DL0096

matrix_integer_selected_point_at_source

Every actual selected cell is its genuine source entry for any matching quotient coordinates and selector values; all coordinate and selector equalities are proved.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ w. ∀ rb. ∀ rc. ∀ cb. ∀ cc. ∀ q. ∀ i. ∀ r. ∀ s. ∀ u. ∀ v. ∀ a. (∃ x. ∃ y. ∃ z. ∃ n. i = q · x + y ∧ (Lt(y,q) ∧ (BetaAt(rb,rc,x,z) ∧ (BetaAt(cb,cc,y,n) ∧ BetaAt(b,c,z · w + n,a))))) → i = q · r + s → Lt(s,q) → BetaAt(rb,rc,r,u) → BetaAt(cb,cc,s,v) → BetaAt(b,c,u · w + v,a)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Lt(a,b) · 2BetaAt(b,c,i,x) · 6

Actual proof prerequisites

division_remainder_unique · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c w rb rc cb cc q i r s u v a. (exists mdr_r_selected_source_point mdr_s_selected_source_point mdr_u_selected_source_point mdr_v_selected_source_point. ((i = (q) * mdr_r_selected_source_point + mdr_s_selected_source_point) /\ ((exists mdr_gap_selected_source_pointcolumn. mdr_gap_selected_source_pointcolumn + S (mdr_s_selected_source_point) = (q)) /\ ((((exists ff_h_mdr_selected_source_pointrow_index. ff_h_mdr_selected_source_pointrow_index + S (mdr_u_selected_source_point) = S ((S (mdr_r_selected_source_point)) * rc)) /\ exists ff_q_mdr_selected_source_pointrow_index. rb = ff_q_mdr_selected_source_pointrow_index * S ((S (mdr_r_selected_source_point)) * rc) + (mdr_u_selected_source_point))) /\ ((((exists ff_h_mdr_selected_source_pointcolumn_index. ff_h_mdr_selected_source_pointcolumn_index + S (mdr_v_selected_source_point) = S ((S (mdr_s_selected_source_point)) * cc)) /\ exists ff_q_mdr_selected_source_pointcolumn_index. cb = ff_q_mdr_selected_source_pointcolumn_index * S ((S (mdr_s_selected_source_point)) * cc) + (mdr_v_selected_source_point))) /\ (((exists ff_h_mdr_selected_source_pointsource. ff_h_mdr_selected_source_pointsource + S (a) = S ((S ((mdr_u_selected_source_point) * (w) + (mdr_v_selected_source_point))) * c)) /\ exists ff_q_mdr_selected_source_pointsource. b = ff_q_mdr_selected_source_pointsource * S ((S ((mdr_u_selected_source_point) * (w) + (mdr_v_selected_source_point))) * c) + (a)))))))) -> i = q * r + s -> (exists mdr_gap_selected_source_column. mdr_gap_selected_source_column + S (s) = (q)) -> (((exists ff_h_mdr_selected_source_row_code. ff_h_mdr_selected_source_row_code + S (u) = S ((S (r)) * rc)) /\ exists ff_q_mdr_selected_source_row_code. rb = ff_q_mdr_selected_source_row_code * S ((S (r)) * rc) + (u))) -> (((exists ff_h_mdr_selected_source_col_code. ff_h_mdr_selected_source_col_code + S (v) = S ((S (s)) * cc)) /\ exists ff_q_mdr_selected_source_col_code. cb = ff_q_mdr_selected_source_col_code * S ((S (s)) * cc) + (v))) -> (((exists ff_h_mdr_selected_source_value. ff_h_mdr_selected_source_value + S (a) = S ((S (u * w + v)) * c)) /\ exists ff_q_mdr_selected_source_value. b = ff_q_mdr_selected_source_value * S ((S (u * w + v)) * c) + (a)))

Complete tactic proof in conservative notation

All 67 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

67 script commands · 12 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro w
  4. L4
    intro rb
  5. L5
    intro rc
  6. L6
    intro cb
  7. L7
    intro cc
  8. L8
    intro q
  9. L9
    intro i
  10. L10
    intro r
02Fix variables and assumptionsL11–19

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

  1. L11
    intro s
  2. L12
    intro u
  3. L13
    intro v
  4. L14
    intro a
  5. L15
    intro hpoint
  6. L16
    intro hcoordinate
  7. L17
    intro hs
  8. L18
    intro hrow
  9. L19
    intro hcolumn
03Separate the logical casesL20–27

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

  1. L20
    cases hpoint
  2. L21
    cases hpoint_witness
  3. L22
    cases hpoint_witness_witness
  4. L23
    cases hpoint_witness_witness_witness
  5. L24
    cases hpoint_witness_witness_witness_witness
  6. L25
    cases hpoint_witness_witness_witness_witness_right
  7. L26
    cases hpoint_witness_witness_witness_witness_right_right
  8. L27
    cases hpoint_witness_witness_witness_witness_right_right_right
04Establish hcoordinatesL28–37

