BT00Y2 · Bertrand theorem

division_successor_quotient_divisor_le

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

A division with successor quotient bounds its divisor by the dividend.

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. ∀ n. ∀ q. ∀ r. DivRem(n,d,S q,r) → Le(d,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

PD0001 Le PD0007 DivRem

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall d n q r. (((n) = (d) * (S q) + (r) /\ (exists bcf_lt_gap_b5ccsqdl_source_bound. bcf_lt_gap_b5ccsqdl_source_bound + S (r) = d))) -> (exists bcf_le_gap_b5ccsqdl_result. bcf_le_gap_b5ccsqdl_result + (d) = n)

Proof neighborhood

Direct theorem prerequisites

BT0003 add_assoc BT0002 add_comm

Direct theorem dependents

BT00Y5 central_binom_prime_power_contribution_le_double

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

21 script commands · 6 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 (2)
add assocadd comm
01Fix variables and assumptionsL1–5

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

  1. L1
    intro d
  2. L2
    intro n
  3. L3
    intro q
  4. L4
    intro r
  5. L5
    intro hdivision
02Separate the logical casesL6–6

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

  1. L6
    cases hdivision
03Construct an explicit witnessL7–7

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

  1. L7
    exists d * q + r
04Calculate and transport equalitiesL8–9

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

  1. L8
    rewrite hdivision_left
  2. L9
    rewrite PA6
05Establish hleft_assocL10–11

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

  1. L10
    have hleft_assoc : (d * q + r) + d = d * q + (r + d)
  2. L11
    apply add_assoc
06Establish hright_assocL12–21

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

  1. L12
    have hright_assoc : (d * q + d) + r = d * q + (d + r)
  2. L13
    apply add_assoc
  3. L14
    trans d * q + (r + d)
  4. L15
    exact hleft_assoc
  5. L16
    trans d * q + (d + r)
  6. L17
    congr
  7. L18
    refl
  8. L19
    apply add_comm
  9. L20
    symm
  10. L21
    exact hright_assoc

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro d
  2. 0002intro n
  3. 0003intro q
  4. 0004intro r
  5. 0005intro hdivision
  6. 0006cases hdivision
  7. 0007exists d * q + r
  8. 0008rewrite hdivision_left
  9. 0009rewrite PA6
  10. 0010have hleft_assoc : (d * q + r) + d = d * q + (r + d)
  11. 0011apply add_assoc
  12. 0012have hright_assoc : (d * q + d) + r = d * q + (d + r)
  13. 0013apply add_assoc
  14. 0014trans d * q + (r + d)
  15. 0015exact hleft_assoc
  16. 0016trans d * q + (d + r)
  17. 0017congr
  18. 0018refl
  19. 0019apply add_comm
  20. 0020symm
  21. 0021exact hright_assoc