PC000E

beta_cutoff_choice_exists

Construct each exact cutoff entry by decidable order and beta decoding.

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. ∀ i. ∃ e. Lt(i,u) ∧ e = 0 ∨ Le(u,i) ∧ BetaAt(b,c,i,e)

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

Definition DAG

Le(a,b) · 2Lt(a,b) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

le_or_lt · checked external prerequisitebeta_at_exists · checked external prerequisite
Original expanded first-order statement 
forall u b c i. exists e. (((exists pc_lt_cutchoice_exists_below. pc_lt_cutchoice_exists_below + S (i) = (u)) /\ e = 0) \/ ((exists pc_le_cutchoice_exists_above. pc_le_cutchoice_exists_above + (u) = (i)) /\ (((exists fs_h_pc_cutchoice_exists_source. fs_h_pc_cutchoice_exists_source + S (e) = S ((S (i)) * c)) /\ exists fs_q_pc_cutchoice_exists_source. b = fs_q_pc_cutchoice_exists_source * S ((S (i)) * c) + (e)))))

Complete tactic proof in conservative notation

All 25 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

25 script commands · 12 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro u
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro i
02Establish hsL5–8

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

  1. L5
    have hs : Le(u,i) ∨ Lt(i,u)Definitions: Le(u,i)Lt(i,u)Original native command in the exact edition
  2. L6
    specialize le_or_lt u
  3. L7
    specialize le_or_lt i
  4. L8
    apply le_or_lt
03Separate the logical casesL9–9

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

  1. L9
    cases hs
04Establish heL10–14

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

  1. L10
    have he : ∃ e. BetaAt(b,c,i,e)Definitions: BetaAt(b,c,i,e)Original native command in the exact edition
  2. L11
    specialize beta_at_exists b
  3. L12
    specialize beta_at_exists c
  4. L13
    specialize beta_at_exists i
  5. L14
    apply beta_at_exists
05Separate the logical casesL15–15

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

  1. L15
    cases he
06Construct an explicit witnessL16–16

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

  1. L16
    exists x
07Separate the logical casesL17–18

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

  1. L17
    right
  2. L18
    split
08Use earlier factsL19–20

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

  1. L19
    exact hs_left
  2. L20
    exact he_witness
09Construct an explicit witnessL21–21

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

  1. L21
    exists 0
10Separate the logical casesL22–23

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

  1. L22
    left
  2. L23
    split
11Use earlier factsL24–24

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

  1. L24
    exact hs_right
12Calculate and transport equalitiesL25–25

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

  1. L25
    refl

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro u
  2. 0002intro b
  3. 0003intro c
  4. 0004intro i
  5. 0005have hs : Le(u,i) ∨ Lt(i,u)
  6. 0006specialize le_or_lt u
  7. 0007specialize le_or_lt i
  8. 0008apply le_or_lt
  9. 0009cases hs
  10. 0010have he : ∃ e. BetaAt(b,c,i,e)
  11. 0011specialize beta_at_exists b
  12. 0012specialize beta_at_exists c
  13. 0013specialize beta_at_exists i
  14. 0014apply beta_at_exists
  15. 0015cases he
  16. 0016exists x
  17. 0017right
  18. 0018split
  19. 0019exact hs_left
  20. 0020exact he_witness
  21. 0021exists 0
  22. 0022left
  23. 0023split
  24. 0024exact hs_right
  25. 0025refl