DL0019

matrix_recursive_lt_add_left

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_le_add_left Stable theorem; checked-use authorized

Direct dependents

DL001A matrix_recursive_flattened_index_bound DL0095 matrix_integer_rectangular_index_bound

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 : exists mdr_gap_add_middle. mdr_gap_add_middle + (c + S a) = (c + b)
  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 exact 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 : exists mdr_gap_add_middle. mdr_gap_add_middle + (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