DL00A0

matrix_lattice_absolute_difference_exists

Construct an actual natural absolute difference of every signed pair using constructive natural order.

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

∀ p. ∀ n. ∃ D. p = n + D ∨ n = p + D

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

Definition DAG

Le(a,b) · 1

Actual proof prerequisites

le_or_lt · checked external prerequisitelt_to_le · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order statement 
forall p n. exists D. (((p) = (n) + (D)) \/ ((n) = (p) + (D)))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 13 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 assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro n
02Use earlier factsL3–4

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

  1. L3
    specialize le_or_lt p
  2. L4
    specialize le_or_lt n
03Separate the logical casesL5–6

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

  1. L5
    cases le_or_lt
  2. L6
    cases le_or_lt_left
04Construct an explicit witnessL7–7

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

  1. L7
    exists x
05Separate the logical casesL8–8

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

  1. L8
    right
06Calculate and transport equalitiesL9–10

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

  1. L9
    trans x + p
  2. L10
    symm
07Use earlier factsL11–12

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

  1. L11
    exact le_or_lt_left_witness
  2. L12
    apply add_comm
08Establish hreverseL13–17

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

  1. L13
    have hreverse : Le(n,p)Definitions: Le(n,p)Original native command in the exact edition
  2. L14
    specialize lt_to_le (n)
  3. L15
    specialize lt_to_le (p)
  4. L16
    apply lt_to_le
  5. L17
    exact le_or_lt_right
09Separate the logical casesL18–18

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

  1. L18
    cases hreverse
10Construct an explicit witnessL19–19

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

  1. L19
    exists x
11Separate the logical casesL20–20

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

  1. L20
    left
12Calculate and transport equalitiesL21–22

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

  1. L21
    trans x + n
  2. L22
    symm
13Use earlier factsL23–24

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

  1. L23
    exact hreverse_witness
  2. L24
    apply add_comm

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003specialize le_or_lt p
  4. 0004specialize le_or_lt n
  5. 0005cases le_or_lt
  6. 0006cases le_or_lt_left
  7. 0007exists x
  8. 0008right
  9. 0009trans x + p
  10. 0010symm
  11. 0011exact le_or_lt_left_witness
  12. 0012apply add_comm
  13. 0013have hreverse : Le(n,p)
  14. 0014specialize lt_to_le (n)
  15. 0015specialize lt_to_le (p)
  16. 0016apply lt_to_le
  17. 0017exact le_or_lt_right
  18. 0018cases hreverse
  19. 0019exists x
  20. 0020left
  21. 0021trans x + n
  22. 0022symm
  23. 0023exact hreverse_witness
  24. 0024apply add_comm