PC0003

prime_bit_prefix_drop_last

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

A primality mask restricts to its preceding prefix.

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_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))))

Constructive proof overview

Generated structural guide

A primality mask restricts to its preceding prefix.

The unchanged tactic script uses 1 declared prerequisite and contains 12 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_succ Stable theorem; checked-use authorized

Direct dependents

none

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

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.

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 exact 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