DL0039

matrix_rank_bounded_prefix_drop_last

Deleting the last index preserves a finite selector value bound.

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. ∀ l. ∀ B. (∀ x. Lt(x,S l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B)) → ∀ x. Lt(x,l) → ∃ 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) · 4BetaAt(b,c,i,x) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order statement 
forall b c l B. (forall fom_index_mrf_drop_source. (exists fom_gap_mrf_drop_source_index_bound. fom_gap_mrf_drop_source_index_bound + S (fom_index_mrf_drop_source) = S l) -> exists fom_value_mrf_drop_source. ((((exists fom_beta_height_mrf_drop_source_entry. fom_beta_height_mrf_drop_source_entry + S (fom_value_mrf_drop_source) = S ((S (fom_index_mrf_drop_source)) * c)) /\ exists fom_beta_quotient_mrf_drop_source_entry. b = fom_beta_quotient_mrf_drop_source_entry * S ((S (fom_index_mrf_drop_source)) * c) + (fom_value_mrf_drop_source))) /\ (exists fom_gap_mrf_drop_source_value_bound. fom_gap_mrf_drop_source_value_bound + S (fom_value_mrf_drop_source) = B))) -> (forall fom_index_mrf_drop_result. (exists fom_gap_mrf_drop_result_index_bound. fom_gap_mrf_drop_result_index_bound + S (fom_index_mrf_drop_result) = l) -> exists fom_value_mrf_drop_result. ((((exists fom_beta_height_mrf_drop_result_entry. fom_beta_height_mrf_drop_result_entry + S (fom_value_mrf_drop_result) = S ((S (fom_index_mrf_drop_result)) * c)) /\ exists fom_beta_quotient_mrf_drop_result_entry. b = fom_beta_quotient_mrf_drop_result_entry * S ((S (fom_index_mrf_drop_result)) * c) + (fom_value_mrf_drop_result))) /\ (exists fom_gap_mrf_drop_result_value_bound. fom_gap_mrf_drop_result_value_bound + S (fom_value_mrf_drop_result) = B)))

Complete tactic proof in conservative notation

All 13 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

13 script commands · 2 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro B
  5. L5
    intro hbounded
  6. L6
    intro i
  7. L7
    intro hi
02Use earlier factsL8–13

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

  1. L8
    specialize hbounded (i)
  2. L9
    apply hbounded
  3. L10
    specialize le_succ (S i)
  4. L11
    specialize le_succ (l)
  5. L12
    apply le_succ
  6. L13
    exact hi

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro B
  5. 0005intro hbounded
  6. 0006intro i
  7. 0007intro hi
  8. 0008specialize hbounded (i)
  9. 0009apply hbounded
  10. 0010specialize le_succ (S i)
  11. 0011specialize le_succ (l)
  12. 0012apply le_succ
  13. 0013exact hi