PC0002

prime_bit_prefix_empty

The empty primality bit prefix is valid.

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. PrimeBitPrefix(b,c,0)

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

Definition DAG

PrimeBitPrefix(b,c,l) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order 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)))

Complete tactic proof in conservative notation

All 11 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

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.

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 assumptionsL1–4

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 defined 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