PC001A

primorial_prefix_decoded_choice

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

Every actually decoded dense primorial factor has its exact prime-or-one choice.

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 i a. (forall bpr_index_pc_prim_entry_source. (exists bpr_gap_pc_prim_entry_source_bound. bpr_gap_pc_prim_entry_source_bound + S (bpr_index_pc_prim_entry_source) = l) -> exists bpr_value_pc_prim_entry_source. ((((exists bpr_height_pc_prim_entry_source_decoded. bpr_height_pc_prim_entry_source_decoded + S (bpr_value_pc_prim_entry_source) = S ((S (bpr_index_pc_prim_entry_source)) * c)) /\ exists bpr_quotient_pc_prim_entry_source_decoded. b = bpr_quotient_pc_prim_entry_source_decoded * S ((S (bpr_index_pc_prim_entry_source)) * c) + (bpr_value_pc_prim_entry_source))) /\ (((((~(S (bpr_index_pc_prim_entry_source) = 1) /\ forall bpr_left_pc_prim_entry_source_choice_prime bpr_right_pc_prim_entry_source_choice_prime. S (bpr_index_pc_prim_entry_source) = bpr_left_pc_prim_entry_source_choice_prime * bpr_right_pc_prim_entry_source_choice_prime -> bpr_left_pc_prim_entry_source_choice_prime = 1 \/ bpr_right_pc_prim_entry_source_choice_prime = 1)) /\ bpr_value_pc_prim_entry_source = S (bpr_index_pc_prim_entry_source)) \/ (~((~(S (bpr_index_pc_prim_entry_source) = 1) /\ forall bpr_left_pc_prim_entry_source_choice_prime bpr_right_pc_prim_entry_source_choice_prime. S (bpr_index_pc_prim_entry_source) = bpr_left_pc_prim_entry_source_choice_prime * bpr_right_pc_prim_entry_source_choice_prime -> bpr_left_pc_prim_entry_source_choice_prime = 1 \/ bpr_right_pc_prim_entry_source_choice_prime = 1)) /\ bpr_value_pc_prim_entry_source = 1))))) -> (exists pc_lt_prim_entry_bound. pc_lt_prim_entry_bound + S (i) = (l)) -> (((exists fs_h_pc_prim_entry_given. fs_h_pc_prim_entry_given + S (a) = S ((S (i)) * c)) /\ exists fs_q_pc_prim_entry_given. b = fs_q_pc_prim_entry_given * S ((S (i)) * c) + (a))) -> (((((~(S (i) = 1) /\ forall bpr_left_pc_prim_entry_result_prime bpr_right_pc_prim_entry_result_prime. S (i) = bpr_left_pc_prim_entry_result_prime * bpr_right_pc_prim_entry_result_prime -> bpr_left_pc_prim_entry_result_prime = 1 \/ bpr_right_pc_prim_entry_result_prime = 1)) /\ a = S (i)) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_prim_entry_result_prime bpr_right_pc_prim_entry_result_prime. S (i) = bpr_left_pc_prim_entry_result_prime * bpr_right_pc_prim_entry_result_prime -> bpr_left_pc_prim_entry_result_prime = 1 \/ bpr_right_pc_prim_entry_result_prime = 1)) /\ a = 1)))

Constructive proof overview

Generated structural guide

Every actually decoded dense primorial factor has its exact prime-or-one choice.

The unchanged tactic script uses 1 declared prerequisite and contains 26 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

PC001E primorial_cutoff_weighted_lower

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

26 script commands · 6 reading checkpoints · 2 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 assumptionsL1–8

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 i
  5. L5
    intro a
  6. L6
    intro h
  7. L7
    intro hi
  8. L8
    intro ha
02Establish hpL9–12

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

  1. L9
    have hp : ∃ v. BetaAt(b,c,i,v) ∧ (Prime(S i) ∧ v = S i ∨ ¬Prime(S i) ∧ v = 1)Definitions: PrimeBetaAt
  2. L10
    specialize h i
  3. L11
    apply h
  4. L12
    exact hi
03Separate the logical casesL13–14

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

  1. L13
    cases hp
  2. L14
    cases hp_witness
04Establish heqL15–24

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

  1. L15
    have heq : x = a
  2. L16
    specialize beta_at_unique b
  3. L17
    specialize beta_at_unique c
  4. L18
    specialize beta_at_unique i
  5. L19
    specialize beta_at_unique x
  6. L20
    specialize beta_at_unique a
  7. L21
    apply beta_at_unique
  8. L22
    exact hp_witness_left
  9. L23
    exact ha
  10. L24
    rewrite heq at hp_witness_right
05Calculate and transport equalitiesL25–25

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L25
    rewrite heq at hp_witness_right
06Use earlier factsL26–26

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

  1. L26
    exact hp_witness_right

Library-wide reading audit

Original exact command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro a
  6. 0006intro h
  7. 0007intro hi
  8. 0008intro ha
  9. 0009have hp : exists v. (((exists fs_h_pc_prim_entry_actual. fs_h_pc_prim_entry_actual + S (v) = S ((S (i)) * c)) /\ exists fs_q_pc_prim_entry_actual. b = fs_q_pc_prim_entry_actual * S ((S (i)) * c) + (v))) /\ (((((~(S (i) = 1) /\ forall bpr_left_pc_prim_entry_choice_prime bpr_right_pc_prim_entry_choice_prime. S (i) = bpr_left_pc_prim_entry_choice_prime * bpr_right_pc_prim_entry_choice_prime -> bpr_left_pc_prim_entry_choice_prime = 1 \/ bpr_right_pc_prim_entry_choice_prime = 1)) /\ v = S (i)) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_prim_entry_choice_prime bpr_right_pc_prim_entry_choice_prime. S (i) = bpr_left_pc_prim_entry_choice_prime * bpr_right_pc_prim_entry_choice_prime -> bpr_left_pc_prim_entry_choice_prime = 1 \/ bpr_right_pc_prim_entry_choice_prime = 1)) /\ v = 1)))
  10. 0010specialize h i
  11. 0011apply h
  12. 0012exact hi
  13. 0013cases hp
  14. 0014cases hp_witness
  15. 0015have heq : x = a
  16. 0016specialize beta_at_unique b
  17. 0017specialize beta_at_unique c
  18. 0018specialize beta_at_unique i
  19. 0019specialize beta_at_unique x
  20. 0020specialize beta_at_unique a
  21. 0021apply beta_at_unique
  22. 0022exact hp_witness_left
  23. 0023exact ha
  24. 0024rewrite heq at hp_witness_right
  25. 0025rewrite heq at hp_witness_right
  26. 0026exact hp_witness_right