DL001A

matrix_recursive_flattened_index_bound

Every in-range row and column has an actual flattened index below the square matrix length.

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

∀ w. ∀ r. ∀ s. Lt(r,w) → Lt(s,w) → Lt(r · w + s,w · w)

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

Definition DAG

Le(a,b) · 1Lt(a,b) · 3

Actual proof prerequisites

matrix_recursive_lt_add_leftmul_le_mul_right · checked external prerequisitemul_succ_left · checked external prerequisitelt_of_lt_of_le · checked external prerequisite
Original expanded first-order statement 
forall w r s. (exists mdr_gap_flat_row. mdr_gap_flat_row + S (r) = (w)) -> (exists mdr_gap_flat_column. mdr_gap_flat_column + S (s) = (w)) -> (exists mdr_gap_flat_result. mdr_gap_flat_result + S (r * w + s) = (w * w))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 4 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 (1)
matrix recursive lt add left
01Fix variables and assumptionsL1–5

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

  1. L1
    intro w
  2. L2
    intro r
  3. L3
    intro s
  4. L4
    intro hr
  5. L5
    intro hs
02Establish heqL6–7

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

  1. L6
    have heq : (S r) * w = r * w + w
  2. L7
    apply mul_succ_left
03Establish hrowendL8–17

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

  1. L8
    have hrowend : Le(S r · w,w · w)Definitions: Le(S r · w,w · w)Original native command in the exact edition
  2. L9
    specialize mul_le_mul_right (S r)
  3. L10
    specialize mul_le_mul_right (w)
  4. L11
    specialize mul_le_mul_right (w)
  5. L12
    apply mul_le_mul_right
  6. L13
    exact hr
  7. L14
    rewrite heq at hrowend
  8. L15
    specialize lt_of_lt_of_le (r * w + s)
  9. L16
    specialize lt_of_lt_of_le (r * w + w)
  10. L17
    specialize lt_of_lt_of_le (w * w)
04Use earlier factsL18–24

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

  1. L18
    apply lt_of_lt_of_le
  2. L19
    specialize matrix_recursive_lt_add_left (s)
  3. L20
    specialize matrix_recursive_lt_add_left (w)
  4. L21
    specialize matrix_recursive_lt_add_left (r * w)
  5. L22
    apply matrix_recursive_lt_add_left
  6. L23
    exact hs
  7. L24
    exact hrowend

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro w
  2. 0002intro r
  3. 0003intro s
  4. 0004intro hr
  5. 0005intro hs
  6. 0006have heq : (S r) * w = r * w + w
  7. 0007apply mul_succ_left
  8. 0008have hrowend : Le(S r · w,w · w)
  9. 0009specialize mul_le_mul_right (S r)
  10. 0010specialize mul_le_mul_right (w)
  11. 0011specialize mul_le_mul_right (w)
  12. 0012apply mul_le_mul_right
  13. 0013exact hr
  14. 0014rewrite heq at hrowend
  15. 0015specialize lt_of_lt_of_le (r * w + s)
  16. 0016specialize lt_of_lt_of_le (r * w + w)
  17. 0017specialize lt_of_lt_of_le (w * w)
  18. 0018apply lt_of_lt_of_le
  19. 0019specialize matrix_recursive_lt_add_left (s)
  20. 0020specialize matrix_recursive_lt_add_left (w)
  21. 0021specialize matrix_recursive_lt_add_left (r * w)
  22. 0022apply matrix_recursive_lt_add_left
  23. 0023exact hs
  24. 0024exact hrowend