DL00A0

matrix_lattice_absolute_difference_exists

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

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

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 p n. exists D. (((p) = (n) + (D)) \/ ((n) = (p) + (D)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 24 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_or_lt Stable theorem; checked-use authorized lt_to_le Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

DL00A7 absolute_recursive_determinant_exists DL00AA positive_determinant_matrix_data_from_nonzero

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

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.

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 : exists mdr_gap_absolute_reverse. mdr_gap_absolute_reverse + (n) = (p)
  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 exact 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 : exists mdr_gap_absolute_reverse. mdr_gap_absolute_reverse + (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