BT001R · Bertrand theorem

division_block_upper

Stable checked-use theorem · independently kernel verified

A bounded remainder keeps its decomposition below the next divisor block.

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

∀ d. ∀ q. ∀ r. Lt(r,d) → Lt(d · q + r,d · S q)

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

PD0002 Lt

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall d q r. (exists k. k + S r = d) -> exists k. k + S (d * q + r) = d * S q

Proof neighborhood

Direct theorem prerequisites

BT0003 add_assoc BT0002 add_comm

Direct theorem dependents

BT00YF division_three_scaled_upper_of_quotient_lt BT010E division_quotient_lower_of_scaled_le

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

34 script commands · 19 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 (2)
add assocadd comm
01Fix variables and assumptionsL1–4

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

  1. L1
    intro d
  2. L2
    intro q
  3. L3
    intro r
  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 x
04Calculate and transport equalitiesL7–7

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

  1. L7
    trans S (x + (d * q + r))
05Use earlier factsL8–8

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

  1. L8
    apply PA4
06Calculate and transport equalitiesL9–11

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

  1. L9
    trans S ((x + d * q) + r)
  2. L10
    congr
  3. L11
    symm
07Use earlier factsL12–12

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

  1. L12
    apply add_assoc
08Calculate and transport equalitiesL13–15

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

  1. L13
    trans S ((d * q + x) + r)
  2. L14
    congr
  3. L15
    congr
09Use earlier factsL16–16

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

  1. L16
    apply add_comm
10Calculate and transport equalitiesL17–19

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

  1. L17
    refl
  2. L18
    trans S (d * q + (x + r))
  3. L19
    congr
11Use earlier factsL20–20

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

  1. L20
    apply add_assoc
12Calculate and transport equalitiesL21–22

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

  1. L21
    trans d * q + S (x + r)
  2. L22
    symm
13Use earlier factsL23–23

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

  1. L23
    apply PA4
14Calculate and transport equalitiesL24–27

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

  1. L24
    trans d * q + (x + S r)
  2. L25
    congr
  3. L26
    refl
  4. L27
    symm
15Use earlier factsL28–28

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

  1. L28
    apply PA4
16Calculate and transport equalitiesL29–31

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

  1. L29
    trans d * q + d
  2. L30
    congr
  3. L31
    refl
17Use earlier factsL32–32

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

  1. L32
    exact h_witness
18Calculate and transport equalitiesL33–33

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

  1. L33
    symm
19Use earlier factsL34–34

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

  1. L34
    apply PA6

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro d
  2. 0002intro q
  3. 0003intro r
  4. 0004intro h
  5. 0005cases h
  6. 0006exists x
  7. 0007trans S (x + (d * q + r))
  8. 0008apply PA4
  9. 0009trans S ((x + d * q) + r)
  10. 0010congr
  11. 0011symm
  12. 0012apply add_assoc
  13. 0013trans S ((d * q + x) + r)
  14. 0014congr
  15. 0015congr
  16. 0016apply add_comm
  17. 0017refl
  18. 0018trans S (d * q + (x + r))
  19. 0019congr
  20. 0020apply add_assoc
  21. 0021trans d * q + S (x + r)
  22. 0022symm
  23. 0023apply PA4
  24. 0024trans d * q + (x + S r)
  25. 0025congr
  26. 0026refl
  27. 0027symm
  28. 0028apply PA4
  29. 0029trans d * q + d
  30. 0030congr
  31. 0031refl
  32. 0032exact h_witness
  33. 0033symm
  34. 0034apply PA6