DL00A4

matrix_lattice_pair_nonzero_of_absolute

A positive actual absolute difference rules out equal signed components, without assuming a canonical representative.

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. ¬D = 0 → p = n + D ∨ n = p + D → ¬p = n

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

Definition DAG

none

Actual proof prerequisites

add_left_cancel · checked external prerequisite
Original expanded first-order statement 
forall p n D. ~(D = 0) -> (((p) = (n) + (D)) \/ ((n) = (p) + (D))) -> ~(p = n)

Complete tactic proof in conservative notation

All 31 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

31 script commands · 16 reading checkpoints · 0 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 assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro D
  4. L4
    intro habsolute
  5. L5
    intro hgap
  6. L6
    intro hpair
02Use earlier factsL7–7

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

  1. L7
    apply habsolute
03Separate the logical casesL8–8

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

  1. L8
    cases hgap
04Use earlier factsL9–12

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

  1. L9
    specialize add_left_cancel (n)
  2. L10
    specialize add_left_cancel (D)
  3. L11
    specialize add_left_cancel (0)
  4. L12
    apply add_left_cancel
05Calculate and transport equalitiesL13–14

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

  1. L13
    trans p
  2. L14
    symm
06Use earlier factsL15–15

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

  1. L15
    exact hgap_left
07Calculate and transport equalitiesL16–16

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

  1. L16
    trans n
08Use earlier factsL17–17

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

  1. L17
    exact hpair
09Calculate and transport equalitiesL18–18

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

  1. L18
    symm
10Use earlier factsL19–23

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

  1. L19
    apply PA3
  2. L20
    specialize add_left_cancel (p)
  3. L21
    specialize add_left_cancel (D)
  4. L22
    specialize add_left_cancel (0)
  5. L23
    apply add_left_cancel
11Calculate and transport equalitiesL24–25

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

  1. L24
    trans n
  2. L25
    symm
12Use earlier factsL26–26

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

  1. L26
    exact hgap_right
13Calculate and transport equalitiesL27–28

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

  1. L27
    trans p
  2. L28
    symm
14Use earlier factsL29–29

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

  1. L29
    exact hpair
15Calculate and transport equalitiesL30–30

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

  1. L30
    symm
16Use earlier factsL31–31

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

  1. L31
    apply PA3

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro D
  4. 0004intro habsolute
  5. 0005intro hgap
  6. 0006intro hpair
  7. 0007apply habsolute
  8. 0008cases hgap
  9. 0009specialize add_left_cancel (n)
  10. 0010specialize add_left_cancel (D)
  11. 0011specialize add_left_cancel (0)
  12. 0012apply add_left_cancel
  13. 0013trans p
  14. 0014symm
  15. 0015exact hgap_left
  16. 0016trans n
  17. 0017exact hpair
  18. 0018symm
  19. 0019apply PA3
  20. 0020specialize add_left_cancel (p)
  21. 0021specialize add_left_cancel (D)
  22. 0022specialize add_left_cancel (0)
  23. 0023apply add_left_cancel
  24. 0024trans n
  25. 0025symm
  26. 0026exact hgap_right
  27. 0027trans p
  28. 0028symm
  29. 0029exact hpair
  30. 0030symm
  31. 0031apply PA3