BT00WE · Bertrand theorem

ceil_div_six_budget_of_scaled_le

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

A scaled lower bound cancels against the lower half of CeilDivSix.

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

∀ x. ∀ k. ∀ e. CeilDivSix(x,e) → Le(6 · k,x) → Le(k,e)

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 PD0052 CeilDivSix

3 occurrences

In local proof propositions

PD0001 Le

1 occurrences

Exact expanded native-PA statement
forall x k e. (((exists bcs_lower_gap_hj32_ceil_budget_source. bcs_lower_gap_hj32_ceil_budget_source + (x) = 6 * (e)) /\ exists bcs_upper_gap_hj32_ceil_budget_source. bcs_upper_gap_hj32_ceil_budget_source + S (6 * (e)) = (x) + 6)) -> (exists bqb_le_gap_hj32_ceil_budget_scaled. bqb_le_gap_hj32_ceil_budget_scaled + (6 * k) = (x)) -> (exists bqb_le_gap_hj32_ceil_budget_result. bqb_le_gap_hj32_ceil_budget_result + (k) = (e))

Proof neighborhood

Direct theorem prerequisites

BT000F le_trans BT000C succ_ne_zero BT00RD mul_le_cancel_left_nonzero

Direct theorem dependents

BT00WP bertrand_h_root_32_from_total BT00WQ bertrand_h_root_33_from_total BT00WR bertrand_h_root_34_from_total BT00WS bertrand_h_root_35_from_total BT00WT bertrand_h_root_36_from_total BT00WU bertrand_h_root_37_from_total

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

20 script commands · 4 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 (3)
le transsucc ne zeromul le cancel left nonzero
01Fix variables and assumptionsL1–5

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

  1. L1
    intro x
  2. L2
    intro k
  3. L3
    intro e
  4. L4
    intro hceiling
  5. L5
    intro hscaled
02Separate the logical casesL6–6

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

  1. L6
    cases hceiling
03Establish hscaled_to_ceilingL7–16

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

  1. L7
    have hscaled_to_ceiling : Le(6 · k,6 · e)Definitions: Le(6 · k,6 · e)Original native command in the exact edition
  2. L8
    specialize le_trans (6 * k)
  3. L9
    specialize le_trans x
  4. L10
    specialize le_trans (6 * e)
  5. L11
    apply le_trans
  6. L12
    exact hscaled
  7. L13
    exact hceiling_left
  8. L14
    specialize mul_le_cancel_left_nonzero 6
  9. L15
    specialize mul_le_cancel_left_nonzero k
  10. L16
    specialize mul_le_cancel_left_nonzero e
04Use earlier factsL17–20

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

  1. L17
    apply mul_le_cancel_left_nonzero
  2. L18
    specialize succ_ne_zero 5
  3. L19
    exact succ_ne_zero
  4. L20
    exact hscaled_to_ceiling

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro x
  2. 0002intro k
  3. 0003intro e
  4. 0004intro hceiling
  5. 0005intro hscaled
  6. 0006cases hceiling
  7. 0007have hscaled_to_ceiling : Le(6 · k,6 · e)
    Exact native replay linehave hscaled_to_ceiling : exists bqb_le_gap_hj32_ceil_budget_chain. bqb_le_gap_hj32_ceil_budget_chain + (6 * k) = (6 * e)
  8. 0008specialize le_trans (6 * k)
  9. 0009specialize le_trans x
  10. 0010specialize le_trans (6 * e)
  11. 0011apply le_trans
  12. 0012exact hscaled
  13. 0013exact hceiling_left
  14. 0014specialize mul_le_cancel_left_nonzero 6
  15. 0015specialize mul_le_cancel_left_nonzero k
  16. 0016specialize mul_le_cancel_left_nonzero e
  17. 0017apply mul_le_cancel_left_nonzero
  18. 0018specialize succ_ne_zero 5
  19. 0019exact succ_ne_zero
  20. 0020exact hscaled_to_ceiling