BT001L · Bertrand theorem

mul_le_mul_left

Stable checked-use theorem · independently kernel verified

Left multiplication preserves the witness-defined order.

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. ∀ c. Le(a,b) → Le(c · a,c · 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

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a b c. (exists k. k + a = b) -> exists r. r + c * a = c * b

Proof neighborhood

Direct theorem prerequisites

BT0007 mul_add

Direct theorem dependents

BT00PV mul_le_mul BT00PW le_mul_of_one_le_right BT00QE succ_le_mul_of_two_le_right BT00R2 ceil_div_six_functional BT00RC floor_sqrt_monotone BT00RF ceil_div_six_le_of_upper BT00RG double_triple_remainder_complement_budget BT00VK central_binom_strong_upper_step BT00X0 floor_sqrt_factorized_threshold_thirty_two BT00YA prime_square_tail_of_two_three_range BT00YF division_three_scaled_upper_of_quotient_lt BT00YH floor_sqrt_above_root_power_two_strict BT010E division_quotient_lower_of_scaled_le BT0115 bertrand_eventually_closed_upper BT0117 factor_pair_has_small_member_below_square

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 · 7 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 (1)
mul add
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro h
02Separate the logical casesL5–5

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

  1. L5
    cases h
03Construct an explicit witnessL6–6

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists c * x
04Calculate and transport equalitiesL7–8

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L7
    trans c * (x + a)
  2. L8
    symm
05Use earlier factsL9–9

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

  1. L9
    apply mul_add
06Calculate and transport equalitiesL10–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L10
    congr
  2. L11
    refl
07Use earlier factsL12–12

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

  1. L12
    exact h_witness

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005cases h
  6. 0006exists c * x
  7. 0007trans c * (x + a)
  8. 0008symm
  9. 0009apply mul_add
  10. 0010congr
  11. 0011refl
  12. 0012exact h_witness