BT0125 · Bertrand theorem

bertrand_closed_upper

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Every nonzero natural has a prime in its open-closed Bertrand interval.

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.

Statement with defined notation

∀ n. ¬n = 0 → ∃ x. Prime(x) ∧ (Lt(n,x) ∧ Le(x,n + n))

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0001 Le PD0002 Lt PD0004 Prime

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n. ~(n = 0) -> exists p. ((~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1) /\ ((exists u. u + S n = p) /\ (exists v. v + p = n + n)))

Proof neighborhood

Direct theorem prerequisites

BT001G le_or_lt BT0115 bertrand_eventually_closed_upper BT0124 bertrand_small_closed_upper

Direct theorem dependents

BT0127 bertrand_strict

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

12 script commands · 4 reading checkpoints · 0 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)
le or ltbertrand eventually closed upperbertrand small closed upper
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hnonzero
02Use earlier factsL3–4

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

  1. L3
    specialize le_or_lt (16 * 32)
  2. L4
    specialize le_or_lt n
03Separate the logical casesL5–5

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

  1. L5
    cases le_or_lt
04Use earlier factsL6–12

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

  1. L6
    specialize bertrand_eventually_closed_upper n
  2. L7
    apply bertrand_eventually_closed_upper
  3. L8
    exact le_or_lt_left
  4. L9
    specialize bertrand_small_closed_upper n
  5. L10
    apply bertrand_small_closed_upper
  6. L11
    exact hnonzero
  7. L12
    exact le_or_lt_right

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro n
  2. 0002intro hnonzero
  3. 0003specialize le_or_lt (16 * 32)
  4. 0004specialize le_or_lt n
  5. 0005cases le_or_lt
  6. 0006specialize bertrand_eventually_closed_upper n
  7. 0007apply bertrand_eventually_closed_upper
  8. 0008exact le_or_lt_left
  9. 0009specialize bertrand_small_closed_upper n
  10. 0010apply bertrand_small_closed_upper
  11. 0011exact hnonzero
  12. 0012exact le_or_lt_right