BT00X0 · Bertrand theorem

floor_sqrt_factorized_threshold_thirty_two

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

The factorized large-input threshold forces every selected root to be at least 32.

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

∀ n. ∀ s. Le(16 · 32,n) → FloorSqrt(2 · n,s) → Lt(31,s)

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 PD0051 FloorSqrt

3 occurrences

In local proof propositions

PD0001 Le PD0051 FloorSqrt

2 occurrences

Exact expanded native-PA statement
forall n s. (exists bqb_le_gap_hjas_threshold_input. bqb_le_gap_hjas_threshold_input + (16 * 32) = (n)) -> (((exists bcs_sqrt_lower_gap_hjas_threshold_floor. bcs_sqrt_lower_gap_hjas_threshold_floor + (s) * (s) = (2 * n)) /\ exists bcs_sqrt_upper_gap_hjas_threshold_floor. bcs_sqrt_upper_gap_hjas_threshold_floor + S (2 * n) = S (s) * S (s))) -> (exists bqb_le_gap_hjas_threshold_result. bqb_le_gap_hjas_threshold_result + (32) = (s))

Proof neighborhood

Direct theorem prerequisites

BT00WY thirty_two_square_eq_twice_sixteen_times_thirty_two BT0000 zero_add BT00R9 square_lt_successor_square BT001L mul_le_mul_left BT00RC floor_sqrt_monotone

Direct theorem dependents

BT00X6 bertrand_main_inequality_factorized_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

27 script commands · 10 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 (5)
thirty two square eq twice sixteen times thirty twozero addsquare lt successor squaremul le mul leftfloor sqrt monotone
01Fix variables and assumptionsL1–4

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

  1. L1
    intro n
  2. L2
    intro s
  3. L3
    intro hthreshold
  4. L4
    intro hfloor
02Establish hbaseL5–5

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hbase : FloorSqrt(2 · (16 · 32),32)Definitions: FloorSqrt(2 · (16 · 32),32)Original native command in the exact edition
03Separate the logical casesL6–6

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

  1. L6
    split
04Construct an explicit witnessL7–7

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

  1. L7
    exists 0
05Calculate and transport equalitiesL8–8

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

  1. L8
    trans 32 * 32
06Use earlier factsL9–11

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

  1. L9
    apply zero_add
  2. L10
    exact thirty_two_square_eq_twice_sixteen_times_thirty_two
  3. L11
    specialize square_lt_successor_square 32
07Calculate and transport equalitiesL12–12

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

  1. L12
    rewrite thirty_two_square_eq_twice_sixteen_times_thirty_two at square_lt_successor_square
08Use earlier factsL13–13

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

  1. L13
    exact square_lt_successor_square
09Establish hscaledL14–23

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

  1. L14
    have hscaled : Le(2 · (16 · 32),2 · n)Definitions: Le(2 · (16 · 32),2 · n)Original native command in the exact edition
  2. L15
    specialize mul_le_mul_left (16 * 32)
  3. L16
    specialize mul_le_mul_left n
  4. L17
    specialize mul_le_mul_left 2
  5. L18
    apply mul_le_mul_left
  6. L19
    exact hthreshold
  7. L20
    specialize floor_sqrt_monotone (2 * (16 * 32))
  8. L21
    specialize floor_sqrt_monotone (2 * n)
  9. L22
    specialize floor_sqrt_monotone 32
  10. L23
    specialize floor_sqrt_monotone s
10Use earlier factsL24–27

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

  1. L24
    apply floor_sqrt_monotone
  2. L25
    exact hbase
  3. L26
    exact hfloor
  4. L27
    exact hscaled

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro n
  2. 0002intro s
  3. 0003intro hthreshold
  4. 0004intro hfloor
  5. 0005have hbase : FloorSqrt(2 · (16 · 32),32)
    Exact native replay linehave hbase : ((exists bcs_sqrt_lower_gap_hjas_threshold_base_floor. bcs_sqrt_lower_gap_hjas_threshold_base_floor + (32) * (32) = (2 * (16 * 32))) /\ exists bcs_sqrt_upper_gap_hjas_threshold_base_floor. bcs_sqrt_upper_gap_hjas_threshold_base_floor + S (2 * (16 * 32)) = S (32) * S (32))
  6. 0006split
  7. 0007exists 0
  8. 0008trans 32 * 32
  9. 0009apply zero_add
  10. 0010exact thirty_two_square_eq_twice_sixteen_times_thirty_two
  11. 0011specialize square_lt_successor_square 32
  12. 0012rewrite thirty_two_square_eq_twice_sixteen_times_thirty_two at square_lt_successor_square
  13. 0013exact square_lt_successor_square
  14. 0014have hscaled : Le(2 · (16 · 32),2 · n)
    Exact native replay linehave hscaled : exists bqb_le_gap_hjas_threshold_scaled. bqb_le_gap_hjas_threshold_scaled + (2 * (16 * 32)) = (2 * n)
  15. 0015specialize mul_le_mul_left (16 * 32)
  16. 0016specialize mul_le_mul_left n
  17. 0017specialize mul_le_mul_left 2
  18. 0018apply mul_le_mul_left
  19. 0019exact hthreshold
  20. 0020specialize floor_sqrt_monotone (2 * (16 * 32))
  21. 0021specialize floor_sqrt_monotone (2 * n)
  22. 0022specialize floor_sqrt_monotone 32
  23. 0023specialize floor_sqrt_monotone s
  24. 0024apply floor_sqrt_monotone
  25. 0025exact hbase
  26. 0026exact hfloor
  27. 0027exact hscaled