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

  1. L28
    have hcoordinates : r = x /\ s = x1
  2. L29
    specialize division_remainder_unique (q)
  3. L30
    specialize division_remainder_unique (i)
  4. L31
    specialize division_remainder_unique (r)
  5. L32
    specialize division_remainder_unique (s)
  6. L33
    specialize division_remainder_unique (x)
  7. L34
    specialize division_remainder_unique (x1)
  8. L35
    apply division_remainder_unique
  9. L36
    exact hcoordinate
  10. L37
    exact hs
05Use earlier factsL38–39

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

  1. L38
    exact hpoint_witness_witness_witness_witness_left
  2. L39
    exact hpoint_witness_witness_witness_witness_right_left
06Separate the logical casesL40–40

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

  1. L40
    cases hcoordinates
07Establish hrowvalueL41–50

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L41
    have hrowvalue : u = x2
  2. L42
    specialize beta_at_unique (rb)
  3. L43
    specialize beta_at_unique (rc)
  4. L44
    specialize beta_at_unique (r)
  5. L45
    specialize beta_at_unique (u)
  6. L46
    specialize beta_at_unique (x2)
  7. L47
    apply beta_at_unique
  8. L48
    exact hrow
  9. L49
    rewrite hcoordinates_left
  10. L50
    rewrite hcoordinates_left
08Use earlier factsL51–51

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

  1. L51
    exact hpoint_witness_witness_witness_witness_right_right_left
09Establish hcolvalueL52–61

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L52
    have hcolvalue : v = x3
  2. L53
    specialize beta_at_unique (cb)
  3. L54
    specialize beta_at_unique (cc)
  4. L55
    specialize beta_at_unique (s)
  5. L56
    specialize beta_at_unique (v)
  6. L57
    specialize beta_at_unique (x3)
  7. L58
    apply beta_at_unique
  8. L59
    exact hcolumn
  9. L60
    rewrite hcoordinates_right
  10. L61
    rewrite hcoordinates_right
10Use earlier factsL62–62

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

  1. L62
    exact hpoint_witness_witness_witness_witness_right_right_right_left
11Calculate and transport equalitiesL63–66

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

  1. L63
    rewrite hrowvalue
  2. L64
    rewrite hrowvalue
  3. L65
    rewrite hcolvalue
  4. L66
    rewrite hcolvalue
12Use earlier factsL67–67

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

  1. L67
    exact hpoint_witness_witness_witness_witness_right_right_right_right

Library-wide reading audit

Original defined command ledger · 67 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro rb
  5. 0005intro rc
  6. 0006intro cb
  7. 0007intro cc
  8. 0008intro q
  9. 0009intro i
  10. 0010intro r
  11. 0011intro s
  12. 0012intro u
  13. 0013intro v
  14. 0014intro a
  15. 0015intro hpoint
  16. 0016intro hcoordinate
  17. 0017intro hs
  18. 0018intro hrow
  19. 0019intro hcolumn
  20. 0020cases hpoint
  21. 0021cases hpoint_witness
  22. 0022cases hpoint_witness_witness
  23. 0023cases hpoint_witness_witness_witness
  24. 0024cases hpoint_witness_witness_witness_witness
  25. 0025cases hpoint_witness_witness_witness_witness_right
  26. 0026cases hpoint_witness_witness_witness_witness_right_right
  27. 0027cases hpoint_witness_witness_witness_witness_right_right_right
  28. 0028have hcoordinates : r = x /\ s = x1
  29. 0029specialize division_remainder_unique (q)
  30. 0030specialize division_remainder_unique (i)
  31. 0031specialize division_remainder_unique (r)
  32. 0032specialize division_remainder_unique (s)
  33. 0033specialize division_remainder_unique (x)
  34. 0034specialize division_remainder_unique (x1)
  35. 0035apply division_remainder_unique
  36. 0036exact hcoordinate
  37. 0037exact hs
  38. 0038exact hpoint_witness_witness_witness_witness_left
  39. 0039exact hpoint_witness_witness_witness_witness_right_left
  40. 0040cases hcoordinates
  41. 0041have hrowvalue : u = x2
  42. 0042specialize beta_at_unique (rb)
  43. 0043specialize beta_at_unique (rc)
  44. 0044specialize beta_at_unique (r)
  45. 0045specialize beta_at_unique (u)
  46. 0046specialize beta_at_unique (x2)
  47. 0047apply beta_at_unique
  48. 0048exact hrow
  49. 0049rewrite hcoordinates_left
  50. 0050rewrite hcoordinates_left
  51. 0051exact hpoint_witness_witness_witness_witness_right_right_left
  52. 0052have hcolvalue : v = x3
  53. 0053specialize beta_at_unique (cb)
  54. 0054specialize beta_at_unique (cc)
  55. 0055specialize beta_at_unique (s)
  56. 0056specialize beta_at_unique (v)
  57. 0057specialize beta_at_unique (x3)
  58. 0058apply beta_at_unique
  59. 0059exact hcolumn
  60. 0060rewrite hcoordinates_right
  61. 0061rewrite hcoordinates_right
  62. 0062exact hpoint_witness_witness_witness_witness_right_right_right_left
  63. 0063rewrite hrowvalue
  64. 0064rewrite hrowvalue
  65. 0065rewrite hcolvalue
  66. 0066rewrite hcolvalue
  67. 0067exact hpoint_witness_witness_witness_witness_right_right_right_right