PC0013

beta_cutoff_prefix_exists

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

Construct the complete finite cutoff table by actual length induction.

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 u b c l. exists d f. forall pc_index_cut_exists. (exists pc_lt_cut_exists_bound. pc_lt_cut_exists_bound + S (pc_index_cut_exists) = (l)) -> exists pc_bit_cut_exists. (((exists fs_h_pc_cut_exists_entry. fs_h_pc_cut_exists_entry + S (pc_bit_cut_exists) = S ((S (pc_index_cut_exists)) * f)) /\ exists fs_q_pc_cut_exists_entry. d = fs_q_pc_cut_exists_entry * S ((S (pc_index_cut_exists)) * f) + (pc_bit_cut_exists))) /\ ((((exists pc_lt_cut_exists_choice_below. pc_lt_cut_exists_choice_below + S (pc_index_cut_exists) = (u)) /\ pc_bit_cut_exists = 0) \/ ((exists pc_le_cut_exists_choice_above. pc_le_cut_exists_choice_above + (u) = (pc_index_cut_exists)) /\ (((exists fs_h_pc_cut_exists_choice_source. fs_h_pc_cut_exists_choice_source + S (pc_bit_cut_exists) = S ((S (pc_index_cut_exists)) * c)) /\ exists fs_q_pc_cut_exists_choice_source. b = fs_q_pc_cut_exists_choice_source * S ((S (pc_index_cut_exists)) * c) + (pc_bit_cut_exists))))))

Constructive proof overview

Generated structural guide

Construct the complete finite cutoff table by actual length induction.

The unchanged tactic script uses 3 declared prerequisites and contains 33 exact native proof lines.

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

Proof neighborhood

Direct dependencies

PC000F beta_cutoff_prefix_empty PC000E beta_cutoff_choice_exists PC0012 beta_cutoff_prefix_extend

Direct dependents

PC002A prime_count_chebyshev_upper

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

33 script commands · 9 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.

Named ingredients (3)
beta cutoff prefix emptybeta cutoff choice existsbeta cutoff prefix extend

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 assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro u
  2. L2
    intro b
  3. L3
    intro c
02Induction on lL4–4

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L4
    induction l
03Construct an explicit witnessL5–6

Supply the displayed value, then prove that it has the required property.

  1. L5
    exists 0
  2. L6
    exists 0
04Use earlier factsL7–12

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

  1. L7
    specialize beta_cutoff_prefix_empty u
  2. L8
    specialize beta_cutoff_prefix_empty b
  3. L9
    specialize beta_cutoff_prefix_empty c
  4. L10
    specialize beta_cutoff_prefix_empty 0
  5. L11
    specialize beta_cutoff_prefix_empty 0
  6. L12
    apply beta_cutoff_prefix_empty
05Establish hpL13–14

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

  1. L13
    have hp : ∃ d. ∃ f. BetaCutoffPrefix(u,b,c,d,f,l)Definitions: BetaCutoffPrefix
  2. L14
    apply IH
06Separate the logical casesL15–16

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

  1. L15
    cases hp
  2. L16
    cases hp_witness
07Establish hcL17–22

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

  1. L17
    have hc : exists e. (((exists pc_lt_cut_exists_choice_below. pc_lt_cut_exists_choice_below + S (l) = (u)) /\ e = 0) \/ ((exists pc_le_cut_exists_choice_above. pc_le_cut_exists_choice_above + (u) = (l)) /\ (((exists fs_h_pc_cut_exists_choice_source. fs_h_pc_cut_exists_choice_source + S (e) = S ((S (l)) * c)) /\ exists fs_q_pc_cut_exists_choice_source. b = fs_q_pc_cut_exists_choice_source * S ((S (l)) * c) + (e)))))
  2. L18
    specialize beta_cutoff_choice_exists u
  3. L19
    specialize beta_cutoff_choice_exists b
  4. L20
    specialize beta_cutoff_choice_exists c
  5. L21
    specialize beta_cutoff_choice_exists l
  6. L22
    apply beta_cutoff_choice_exists
08Separate the logical casesL23–23

