DL0033

matrix_rank_no_index_below_zero

The empty finite domain has no index, by natural successor nonzeroness.

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

∀ i. ¬Lt(i,0)

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

le_zero · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall i. ~(exists mdr_gap_no_index. mdr_gap_no_index + S (i) = (0))

Complete tactic proof in conservative notation

All 9 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

9 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro i
  2. L2
    intro hi
02Establish hzeroL3–9

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

  1. L3
    have hzero : S i = 0
  2. L4
    specialize le_zero (S i)
  3. L5
    apply le_zero
  4. L6
    exact hi
  5. L7
    specialize succ_ne_zero (i)
  6. L8
    apply succ_ne_zero
  7. L9
    exact hzero

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro i
  2. 0002intro hi
  3. 0003have hzero : S i = 0
  4. 0004specialize le_zero (S i)
  5. 0005apply le_zero
  6. 0006exact hi
  7. 0007specialize succ_ne_zero (i)
  8. 0008apply succ_ne_zero
  9. 0009exact hzero