BT010I · Bertrand theorem

third_quotient_double_gap_exists

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

The third quotient has an exact additive gap to the doubled input.

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. ∀ q. ∀ r. DivRem(n + n,3,q,r) → ∃ x. q + x = n + n

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

PD0007 DivRem

1 occurrences

In local proof propositions

PD0001 Le

1 occurrences

Exact expanded native-PA statement
forall n q r. (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_b5rbdqld_division_bound. bcf_lt_gap_b5rbdqld_division_bound + S (r) = 3))) -> exists h. q + h = n + n

Proof neighborhood

Direct theorem prerequisites

BT0002 add_comm BT010H division_quotient_le_dividend

Direct theorem dependents

BT010J floor_third_double_gap_package

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

16 script commands · 7 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 (2)
add commdivision quotient le dividend
01Fix variables and assumptionsL1–4

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

  1. L1
    intro n
  2. L2
    intro q
  3. L3
    intro r
  4. L4
    intro hdivision
02Establish hboundL5–10

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

  1. L5
    have hbound : Le(q,n + n)Definitions: Le(q,n + n)Original native command in the exact edition
  2. L6
    specialize division_quotient_le_dividend n
  3. L7
    specialize division_quotient_le_dividend q
  4. L8
    specialize division_quotient_le_dividend r
  5. L9
    apply division_quotient_le_dividend
  6. L10
    exact hdivision
03Separate the logical casesL11–11

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

  1. L11
    cases hbound
04Construct an explicit witnessL12–12

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

  1. L12
    exists x
05Use earlier factsL13–14

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

  1. L13
    specialize add_comm q
  2. L14
    specialize add_comm x
06Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite add_comm
07Use earlier factsL16–16

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

  1. L16
    exact hbound_witness

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro n
  2. 0002intro q
  3. 0003intro r
  4. 0004intro hdivision
  5. 0005have hbound : Le(q,n + n)
    Exact native replay linehave hbound : exists bcf_le_gap_b5rbdqld_result. bcf_le_gap_b5rbdqld_result + (q) = n + n
  6. 0006specialize division_quotient_le_dividend n
  7. 0007specialize division_quotient_le_dividend q
  8. 0008specialize division_quotient_le_dividend r
  9. 0009apply division_quotient_le_dividend
  10. 0010exact hdivision
  11. 0011cases hbound
  12. 0012exists x
  13. 0013specialize add_comm q
  14. 0014specialize add_comm x
  15. 0015rewrite add_comm
  16. 0016exact hbound_witness