PC0001

prime_bit_choice_exists

Construct the zero/one primality indicator at each dense index.

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

∀ i. ∃ e. Prime(S i) ∧ e = 1 ∨ ¬Prime(S i) ∧ e = 0

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

Definition DAG

Prime(p) · 4

Actual proof prerequisites

prime_decidable · checked external prerequisite
Original expanded first-order statement 
forall i. exists e. ((((~(S (i) = 1) /\ forall bpr_left_pc_choice_exists_prime bpr_right_pc_choice_exists_prime. S (i) = bpr_left_pc_choice_exists_prime * bpr_right_pc_choice_exists_prime -> bpr_left_pc_choice_exists_prime = 1 \/ bpr_right_pc_choice_exists_prime = 1)) /\ e = 1) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_choice_exists_prime bpr_right_pc_choice_exists_prime. S (i) = bpr_left_pc_choice_exists_prime * bpr_right_pc_choice_exists_prime -> bpr_left_pc_choice_exists_prime = 1 \/ bpr_right_pc_choice_exists_prime = 1)) /\ e = 0))

Complete tactic proof in conservative notation

All 15 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

15 script commands · 11 reading checkpoints · 1 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 assumptionsL1–1

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

  1. L1
    intro i
02Establish hpL2–4

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

  1. L2
    have hp : Prime(S i) ∨ ¬Prime(S i)Definitions: Prime(S i)Original native command in the exact edition
  2. L3
    specialize prime_decidable (S i)
  3. L4
    apply prime_decidable
03Separate the logical casesL5–5

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

  1. L5
    cases hp
04Construct an explicit witnessL6–6

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

  1. L6
    exists 1
05Separate the logical casesL7–8

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

  1. L7
    left
  2. L8
    split
06Use earlier factsL9–9

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

  1. L9
    exact hp_left
07Calculate and transport equalitiesL10–10

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

  1. L10
    refl
08Construct an explicit witnessL11–11

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

  1. L11
    exists 0
09Separate the logical casesL12–13

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

  1. L12
    right
  2. L13
    split
10Use earlier factsL14–14

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

  1. L14
    exact hp_right
11Calculate and transport equalitiesL15–15

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

  1. L15
    refl

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro i
  2. 0002have hp : Prime(S i) ∨ ¬Prime(S i)
  3. 0003specialize prime_decidable (S i)
  4. 0004apply prime_decidable
  5. 0005cases hp
  6. 0006exists 1
  7. 0007left
  8. 0008split
  9. 0009exact hp_left
  10. 0010refl
  11. 0011exists 0
  12. 0012right
  13. 0013split
  14. 0014exact hp_right
  15. 0015refl