MX002D

signed_cartesian_quotient_row_bound

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

An actual flattened index below a rectangular area has its quotient row below the height, including vacuous zero-area cases.

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 expanded first-order arithmetic statement

forall m n k i j. k=n*i+j -> (exists pvs_gap_quotient_source. pvs_gap_quotient_source + S (k) = (m*n)) -> (exists pvs_gap_quotient_result. pvs_gap_quotient_result + S (i) = (m))

Constructive proof overview

Generated structural guide

An actual flattened index below a rectangular area has its quotient row below the height, including vacuous zero-area cases.

The unchanged tactic script uses 6 declared prerequisites and contains 36 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

le_or_lt Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized le_trans Stable theorem; checked-use authorized mul_le_mul_left Stable theorem; checked-use authorized le_add_right Stable theorem; checked-use authorized lt_not_le Stable theorem; checked-use authorized

Direct dependents

MX002E signed_cartesian_coordinates_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

36 script commands · 7 reading checkpoints · 3 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro k
  4. L4
    intro i
  5. L5
    intro j
  6. L6
    intro hk
  7. L7
    intro hbound
02Establish hcaseL8–11

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

  1. L8
    have hcase : (exists pvs_le_gap_quotient_case. pvs_le_gap_quotient_case + (m) = (i)) \/ (exists pvs_gap_quotient_other. pvs_gap_quotient_other + S (i) = (m))
  2. L9
    specialize le_or_lt (m)
  3. L10
    specialize le_or_lt (i)
  4. L11
    apply le_or_lt
03Separate the logical casesL12–13

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

  1. L12
    cases hcase
  2. L13
    exfalso
04Establish hreverseL14–14

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

  1. L14
    have hreverse : exists pvs_le_gap_quotient_reverse. pvs_le_gap_quotient_reverse + (m*n) = (k)
05Establish hcommL15–24

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

  1. L15
    have hcomm : m*n=n*m
  2. L16
    apply mul_comm
  3. L17
    rewrite hcomm
  4. L18
    rewrite hk
  5. L19
    specialize le_trans (n*m)
  6. L20
    specialize le_trans (n*i)
  7. L21
    specialize le_trans (n*i+j)
  8. L22
    apply le_trans
  9. L23
    specialize mul_le_mul_left (m)
  10. L24
    specialize mul_le_mul_left (i)
06Use earlier factsL25–34

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

  1. L25
    specialize mul_le_mul_left (n)
  2. L26
    apply mul_le_mul_left
  3. L27
    exact hcase_left
  4. L28
    specialize le_add_right (n*i)
  5. L29
    specialize le_add_right (j)
  6. L30
    apply le_add_right
  7. L31
    specialize lt_not_le (k)
  8. L32
    specialize lt_not_le (m*n)
  9. L33
    apply lt_not_le
  10. L34
    exact hbound
07Use earlier factsL35–36

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

  1. L35
    exact hreverse
  2. L36
    exact hcase_right

Library-wide reading audit

Original exact command ledger · 36 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro k
  4. 0004intro i
  5. 0005intro j
  6. 0006intro hk
  7. 0007intro hbound
  8. 0008have hcase : (exists pvs_le_gap_quotient_case. pvs_le_gap_quotient_case + (m) = (i)) \/ (exists pvs_gap_quotient_other. pvs_gap_quotient_other + S (i) = (m))
  9. 0009specialize le_or_lt (m)
  10. 0010specialize le_or_lt (i)
  11. 0011apply le_or_lt
  12. 0012cases hcase
  13. 0013exfalso
  14. 0014have hreverse : exists pvs_le_gap_quotient_reverse. pvs_le_gap_quotient_reverse + (m*n) = (k)
  15. 0015have hcomm : m*n=n*m
  16. 0016apply mul_comm
  17. 0017rewrite hcomm
  18. 0018rewrite hk
  19. 0019specialize le_trans (n*m)
  20. 0020specialize le_trans (n*i)
  21. 0021specialize le_trans (n*i+j)
  22. 0022apply le_trans
  23. 0023specialize mul_le_mul_left (m)
  24. 0024specialize mul_le_mul_left (i)
  25. 0025specialize mul_le_mul_left (n)
  26. 0026apply mul_le_mul_left
  27. 0027exact hcase_left
  28. 0028specialize le_add_right (n*i)
  29. 0029specialize le_add_right (j)
  30. 0030apply le_add_right
  31. 0031specialize lt_not_le (k)
  32. 0032specialize lt_not_le (m*n)
  33. 0033apply lt_not_le
  34. 0034exact hbound
  35. 0035exact hreverse
  36. 0036exact hcase_right