DL001B

matrix_recursive_quotient_row_bound

A genuine row-major index below q squared has row coordinate below q, including the vacuous zero-width boundary.

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

∀ q. ∀ k. ∀ r. ∀ s. k = q · r + s → Lt(k,q · q) → Lt(r,q)

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

Definition DAG

Le(a,b) · 1Lt(a,b) · 2

Actual proof prerequisites

le_or_lt · checked external prerequisitemul_le_mul_left · checked external prerequisitele_add_right · checked external prerequisitele_trans · checked external prerequisitelt_not_le · checked external prerequisite
Original expanded first-order statement 
forall q k r s. k = q * r + s -> (exists mdr_gap_quotient_source. mdr_gap_quotient_source + S (k) = (q * q)) -> (exists mdr_gap_quotient_row. mdr_gap_quotient_row + S (r) = (q))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 5 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.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro q
  2. L2
    intro k
  3. L3
    intro r
  4. L4
    intro s
  5. L5
    intro hk
  6. L6
    intro hbound
02Use earlier factsL7–8

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

  1. L7
    specialize le_or_lt q
  2. L8
    specialize le_or_lt r
03Separate the logical casesL9–10

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

  1. L9
    cases le_or_lt
  2. L10
    exfalso
04Establish hreverseL11–20

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

  1. L11
    have hreverse : Le(q · q,k)Definitions: Le(q · q,k)Original native command in the exact edition
  2. L12
    rewrite hk
  3. L13
    specialize le_trans (q * q)
  4. L14
    specialize le_trans (q * r)
  5. L15
    specialize le_trans (q * r + s)
  6. L16
    apply le_trans
  7. L17
    specialize mul_le_mul_left (q)
  8. L18
    specialize mul_le_mul_left (r)
  9. L19
    specialize mul_le_mul_left (q)
  10. L20
    apply mul_le_mul_left
05Use earlier factsL21–30

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

  1. L21
    exact le_or_lt_left
  2. L22
    specialize le_add_right (q * r)
  3. L23
    specialize le_add_right (s)
  4. L24
    apply le_add_right
  5. L25
    specialize lt_not_le (k)
  6. L26
    specialize lt_not_le (q * q)
  7. L27
    apply lt_not_le
  8. L28
    exact hbound
  9. L29
    exact hreverse
  10. L30
    exact le_or_lt_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro q
  2. 0002intro k
  3. 0003intro r
  4. 0004intro s
  5. 0005intro hk
  6. 0006intro hbound
  7. 0007specialize le_or_lt q
  8. 0008specialize le_or_lt r
  9. 0009cases le_or_lt
  10. 0010exfalso
  11. 0011have hreverse : Le(q · q,k)
  12. 0012rewrite hk
  13. 0013specialize le_trans (q * q)
  14. 0014specialize le_trans (q * r)
  15. 0015specialize le_trans (q * r + s)
  16. 0016apply le_trans
  17. 0017specialize mul_le_mul_left (q)
  18. 0018specialize mul_le_mul_left (r)
  19. 0019specialize mul_le_mul_left (q)
  20. 0020apply mul_le_mul_left
  21. 0021exact le_or_lt_left
  22. 0022specialize le_add_right (q * r)
  23. 0023specialize le_add_right (s)
  24. 0024apply le_add_right
  25. 0025specialize lt_not_le (k)
  26. 0026specialize lt_not_le (q * q)
  27. 0027apply lt_not_le
  28. 0028exact hbound
  29. 0029exact hreverse
  30. 0030exact le_or_lt_right