DL0002

matrix_recursive_prefix_refl

A beta prefix is unchanged under its own exact code pair.

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. ∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(b,c,x,y)

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

Definition DAG

Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c l. (forall mdr_i_refl mdr_a_refl. (exists mdr_gap_reflb. mdr_gap_reflb + S (mdr_i_refl) = (l)) -> (((exists ff_h_mdr_reflo. ff_h_mdr_reflo + S (mdr_a_refl) = S ((S (mdr_i_refl)) * c)) /\ exists ff_q_mdr_reflo. b = ff_q_mdr_reflo * S ((S (mdr_i_refl)) * c) + (mdr_a_refl))) -> (((exists ff_h_mdr_refln. ff_h_mdr_refln + S (mdr_a_refl) = S ((S (mdr_i_refl)) * c)) /\ exists ff_q_mdr_refln. b = ff_q_mdr_refln * S ((S (mdr_i_refl)) * c) + (mdr_a_refl))))

Complete tactic proof in conservative notation

All 8 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

8 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 i
  5. L5
    intro a
  6. L6
    intro hi
  7. L7
    intro ha
02Use earlier factsL8–8

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

  1. L8
    exact ha

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro a
  6. 0006intro hi
  7. 0007intro ha
  8. 0008exact ha