BT00PX

le_mul_of_one_le_left

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

A factor at least one makes left multiplication extensive.

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 a b. (exists bpo_gap_left_factor. bpo_gap_left_factor + (1) = (a)) -> (exists bpo_gap_left_result. bpo_gap_left_result + (b) = (a * b))

Structural proof guide

A factor at least one makes left multiplication extensive.

Direct prerequisites: mul_le_mul_right, one_mul. The authored body proceeds by intermediate claims (1), equality transport (1).

Proof neighborhood

Direct dependencies

BT001M mul_le_mul_right BT0009 one_mul

Direct dependents

BT010H division_quotient_le_dividend BT0111 no_bertrand_middle_contribution_interval_le_four_pow

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

12 script commands · 2 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.

Named ingredients (2)
mul le mul rightone mul
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro ha
02Establish hscaledL4–12

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

  1. L4
    have hscaled : exists k. k + 1 * b = a * b
  2. L5
    specialize mul_le_mul_right 1
  3. L6
    specialize mul_le_mul_right a
  4. L7
    specialize mul_le_mul_right b
  5. L8
    apply mul_le_mul_right
  6. L9
    exact ha
  7. L10
    specialize one_mul b
  8. L11
    rewrite one_mul at hscaled
  9. L12
    exact hscaled

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro ha
  4. 0004have hscaled : exists k. k + 1 * b = a * b
  5. 0005specialize mul_le_mul_right 1
  6. 0006specialize mul_le_mul_right a
  7. 0007specialize mul_le_mul_right b
  8. 0008apply mul_le_mul_right
  9. 0009exact ha
  10. 0010specialize one_mul b
  11. 0011rewrite one_mul at hscaled
  12. 0012exact hscaled