DL002E

matrix_rank_common_multiple_divides

A common multiple contains each explicitly bounded positive natural divisor, with the exact legacy bound convention.

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

∀ T. ∀ B. ∀ t. (∀ x. (∃ y. S x + S y = S B) → ∃ y. T = S x · y) → Lt(t,B) → ∃ x. T = S t · x

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

add_comm · checked external prerequisite
Original expanded first-order statement 
forall T B t. (forall mdr_t_common_source. (exists mdr_h_common_source. S mdr_t_common_source + S mdr_h_common_source = S (B)) -> exists mdr_q_common_source. T = S mdr_t_common_source * mdr_q_common_source) -> (exists mdr_gap_factor_bound. mdr_gap_factor_bound + S (t) = (B)) -> (exists mdr_q_factor_result. T = (S t) * mdr_q_factor_result)

Complete tactic proof in conservative notation

All 15 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

15 script commands · 6 reading checkpoints · 1 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 assumptionsL1–5

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

  1. L1
    intro T
  2. L2
    intro B
  3. L3
    intro t
  4. L4
    intro hcommon
  5. L5
    intro ht
02Separate the logical casesL6–6

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

  1. L6
    cases ht
03Use earlier factsL7–8

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

  1. L7
    specialize hcommon (t)
  2. L8
    apply hcommon
04Construct an explicit witnessL9–9

Supply the displayed value, then prove that it has the required property.

  1. L9
    exists x
05Calculate and transport equalitiesL10–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L10
    rewrite PA4
06Establish hsumL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add comm.

  1. L11
    have hsum : S t + x = x + S t
  2. L12
    apply add_comm
  3. L13
    rewrite hsum
  4. L14
    rewrite ht_witness
  5. L15
    refl

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro T
  2. 0002intro B
  3. 0003intro t
  4. 0004intro hcommon
  5. 0005intro ht
  6. 0006cases ht
  7. 0007specialize hcommon (t)
  8. 0008apply hcommon
  9. 0009exists x
  10. 0010rewrite PA4
  11. 0011have hsum : S t + x = x + S t
  12. 0012apply add_comm
  13. 0013rewrite hsum
  14. 0014rewrite ht_witness
  15. 0015refl