PC0007

prime_bit_prefix_all_bits

A primality mask consists of actual zero/one entries.

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

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.

These are the exact finite integer inequalities with constant 8, for every N≥2. The proof uses constructive binomial and primorial infrastructure; it does not assume logarithms, asymptotic estimates, the prime number theorem, or a factorization oracle.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. PrimeBitPrefix(b,c,l) → ∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ (y = 0 ∨ y = 1)

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

Definition DAG

PrimeBitPrefix(b,c,l) · 1Lt(a,b) · 1Prime(p) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

none
Original expanded first-order 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))

Complete tactic proof in conservative notation

All 22 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

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.

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 assumptionsL1–6

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: BetaAt(b,c,i,e)Prime(S i)Original native command in the exact edition
  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 defined 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 : ∃ e. BetaAt(b,c,i,e) ∧ (Prime(S i) ∧ e = 1 ∨ ¬Prime(S i) ∧ 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