DL0004

matrix_recursive_prefix_restrict

An exact preserved beta prefix restricts to every constructively bounded shorter 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. ∀ u. ∀ v. ∀ l. ∀ k. Le(k,l) → (∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(u,v,x,y)) → ∀ x. ∀ y. Lt(x,k) → BetaAt(b,c,x,y) → BetaAt(u,v,x,y)

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

Definition DAG

Le(a,b) · 1Lt(a,b) · 2BetaAt(b,c,i,x) · 4

Actual proof prerequisites

lt_of_lt_of_le · checked external prerequisite
Original expanded first-order statement 
forall b c u v l k. (exists mdr_gap_bound. mdr_gap_bound + (k) = (l)) -> (forall mdr_i_source mdr_a_source. (exists mdr_gap_sourceb. mdr_gap_sourceb + S (mdr_i_source) = (l)) -> (((exists ff_h_mdr_sourceo. ff_h_mdr_sourceo + S (mdr_a_source) = S ((S (mdr_i_source)) * c)) /\ exists ff_q_mdr_sourceo. b = ff_q_mdr_sourceo * S ((S (mdr_i_source)) * c) + (mdr_a_source))) -> (((exists ff_h_mdr_sourcen. ff_h_mdr_sourcen + S (mdr_a_source) = S ((S (mdr_i_source)) * v)) /\ exists ff_q_mdr_sourcen. u = ff_q_mdr_sourcen * S ((S (mdr_i_source)) * v) + (mdr_a_source)))) -> (forall mdr_i_short mdr_a_short. (exists mdr_gap_shortb. mdr_gap_shortb + S (mdr_i_short) = (k)) -> (((exists ff_h_mdr_shorto. ff_h_mdr_shorto + S (mdr_a_short) = S ((S (mdr_i_short)) * c)) /\ exists ff_q_mdr_shorto. b = ff_q_mdr_shorto * S ((S (mdr_i_short)) * c) + (mdr_a_short))) -> (((exists ff_h_mdr_shortn. ff_h_mdr_shortn + S (mdr_a_short) = S ((S (mdr_i_short)) * v)) /\ exists ff_q_mdr_shortn. u = ff_q_mdr_shortn * S ((S (mdr_i_short)) * v) + (mdr_a_short))))

Complete tactic proof in conservative notation

All 22 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

22 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.

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 u
  4. L4
    intro v
  5. L5
    intro l
  6. L6
    intro k
  7. L7
    intro hk
  8. L8
    intro hprefix
  9. L9
    intro i
  10. L10
    intro a
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hi
  2. L12
    intro ha
03Use earlier factsL13–22

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

  1. L13
    specialize hprefix (i)
  2. L14
    specialize hprefix (a)
  3. L15
    apply hprefix
  4. L16
    specialize lt_of_lt_of_le (i)
  5. L17
    specialize lt_of_lt_of_le (k)
  6. L18
    specialize lt_of_lt_of_le (l)
  7. L19
    apply lt_of_lt_of_le
  8. L20
    exact hi
  9. L21
    exact hk
  10. L22
    exact ha

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro u
  4. 0004intro v
  5. 0005intro l
  6. 0006intro k
  7. 0007intro hk
  8. 0008intro hprefix
  9. 0009intro i
  10. 0010intro a
  11. 0011intro hi
  12. 0012intro ha
  13. 0013specialize hprefix (i)
  14. 0014specialize hprefix (a)
  15. 0015apply hprefix
  16. 0016specialize lt_of_lt_of_le (i)
  17. 0017specialize lt_of_lt_of_le (k)
  18. 0018specialize lt_of_lt_of_le (l)
  19. 0019apply lt_of_lt_of_le
  20. 0020exact hi
  21. 0021exact hk
  22. 0022exact ha