BT010A

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.

Exact expanded 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)

Structural proof guide

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

Direct prerequisites: add_le_add_right, add_le_add_left, le_trans. The authored body proceeds by intermediate claims (2).

Proof neighborhood

Direct dependencies

BT0014 add_le_add_right BT0015 add_le_add_left BT000F le_trans

Direct dependents

BT010B floor_sqrt_two_le_of_two_lt BT010D floor_sqrt_three_mul_le_double

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

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.

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 : exists bcf_le_gap_b5rbtdls_left. bcf_le_gap_b5rbtdls_left + (3 + 3) = n + 3
  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 : exists bcf_le_gap_b5rbtdls_right. bcf_le_gap_b5rbtdls_right + (n + 3) = n + n
  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 exact command ledger · 20 lines
  1. 0001intro n
  2. 0002intro hpositive
  3. 0003have 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 : 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