DL0038

matrix_rank_bounded_prefix_empty

Every beta code describes the empty bounded selector prefix.

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. ∀ B. ∀ x. Lt(x,0) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B)

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

Definition DAG

Lt(a,b) · 2BetaAt(b,c,i,x) · 1

Actual proof prerequisites

matrix_rank_no_index_below_zero
Original expanded first-order statement 
forall b c B. (forall fom_index_mrf_empty_bounded. (exists fom_gap_mrf_empty_bounded_index_bound. fom_gap_mrf_empty_bounded_index_bound + S (fom_index_mrf_empty_bounded) = 0) -> exists fom_value_mrf_empty_bounded. ((((exists fom_beta_height_mrf_empty_bounded_entry. fom_beta_height_mrf_empty_bounded_entry + S (fom_value_mrf_empty_bounded) = S ((S (fom_index_mrf_empty_bounded)) * c)) /\ exists fom_beta_quotient_mrf_empty_bounded_entry. b = fom_beta_quotient_mrf_empty_bounded_entry * S ((S (fom_index_mrf_empty_bounded)) * c) + (fom_value_mrf_empty_bounded))) /\ (exists fom_gap_mrf_empty_bounded_value_bound. fom_gap_mrf_empty_bounded_value_bound + S (fom_value_mrf_empty_bounded) = B)))

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 · 3 reading checkpoints · 0 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 assumptionsL1–5

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro B
  4. L4
    intro i
  5. L5
    intro hi
02Separate the logical casesL6–6

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

  1. L6
    exfalso
03Use earlier factsL7–9

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

  1. L7
    specialize matrix_rank_no_index_below_zero (i)
  2. L8
    apply matrix_rank_no_index_below_zero
  3. L9
    exact hi

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro B
  4. 0004intro i
  5. 0005intro hi
  6. 0006exfalso
  7. 0007specialize matrix_rank_no_index_below_zero (i)
  8. 0008apply matrix_rank_no_index_below_zero
  9. 0009exact hi