BT010A · Bertrand theorem

two_lt_double_lower_six

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

A natural above two has double at least three plus three.

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. Lt(2,n) → Le(3 + 3,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

2 occurrences

In local proof propositions

PD0001 Le

2 occurrences

Exact expanded native-PA statement
forall n. (exists bcf_lt_gap_b5rbtdls_positive. bcf_lt_gap_b5rbtdls_positive + S (2) = n) -> (exists bcf_le_gap_b5rbtdls_result. bcf_le_gap_b5rbtdls_result + (3 + 3) = n + n)

Proof neighborhood

Direct theorem prerequisites

BT0014 add_le_add_right BT0015 add_le_add_left BT000F le_trans

Direct theorem dependents

BT010B floor_sqrt_two_le_of_two_lt BT010D floor_sqrt_three_mul_le_double

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

20 script commands · 4 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)
add le add rightadd le add leftle trans
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hpositive
02Establish hleftL3–8

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

  1. L3
    have hleft : Le(3 + 3,n + 3)Definitions: Le(3 + 3,n + 3)Original native command in the exact edition
  2. L4
    specialize add_le_add_right 3
  3. L5
    specialize add_le_add_right n
  4. L6
    specialize add_le_add_right 3
  5. L7
    apply add_le_add_right
  6. L8
    exact hpositive
03Establish hrightL9–18

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

  1. L9
    have hright : Le(n + 3,n + n)Definitions: Le(n + 3,n + n)Original native command in the exact edition
  2. L10
    specialize add_le_add_left 3
  3. L11
    specialize add_le_add_left n
  4. L12
    specialize add_le_add_left n
  5. L13
    apply add_le_add_left
  6. L14
    exact hpositive
  7. L15
    specialize le_trans (3 + 3)
  8. L16
    specialize le_trans (n + 3)
  9. L17
    specialize le_trans (n + n)
  10. L18
    apply le_trans
04Use earlier factsL19–20

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

  1. L19
    exact hleft
  2. L20
    exact hright

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro n
  2. 0002intro hpositive
  3. 0003have hleft : Le(3 + 3,n + 3)
    Exact native replay linehave hleft : exists bcf_le_gap_b5rbtdls_left. bcf_le_gap_b5rbtdls_left + (3 + 3) = n + 3
  4. 0004specialize add_le_add_right 3
  5. 0005specialize add_le_add_right n
  6. 0006specialize add_le_add_right 3
  7. 0007apply add_le_add_right
  8. 0008exact hpositive
  9. 0009have hright : Le(n + 3,n + n)
    Exact native replay linehave hright : exists bcf_le_gap_b5rbtdls_right. bcf_le_gap_b5rbtdls_right + (n + 3) = n + n
  10. 0010specialize add_le_add_left 3
  11. 0011specialize add_le_add_left n
  12. 0012specialize add_le_add_left n
  13. 0013apply add_le_add_left
  14. 0014exact hpositive
  15. 0015specialize le_trans (3 + 3)
  16. 0016specialize le_trans (n + 3)
  17. 0017specialize le_trans (n + n)
  18. 0018apply le_trans
  19. 0019exact hleft
  20. 0020exact hright