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

  1. L23
    cases hc
09Use earlier factsL24–33

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

  1. L24
    specialize beta_cutoff_prefix_extend u
  2. L25
    specialize beta_cutoff_prefix_extend b
  3. L26
    specialize beta_cutoff_prefix_extend c
  4. L27
    specialize beta_cutoff_prefix_extend x
  5. L28
    specialize beta_cutoff_prefix_extend x1
  6. L29
    specialize beta_cutoff_prefix_extend l
  7. L30
    specialize beta_cutoff_prefix_extend x2
  8. L31
    apply beta_cutoff_prefix_extend
  9. L32
    exact hp_witness_witness
  10. L33
    exact hc_witness

Library-wide reading audit

Original exact command ledger · 33 lines
  1. 0001intro u
  2. 0002intro b
  3. 0003intro c
  4. 0004induction l
  5. 0005exists 0
  6. 0006exists 0
  7. 0007specialize beta_cutoff_prefix_empty u
  8. 0008specialize beta_cutoff_prefix_empty b
  9. 0009specialize beta_cutoff_prefix_empty c
  10. 0010specialize beta_cutoff_prefix_empty 0
  11. 0011specialize beta_cutoff_prefix_empty 0
  12. 0012apply beta_cutoff_prefix_empty
  13. 0013have hp : exists d f. forall pc_index_cut_exists_pre. (exists pc_lt_cut_exists_pre_bound. pc_lt_cut_exists_pre_bound + S (pc_index_cut_exists_pre) = (l)) -> exists pc_bit_cut_exists_pre. (((exists fs_h_pc_cut_exists_pre_entry. fs_h_pc_cut_exists_pre_entry + S (pc_bit_cut_exists_pre) = S ((S (pc_index_cut_exists_pre)) * f)) /\ exists fs_q_pc_cut_exists_pre_entry. d = fs_q_pc_cut_exists_pre_entry * S ((S (pc_index_cut_exists_pre)) * f) + (pc_bit_cut_exists_pre))) /\ ((((exists pc_lt_cut_exists_pre_choice_below. pc_lt_cut_exists_pre_choice_below + S (pc_index_cut_exists_pre) = (u)) /\ pc_bit_cut_exists_pre = 0) \/ ((exists pc_le_cut_exists_pre_choice_above. pc_le_cut_exists_pre_choice_above + (u) = (pc_index_cut_exists_pre)) /\ (((exists fs_h_pc_cut_exists_pre_choice_source. fs_h_pc_cut_exists_pre_choice_source + S (pc_bit_cut_exists_pre) = S ((S (pc_index_cut_exists_pre)) * c)) /\ exists fs_q_pc_cut_exists_pre_choice_source. b = fs_q_pc_cut_exists_pre_choice_source * S ((S (pc_index_cut_exists_pre)) * c) + (pc_bit_cut_exists_pre))))))
  14. 0014apply IH
  15. 0015cases hp
  16. 0016cases hp_witness
  17. 0017have hc : exists e. (((exists pc_lt_cut_exists_choice_below. pc_lt_cut_exists_choice_below + S (l) = (u)) /\ e = 0) \/ ((exists pc_le_cut_exists_choice_above. pc_le_cut_exists_choice_above + (u) = (l)) /\ (((exists fs_h_pc_cut_exists_choice_source. fs_h_pc_cut_exists_choice_source + S (e) = S ((S (l)) * c)) /\ exists fs_q_pc_cut_exists_choice_source. b = fs_q_pc_cut_exists_choice_source * S ((S (l)) * c) + (e)))))
  18. 0018specialize beta_cutoff_choice_exists u
  19. 0019specialize beta_cutoff_choice_exists b
  20. 0020specialize beta_cutoff_choice_exists c
  21. 0021specialize beta_cutoff_choice_exists l
  22. 0022apply beta_cutoff_choice_exists
  23. 0023cases hc
  24. 0024specialize beta_cutoff_prefix_extend u
  25. 0025specialize beta_cutoff_prefix_extend b
  26. 0026specialize beta_cutoff_prefix_extend c
  27. 0027specialize beta_cutoff_prefix_extend x
  28. 0028specialize beta_cutoff_prefix_extend x1
  29. 0029specialize beta_cutoff_prefix_extend l
  30. 0030specialize beta_cutoff_prefix_extend x2
  31. 0031apply beta_cutoff_prefix_extend
  32. 0032exact hp_witness_witness
  33. 0033exact hc_witness