PE0005

least_prime_above_bertrand_bound

Bertrand's theorem bounds the actual consecutive prime, rather than only producing a sparse prime subsequence.

Alpha v34 checked-use · first admitted v28 · 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.

Every successor is the globally least prime above its predecessor. This is not a sparse Bertrand chain. The bound theorem constructs the list and both power witnesses from k≠0 alone; the separate total-list theorem includes k=0.

Exact theorem in conservative defined notation

∀ a. ∀ p. Prime(a) → NextPrime(a,p) → Lt(p,a + a)

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

Definition DAG

NextPrime(a,p) · 1Lt(a,b) · 3Prime(p) · 2

Actual proof prerequisites

prime_two_le · checked external prerequisitebertrand_strict · checked external prerequisitelt_of_le_of_lt · checked external prerequisite
Original expanded first-order statement 
forall a p. ((~(a = 1) /\ forall bpr_left_pc_pen_bertrand_source bpr_right_pc_pen_bertrand_source. a = bpr_left_pc_pen_bertrand_source * bpr_right_pc_pen_bertrand_source -> bpr_left_pc_pen_bertrand_source = 1 \/ bpr_right_pc_pen_bertrand_source = 1)) -> (((~(p = 1) /\ forall bpr_left_pc_pen_bertrand_next_prime bpr_right_pc_pen_bertrand_next_prime. p = bpr_left_pc_pen_bertrand_next_prime * bpr_right_pc_pen_bertrand_next_prime -> bpr_left_pc_pen_bertrand_next_prime = 1 \/ bpr_right_pc_pen_bertrand_next_prime = 1)) /\ ((exists pc_lt_pen_bertrand_next_greater. pc_lt_pen_bertrand_next_greater + S (a) = (p)) /\ forall pen_comparison_bertrand_next. ((~(pen_comparison_bertrand_next = 1) /\ forall bpr_left_pc_pen_bertrand_next_comparison bpr_right_pc_pen_bertrand_next_comparison. pen_comparison_bertrand_next = bpr_left_pc_pen_bertrand_next_comparison * bpr_right_pc_pen_bertrand_next_comparison -> bpr_left_pc_pen_bertrand_next_comparison = 1 \/ bpr_right_pc_pen_bertrand_next_comparison = 1)) -> (exists pc_lt_pen_bertrand_next_above. pc_lt_pen_bertrand_next_above + S (a) = (pen_comparison_bertrand_next)) -> (exists pc_le_pen_bertrand_next_minimal. pc_le_pen_bertrand_next_minimal + (p) = (pen_comparison_bertrand_next)))) -> (exists pc_lt_pen_bertrand_bound. pc_lt_pen_bertrand_bound + S (p) = (a + a))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 5 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro p
  3. L3
    intro ha
  4. L4
    intro hp
02Separate the logical casesL5–6

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

  1. L5
    cases hp
  2. L6
    cases hp_right
03Establish hwL7–12

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

  1. L7
    have hw : ∃ q. Prime(q) ∧ (Lt(a,q) ∧ Lt(q,a + a))Definitions: Prime(q)Lt(a,q)Lt(q,a + a)Original native command in the exact edition
  2. L8
    specialize bertrand_strict a
  3. L9
    apply bertrand_strict
  4. L10
    specialize prime_two_le a
  5. L11
    apply prime_two_le
  6. L12
    exact ha
04Separate the logical casesL13–15

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

  1. L13
    cases hw
  2. L14
    cases hw_witness
  3. L15
    cases hw_witness_right
05Use earlier factsL16–24

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

  1. L16
    specialize lt_of_le_of_lt p
  2. L17
    specialize lt_of_le_of_lt x
  3. L18
    specialize lt_of_le_of_lt (a + a)
  4. L19
    apply lt_of_le_of_lt
  5. L20
    specialize hp_right_right x
  6. L21
    apply hp_right_right
  7. L22
    exact hw_witness_left
  8. L23
    exact hw_witness_right_left
  9. L24
    exact hw_witness_right_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro p
  3. 0003intro ha
  4. 0004intro hp
  5. 0005cases hp
  6. 0006cases hp_right
  7. 0007have hw : ∃ q. Prime(q) ∧ (Lt(a,q) ∧ Lt(q,a + a))
  8. 0008specialize bertrand_strict a
  9. 0009apply bertrand_strict
  10. 0010specialize prime_two_le a
  11. 0011apply prime_two_le
  12. 0012exact ha
  13. 0013cases hw
  14. 0014cases hw_witness
  15. 0015cases hw_witness_right
  16. 0016specialize lt_of_le_of_lt p
  17. 0017specialize lt_of_le_of_lt x
  18. 0018specialize lt_of_le_of_lt (a + a)
  19. 0019apply lt_of_le_of_lt
  20. 0020specialize hp_right_right x
  21. 0021apply hp_right_right
  22. 0022exact hw_witness_left
  23. 0023exact hw_witness_right_left
  24. 0024exact hw_witness_right_right