BT010E

division_quotient_lower_of_scaled_le

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

A scaled lower bound forces the division quotient above its scale index.

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.

Exact expanded PA statement

forall d N q r s. (((N) = (d) * (q) + (r) /\ (exists bcf_lt_gap_b5rbdqlosl_division_bound. bcf_lt_gap_b5rbdqlosl_division_bound + S (r) = d))) -> (exists bcf_le_gap_b5rbdqlosl_scaled. bcf_le_gap_b5rbdqlosl_scaled + (d * s) = N) -> (exists bcf_le_gap_b5rbdqlosl_result. bcf_le_gap_b5rbdqlosl_result + (s) = q)

Structural proof guide

A scaled lower bound forces the division quotient above its scale index.

Direct prerequisites: division_block_upper, le_or_lt, mul_le_mul_left, lt_of_lt_of_le, lt_not_le. The authored body proceeds by case analysis (2), intermediate claims (5), equality transport (1).

Proof neighborhood

Direct dependencies

BT001R division_block_upper BT001G le_or_lt BT001L mul_le_mul_left BT001D lt_of_lt_of_le BT001I lt_not_le

Direct dependents

BT010F floor_sqrt_le_third_quotient

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

42 script commands · 11 reading checkpoints · 5 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.

Named ingredients (5)
division block upperle or ltmul le mul leftlt of lt of lelt not le
01Fix variables and assumptionsL1–7

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 s
  6. L6
    intro hdivision
  7. L7
    intro hscaled
02Separate the logical casesL8–8

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

  1. L8
    cases hdivision
03Establish hcasesL9–12

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

  1. L9
    have hcases : (exists bcf_le_gap_b5rbdqlosl_result. bcf_le_gap_b5rbdqlosl_result + (s) = q) \/ (exists bcf_lt_gap_b5rbdqlosl_reverse. bcf_lt_gap_b5rbdqlosl_reverse + S (q) = s)
  2. L10
    specialize le_or_lt s
  3. L11
    specialize le_or_lt q
  4. L12
    exact le_or_lt
04Separate the logical casesL13–13

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

  1. L13
    cases hcases
05Use earlier factsL14–14

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

  1. L14
    exact hcases_left
06Establish hupperL15–21

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

  1. L15
    have hupper : exists bcf_lt_gap_b5rbdqlosl_upper. bcf_lt_gap_b5rbdqlosl_upper + S (N) = d * S q
  2. L16
    rewrite hdivision_left
  3. L17
    specialize division_block_upper d
  4. L18
    specialize division_block_upper q
  5. L19
    specialize division_block_upper r
  6. L20
    apply division_block_upper
  7. L21
    exact hdivision_right
07Establish hreverseL22–23

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

  1. L22
    have hreverse : exists bcf_le_gap_b5rbdqlosl_reverse_le. bcf_le_gap_b5rbdqlosl_reverse_le + (S q) = s
  2. L23
    exact hcases_right
08Establish hproductL24–29

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

  1. L24
    have hproduct : exists bcf_le_gap_b5rbdqlosl_product_le. bcf_le_gap_b5rbdqlosl_product_le + (d * S q) = d * s
  2. L25
    specialize mul_le_mul_left (S q)
  3. L26
    specialize mul_le_mul_left s
  4. L27
    specialize mul_le_mul_left d
  5. L28
    apply mul_le_mul_left
  6. L29
    exact hreverse
09Establish hstrictL30–36

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

  1. L30
    have hstrict : exists bcf_lt_gap_b5rbdqlosl_contradiction. bcf_lt_gap_b5rbdqlosl_contradiction + S (N) = d * s
  2. L31
    specialize lt_of_lt_of_le N
  3. L32
    specialize lt_of_lt_of_le (d * S q)
  4. L33
    specialize lt_of_lt_of_le (d * s)
  5. L34
    apply lt_of_lt_of_le
  6. L35
    exact hupper
  7. L36
    exact hproduct
10Separate the logical casesL37–37

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

  1. L37
    exfalso
11Use earlier factsL38–42

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

  1. L38
    specialize lt_not_le N
  2. L39
    specialize lt_not_le (d * s)
  3. L40
    apply lt_not_le
  4. L41
    exact hstrict
  5. L42
    exact hscaled

Library-wide reading audit

Original exact command ledger · 42 lines
  1. 0001intro d
  2. 0002intro N
  3. 0003intro q
  4. 0004intro r
  5. 0005intro s
  6. 0006intro hdivision
  7. 0007intro hscaled
  8. 0008cases hdivision
  9. 0009have hcases : (exists bcf_le_gap_b5rbdqlosl_result. bcf_le_gap_b5rbdqlosl_result + (s) = q) \/ (exists bcf_lt_gap_b5rbdqlosl_reverse. bcf_lt_gap_b5rbdqlosl_reverse + S (q) = s)
  10. 0010specialize le_or_lt s
  11. 0011specialize le_or_lt q
  12. 0012exact le_or_lt
  13. 0013cases hcases
  14. 0014exact hcases_left
  15. 0015have hupper : exists bcf_lt_gap_b5rbdqlosl_upper. bcf_lt_gap_b5rbdqlosl_upper + S (N) = d * S q
  16. 0016rewrite hdivision_left
  17. 0017specialize division_block_upper d
  18. 0018specialize division_block_upper q
  19. 0019specialize division_block_upper r
  20. 0020apply division_block_upper
  21. 0021exact hdivision_right
  22. 0022have hreverse : exists bcf_le_gap_b5rbdqlosl_reverse_le. bcf_le_gap_b5rbdqlosl_reverse_le + (S q) = s
  23. 0023exact hcases_right
  24. 0024have hproduct : exists bcf_le_gap_b5rbdqlosl_product_le. bcf_le_gap_b5rbdqlosl_product_le + (d * S q) = d * s
  25. 0025specialize mul_le_mul_left (S q)
  26. 0026specialize mul_le_mul_left s
  27. 0027specialize mul_le_mul_left d
  28. 0028apply mul_le_mul_left
  29. 0029exact hreverse
  30. 0030have hstrict : exists bcf_lt_gap_b5rbdqlosl_contradiction. bcf_lt_gap_b5rbdqlosl_contradiction + S (N) = d * s
  31. 0031specialize lt_of_lt_of_le N
  32. 0032specialize lt_of_lt_of_le (d * S q)
  33. 0033specialize lt_of_lt_of_le (d * s)
  34. 0034apply lt_of_lt_of_le
  35. 0035exact hupper
  36. 0036exact hproduct
  37. 0037exfalso
  38. 0038specialize lt_not_le N
  39. 0039specialize lt_not_le (d * s)
  40. 0040apply lt_not_le
  41. 0041exact hstrict
  42. 0042exact hscaled