BT00PW · Bertrand theorem

le_mul_of_one_le_right

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

A factor at least one makes right 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.

Statement with defined notation

∀ a. ∀ b. Lt(0,b) → Le(a,a · b)

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

1 occurrences

Exact expanded native-PA statement
forall a b. (exists bpo_gap_right_factor. bpo_gap_right_factor + (1) = (b)) -> (exists bpo_gap_right_result. bpo_gap_right_result + (a) = (a * b))

Proof neighborhood

Direct theorem prerequisites

BT001L mul_le_mul_left BT000A mul_one

Direct theorem dependents

BT00Q0 one_le_pow BT00SN pow_exponent_monotone_from_total BT00XJ pow_le_pow_of_exponent_le BT0115 bertrand_eventually_closed_upper

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 · 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.

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 (2)
mul le mul leftmul one
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 hb
02Establish hscaledL4–12

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

  1. L4
    have hscaled : Le(a · 1,a · b)Definitions: Le(a · 1,a · b)Original native command in the exact edition
  2. L5
    specialize mul_le_mul_left 1
  3. L6
    specialize mul_le_mul_left b
  4. L7
    specialize mul_le_mul_left a
  5. L8
    apply mul_le_mul_left
  6. L9
    exact hb
  7. L10
    specialize mul_one a
  8. L11
    rewrite mul_one at hscaled
  9. L12
    exact hscaled

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hb
  4. 0004have hscaled : Le(a · 1,a · b)
    Exact native replay linehave hscaled : exists k. k + a * 1 = a * b
  5. 0005specialize mul_le_mul_left 1
  6. 0006specialize mul_le_mul_left b
  7. 0007specialize mul_le_mul_left a
  8. 0008apply mul_le_mul_left
  9. 0009exact hb
  10. 0010specialize mul_one a
  11. 0011rewrite mul_one at hscaled
  12. 0012exact hscaled