BT00PV · Bertrand theorem

mul_le_mul

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

Multiplication is monotone in both natural-number arguments.

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. ∀ d. Le(a,b) → Le(c,d) → Le(a · c,b · d)

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

3 occurrences

In local proof propositions

PD0001 Le

2 occurrences

Exact expanded native-PA statement
forall a b c d. (exists bpo_gap_mul_left. bpo_gap_mul_left + (a) = (b)) -> (exists bpo_gap_mul_right. bpo_gap_mul_right + (c) = (d)) -> (exists bpo_gap_mul_result. bpo_gap_mul_result + (a * c) = (b * d))

Proof neighborhood

Direct theorem prerequisites

BT001M mul_le_mul_right BT001L mul_le_mul_left BT000F le_trans

Direct theorem dependents

BT00PY pow_base_monotone BT00SW bertrand_h_six_step_transport_from_total BT00SX bertrand_j_six_step_transport_from_total BT00VV primorial_le_four_pow_bounded BT00W6 pow_six_ten_le_pow_four_thirteen_from_total BT00WF pow_six_six_le_pow_four_eight_from_total BT00WG pow_six_four_le_pow_four_six_from_total BT00WH pow_three_five_block_plus_one_le_pow_four_four_block_plus_one_from_total BT00WJ pow_two_successor_double_le_pow_four_successor_from_total BT00WP bertrand_h_root_32_from_total BT00WQ bertrand_h_root_33_from_total BT00WS bertrand_h_root_35_from_total BT00WT bertrand_h_root_36_from_total BT00WU bertrand_h_root_37_from_total BT00WV bertrand_j_base_thirty_two_window_from_total BT00X4 bertrand_floor_power_product_le_h_from_total BT00X9 beta_product_pointwise_le BT0114 central_binom_le_of_no_bertrand_prime

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

24 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)
mul le mul rightmul le mul leftle trans
01Fix variables and assumptionsL1–6

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 d
  5. L5
    intro hab
  6. L6
    intro hcd
02Establish hacbcL7–12

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

  1. L7
    have hacbc : Le(a · c,b · c)Definitions: Le(a · c,b · c)Original native command in the exact edition
  2. L8
    specialize mul_le_mul_right a
  3. L9
    specialize mul_le_mul_right b
  4. L10
    specialize mul_le_mul_right c
  5. L11
    apply mul_le_mul_right
  6. L12
    exact hab
03Establish hbcbdL13–22

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

  1. L13
    have hbcbd : Le(b · c,b · d)Definitions: Le(b · c,b · d)Original native command in the exact edition
  2. L14
    specialize mul_le_mul_left c
  3. L15
    specialize mul_le_mul_left d
  4. L16
    specialize mul_le_mul_left b
  5. L17
    apply mul_le_mul_left
  6. L18
    exact hcd
  7. L19
    specialize le_trans (a * c)
  8. L20
    specialize le_trans (b * c)
  9. L21
    specialize le_trans (b * d)
  10. L22
    apply le_trans
04Use earlier factsL23–24

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

  1. L23
    exact hacbc
  2. L24
    exact hbcbd

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro hab
  6. 0006intro hcd
  7. 0007have hacbc : Le(a · c,b · c)
    Exact native replay linehave hacbc : exists k. k + a * c = b * c
  8. 0008specialize mul_le_mul_right a
  9. 0009specialize mul_le_mul_right b
  10. 0010specialize mul_le_mul_right c
  11. 0011apply mul_le_mul_right
  12. 0012exact hab
  13. 0013have hbcbd : Le(b · c,b · d)
    Exact native replay linehave hbcbd : exists k. k + b * c = b * d
  14. 0014specialize mul_le_mul_left c
  15. 0015specialize mul_le_mul_left d
  16. 0016specialize mul_le_mul_left b
  17. 0017apply mul_le_mul_left
  18. 0018exact hcd
  19. 0019specialize le_trans (a * c)
  20. 0020specialize le_trans (b * c)
  21. 0021specialize le_trans (b * d)
  22. 0022apply le_trans
  23. 0023exact hacbc
  24. 0024exact hbcbd