DL0042

matrix_rank_selected_prefix_empty

Every code witnesses the actual selected-submatrix prefix of length zero.

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. ∀ w. ∀ rb. ∀ rc. ∀ cb. ∀ cc. ∀ q. ∀ ub. ∀ uc. ∀ x. Lt(x,0) → ∃ y. (∃ z. ∃ n. ∃ m. ∃ k. x = q · z + n ∧ (Lt(n,q) ∧ (BetaAt(rb,rc,z,m) ∧ (BetaAt(cb,cc,n,k) ∧ BetaAt(b,c,m · w + k,y))))) ∧ BetaAt(ub,uc,x,y)

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

Definition DAG

Lt(a,b) · 2BetaAt(b,c,i,x) · 4

Actual proof prerequisites

matrix_rank_no_index_below_zero
Original expanded first-order statement 
forall b c w rb rc cb cc q ub uc. (forall mdr_i_empty_prefix. (exists mdr_gap_empty_prefixbound. mdr_gap_empty_prefixbound + S (mdr_i_empty_prefix) = (0)) -> exists mdr_a_empty_prefix. (((exists mdr_r_empty_prefixpoint mdr_s_empty_prefixpoint mdr_u_empty_prefixpoint mdr_v_empty_prefixpoint. ((mdr_i_empty_prefix = (q) * mdr_r_empty_prefixpoint + mdr_s_empty_prefixpoint) /\ ((exists mdr_gap_empty_prefixpointcolumn. mdr_gap_empty_prefixpointcolumn + S (mdr_s_empty_prefixpoint) = (q)) /\ ((((exists ff_h_mdr_empty_prefixpointrow_index. ff_h_mdr_empty_prefixpointrow_index + S (mdr_u_empty_prefixpoint) = S ((S (mdr_r_empty_prefixpoint)) * rc)) /\ exists ff_q_mdr_empty_prefixpointrow_index. rb = ff_q_mdr_empty_prefixpointrow_index * S ((S (mdr_r_empty_prefixpoint)) * rc) + (mdr_u_empty_prefixpoint))) /\ ((((exists ff_h_mdr_empty_prefixpointcolumn_index. ff_h_mdr_empty_prefixpointcolumn_index + S (mdr_v_empty_prefixpoint) = S ((S (mdr_s_empty_prefixpoint)) * cc)) /\ exists ff_q_mdr_empty_prefixpointcolumn_index. cb = ff_q_mdr_empty_prefixpointcolumn_index * S ((S (mdr_s_empty_prefixpoint)) * cc) + (mdr_v_empty_prefixpoint))) /\ (((exists ff_h_mdr_empty_prefixpointsource. ff_h_mdr_empty_prefixpointsource + S (mdr_a_empty_prefix) = S ((S ((mdr_u_empty_prefixpoint) * (w) + (mdr_v_empty_prefixpoint))) * c)) /\ exists ff_q_mdr_empty_prefixpointsource. b = ff_q_mdr_empty_prefixpointsource * S ((S ((mdr_u_empty_prefixpoint) * (w) + (mdr_v_empty_prefixpoint))) * c) + (mdr_a_empty_prefix)))))))) /\ (((exists ff_h_mdr_empty_prefixoutput. ff_h_mdr_empty_prefixoutput + S (mdr_a_empty_prefix) = S ((S (mdr_i_empty_prefix)) * uc)) /\ exists ff_q_mdr_empty_prefixoutput. ub = ff_q_mdr_empty_prefixoutput * S ((S (mdr_i_empty_prefix)) * uc) + (mdr_a_empty_prefix))))))

Complete tactic proof in conservative notation

All 16 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

16 script commands · 4 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 assumptionsL1–10

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro w
  4. L4
    intro rb
  5. L5
    intro rc
  6. L6
    intro cb
  7. L7
    intro cc
  8. L8
    intro q
  9. L9
    intro ub
  10. L10
    intro uc
02Fix variables and assumptionsL11–12

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

  1. L11
    intro i
  2. L12
    intro hi
03Separate the logical casesL13–13

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

  1. L13
    exfalso
04Use earlier factsL14–16

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

  1. L14
    specialize matrix_rank_no_index_below_zero (i)
  2. L15
    apply matrix_rank_no_index_below_zero
  3. L16
    exact hi

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro rb
  5. 0005intro rc
  6. 0006intro cb
  7. 0007intro cc
  8. 0008intro q
  9. 0009intro ub
  10. 0010intro uc
  11. 0011intro i
  12. 0012intro hi
  13. 0013exfalso
  14. 0014specialize matrix_rank_no_index_below_zero (i)
  15. 0015apply matrix_rank_no_index_below_zero
  16. 0016exact hi