PC0013

beta_cutoff_prefix_exists

Construct the complete finite cutoff table by actual length induction.

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

∀ u. ∀ b. ∀ c. ∀ l. ∃ d. ∃ f. BetaCutoffPrefix(u,b,c,d,f,l)

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

Definition DAG

BetaCutoffPrefix(u,b,c,d,f,l) · 2Le(a,b) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

beta_cutoff_prefix_emptybeta_cutoff_choice_existsbeta_cutoff_prefix_extend
Original expanded first-order 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))))))

Complete tactic proof in conservative notation

All 33 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (3)
beta cutoff prefix emptybeta cutoff choice existsbeta cutoff prefix extend
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(u,b,c,d,f,l)Original native command in the exact edition
  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 : ∃ e. Lt(l,u) ∧ e = 0 ∨ Le(u,l) ∧ BetaAt(b,c,l,e)Definitions: Lt(l,u)Le(u,l)BetaAt(b,c,l,e)Original native command in the exact edition
  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 defined 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 : ∃ d. ∃ f. BetaCutoffPrefix(u,b,c,d,f,l)
  14. 0014apply IH
  15. 0015cases hp
  16. 0016cases hp_witness
  17. 0017have hc : ∃ e. Lt(l,u) ∧ e = 0 ∨ Le(u,l) ∧ BetaAt(b,c,l,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