prime bpr_right_pc_mask_bits_source_choice_prime. S (pc_index_mask_bits_source) = bpr_left_pc_mask_bits_source_choice_prime * bpr_right_pc_mask_bits_source_choice_prime -> bpr_left_pc_mask_bits_source_choice_prime = 1 \/ bpr_right_pc_mask_bits_source_choice_prime = 1)) /\ pc_bit_mask_bits_source = 1) \/ (~((~(S (pc_index_mask_bits_source) = 1) /\ forall bpr_left_pc_mask_bits_source_choice_prime bpr_right_pc_mask_bits_source_choice_prime. S (pc_index_mask_bits_source) = bpr_left_pc_mask_bits_source_choice_prime * bpr_right_pc_mask_bits_source_choice_prime -> bpr_left_pc_mask_bits_source_choice_prime = 1 \/ bpr_right_pc_mask_bits_source_choice_prime = 1)) /\ pc_bit_mask_bits_source = 0)))) -> (forall pc_index_mask_bits_target. (exists pc_lt_mask_bits_target_bound. pc_lt_mask_bits_target_bound + S (pc_index_mask_bits_target) = (l)) -> exists pc_bit_mask_bits_target. (((exists fs_h_pc_mask_bits_target_entry. fs_h_pc_mask_bits_target_entry + S (pc_bit_mask_bits_target) = S ((S (pc_index_mask_bits_target)) * c)) /\ exists fs_q_pc_mask_bits_target_entry. b = fs_q_pc_mask_bits_target_entry * S ((S (pc_index_mask_bits_target)) * c) + (pc_bit_mask_bits_target))) /\ (pc_bit_mask_bits_target = 0 \/ pc_bit_mask_bits_target = 1))

Constructive proof overview

Generated structural guide

A primality mask consists of actual zero/one entries.

The unchanged tactic script uses 0 declared prerequisites and contains 22 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

PC0009 prime_count_bounded PC002A prime_count_chebyshev_upper

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

22 script commands · 10 reading checkpoints · 1 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro h
  5. L5
    intro i
  6. L6
    intro hi
02Establish hpL7–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply h.

  1. L7
    have hp : ∃ e. BetaAt(b,c,i,e) ∧ (Prime(S i) ∧ e = 1 ∨ ¬Prime(S i) ∧ e = 0)Definitions: PrimeBetaAt
  2. L8
    specialize h i
  3. L9
    apply h
  4. L10
    exact hi
03Separate the logical casesL11–12

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

  1. L11
    cases hp
  2. L12
    cases hp_witness
04Construct an explicit witnessL13–13

Supply the displayed value, then prove that it has the required property.

  1. L13
    exists x
05Separate the logical casesL14–14

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

  1. L14
    split
06Use earlier factsL15–15

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

  1. L15
    exact hp_witness_left
07Separate the logical casesL16–18

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

  1. L16
    cases hp_witness_right
  2. L17
    cases hp_witness_right_left
  3. L18
    right
08Use earlier factsL19–19

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

  1. L19
    exact hp_witness_right_left_right
09Separate the logical casesL20–21

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

  1. L20
    cases hp_witness_right_right
  2. L21
    left
10Use earlier factsL22–22

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

  1. L22
    exact hp_witness_right_right_right

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro h
  5. 0005intro i
  6. 0006intro hi
  7. 0007have hp : exists e. (((exists fs_h_pc_mask_bits_entry. fs_h_pc_mask_bits_entry + S (e) = S ((S (i)) * c)) /\ exists fs_q_pc_mask_bits_entry. b = fs_q_pc_mask_bits_entry * S ((S (i)) * c) + (e))) /\ (((((~(S (i) = 1) /\ forall bpr_left_pc_mask_bits_choice_prime bpr_right_pc_mask_bits_choice_prime. S (i) = bpr_left_pc_mask_bits_choice_prime * bpr_right_pc_mask_bits_choice_prime -> bpr_left_pc_mask_bits_choice_prime = 1 \/ bpr_right_pc_mask_bits_choice_prime = 1)) /\ e = 1) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_mask_bits_choice_prime bpr_right_pc_mask_bits_choice_prime. S (i) = bpr_left_pc_mask_bits_choice_prime * bpr_right_pc_mask_bits_choice_prime -> bpr_left_pc_mask_bits_choice_prime = 1 \/ bpr_right_pc_mask_bits_choice_prime = 1)) /\ e = 0)))
  8. 0008specialize h i
  9. 0009apply h
  10. 0010exact hi
  11. 0011cases hp
  12. 0012cases hp_witness
  13. 0013exists x
  14. 0014split
  15. 0015exact hp_witness_left
  16. 0016cases hp_witness_right
  17. 0017cases hp_witness_right_left
  18. 0018right
  19. 0019exact hp_witness_right_left_right
  20. 0020cases hp_witness_right_right
  21. 0021left
  22. 0022exact hp_witness_right_right_right