DL0090

matrix_integer_square_index_width_nonzero

An actual index in a square prefix proves its row width is nonzero; zero-dimensional bounds remain genuinely vacuous.

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. ∀ i. Lt(i,q · q) → ¬q = 0

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

matrix_rank_no_index_below_zero
Original expanded first-order statement 
forall q i. (exists mdr_gap_square_index. mdr_gap_square_index + S (i) = (q * q)) -> ~(q = 0)

Complete tactic proof in conservative notation

All 12 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

12 script commands · 2 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.

Named ingredients (1)
matrix rank no index below zero
01Fix variables and assumptionsL1–4

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

  1. L1
    intro q
  2. L2
    intro i
  3. L3
    intro hi
  4. L4
    intro hzero
02Establish hlengthL5–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA5.

  1. L5
    have hlength : q * q = 0
  2. L6
    rewrite hzero
  3. L7
    rewrite hzero
  4. L8
    apply PA5
  5. L9
    rewrite hlength at hi
  6. L10
    specialize matrix_rank_no_index_below_zero (i)
  7. L11
    apply matrix_rank_no_index_below_zero
  8. L12
    exact hi

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro q
  2. 0002intro i
  3. 0003intro hi
  4. 0004intro hzero
  5. 0005have hlength : q * q = 0
  6. 0006rewrite hzero
  7. 0007rewrite hzero
  8. 0008apply PA5
  9. 0009rewrite hlength at hi
  10. 0010specialize matrix_rank_no_index_below_zero (i)
  11. 0011apply matrix_rank_no_index_below_zero
  12. 0012exact hi