DL0019

matrix_recursive_lt_add_left

Strict natural order is preserved by a common left summand, by explicit successor transport.

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

∀ a. ∀ b. ∀ c. Lt(a,b) → Lt(c + a,c + b)

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

Definition DAG

Le(a,b) · 1Lt(a,b) · 2

Actual proof prerequisites

add_le_add_left · checked external prerequisite
Original expanded first-order statement 
forall a b c. (exists mdr_gap_add_source. mdr_gap_add_source + S (a) = (b)) -> (exists mdr_gap_add_result. mdr_gap_add_result + S (c + a) = (c + b))

Complete tactic proof in conservative notation

All 14 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

14 script commands · 3 reading checkpoints · 2 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro hab
02Establish heqL5–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA4.

  1. L5
    have heq : c + S a = S (c + a)
  2. L6
    apply PA4
03Establish hsumL7–14

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

  1. L7
    have hsum : Le(c + S a,c + b)Definitions: Le(c + S a,c + b)Original native command in the exact edition
  2. L8
    specialize add_le_add_left (S a)
  3. L9
    specialize add_le_add_left (b)
  4. L10
    specialize add_le_add_left (c)
  5. L11
    apply add_le_add_left
  6. L12
    exact hab
  7. L13
    rewrite heq at hsum
  8. L14
    exact hsum

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro hab
  5. 0005have heq : c + S a = S (c + a)
  6. 0006apply PA4
  7. 0007have hsum : Le(c + S a,c + b)
  8. 0008specialize add_le_add_left (S a)
  9. 0009specialize add_le_add_left (b)
  10. 0010specialize add_le_add_left (c)
  11. 0011apply add_le_add_left
  12. 0012exact hab
  13. 0013rewrite heq at hsum
  14. 0014exact hsum