PC0003

prime_bit_prefix_drop_last

A primality mask restricts to its preceding prefix.

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,S l) → PrimeBitPrefix(b,c,l)

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

Definition DAG

PrimeBitPrefix(b,c,l) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall b c l. (forall pc_index_mask_drop_source. (exists pc_lt_mask_drop_source_bound. pc_lt_mask_drop_source_bound + S (pc_index_mask_drop_source) = (S l)) -> exists pc_bit_mask_drop_source. (((exists fs_h_pc_mask_drop_source_entry. fs_h_pc_mask_drop_source_entry + S (pc_bit_mask_drop_source) = S ((S (pc_index_mask_drop_source)) * c)) /\ exists fs_q_pc_mask_drop_source_entry. b = fs_q_pc_mask_drop_source_entry * S ((S (pc_index_mask_drop_source)) * c) + (pc_bit_mask_drop_source))) /\ (((((~(S (pc_index_mask_drop_source) = 1) /\ forall bpr_left_pc_mask_drop_source_choice_prime bpr_right_pc_mask_drop_source_choice_prime. S (pc_index_mask_drop_source) = bpr_left_pc_mask_drop_source_choice_prime * bpr_right_pc_mask_drop_source_choice_prime -> bpr_left_pc_mask_drop_source_choice_prime = 1 \/ bpr_right_pc_mask_drop_source_choice_prime = 1)) /\ pc_bit_mask_drop_source = 1) \/ (~((~(S (pc_index_mask_drop_source) = 1) /\ forall bpr_left_pc_mask_drop_source_choice_prime bpr_right_pc_mask_drop_source_choice_prime. S (pc_index_mask_drop_source) = bpr_left_pc_mask_drop_source_choice_prime * bpr_right_pc_mask_drop_source_choice_prime -> bpr_left_pc_mask_drop_source_choice_prime = 1 \/ bpr_right_pc_mask_drop_source_choice_prime = 1)) /\ pc_bit_mask_drop_source = 0)))) -> (forall pc_index_mask_drop_target. (exists pc_lt_mask_drop_target_bound. pc_lt_mask_drop_target_bound + S (pc_index_mask_drop_target) = (l)) -> exists pc_bit_mask_drop_target. (((exists fs_h_pc_mask_drop_target_entry. fs_h_pc_mask_drop_target_entry + S (pc_bit_mask_drop_target) = S ((S (pc_index_mask_drop_target)) * c)) /\ exists fs_q_pc_mask_drop_target_entry. b = fs_q_pc_mask_drop_target_entry * S ((S (pc_index_mask_drop_target)) * c) + (pc_bit_mask_drop_target))) /\ (((((~(S (pc_index_mask_drop_target) = 1) /\ forall bpr_left_pc_mask_drop_target_choice_prime bpr_right_pc_mask_drop_target_choice_prime. S (pc_index_mask_drop_target) = bpr_left_pc_mask_drop_target_choice_prime * bpr_right_pc_mask_drop_target_choice_prime -> bpr_left_pc_mask_drop_target_choice_prime = 1 \/ bpr_right_pc_mask_drop_target_choice_prime = 1)) /\ pc_bit_mask_drop_target = 1) \/ (~((~(S (pc_index_mask_drop_target) = 1) /\ forall bpr_left_pc_mask_drop_target_choice_prime bpr_right_pc_mask_drop_target_choice_prime. S (pc_index_mask_drop_target) = bpr_left_pc_mask_drop_target_choice_prime * bpr_right_pc_mask_drop_target_choice_prime -> bpr_left_pc_mask_drop_target_choice_prime = 1 \/ bpr_right_pc_mask_drop_target_choice_prime = 1)) /\ pc_bit_mask_drop_target = 0))))

Complete tactic proof in conservative notation

All 12 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

12 script commands · 2 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 assumptionsL1–6

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
02Use earlier factsL7–12

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

  1. L7
    specialize h i
  2. L8
    apply h
  3. L9
    specialize le_succ (S i)
  4. L10
    specialize le_succ l
  5. L11
    apply le_succ
  6. L12
    exact hi

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro h
  5. 0005intro i
  6. 0006intro hi
  7. 0007specialize h i
  8. 0008apply h
  9. 0009specialize le_succ (S i)
  10. 0010specialize le_succ l
  11. 0011apply le_succ
  12. 0012exact hi