DL00A1

matrix_lattice_opposite_gaps_zero

Oppositely oriented nonnegative gaps can coexist only when both are zero, by actual cancellative natural arithmetic.

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. ∀ D. ∀ E. a = b + D → b = a + E → D = 0 ∧ E = 0

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

Definition DAG

none

Actual proof prerequisites

add_left_cancel · checked external prerequisiteadd_assoc · checked external prerequisiteadd_eq_zero_left · checked external prerequisiteadd_eq_zero_right · checked external prerequisite
Original expanded first-order statement 
forall a b D E. a = b + D -> b = a + E -> D = 0 /\ E = 0

Complete tactic proof in conservative notation

All 33 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

33 script commands · 10 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 assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro D
  4. L4
    intro E
  5. L5
    intro hfirst
  6. L6
    intro hsecond
02Establish hsumL7–16

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

  1. L7
    have hsum : D + E = 0
  2. L8
    specialize add_left_cancel (b)
  3. L9
    specialize add_left_cancel (D + E)
  4. L10
    specialize add_left_cancel (0)
  5. L11
    apply add_left_cancel
  6. L12
    trans (b + D) + E
  7. L13
    symm
  8. L14
    apply add_assoc
  9. L15
    trans a + E
  10. L16
    congr
03Calculate and transport equalitiesL17–17

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

  1. L17
    symm
04Use earlier factsL18–18

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

  1. L18
    exact hfirst
05Calculate and transport equalitiesL19–21

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

  1. L19
    refl
  2. L20
    trans b
  3. L21
    symm
06Use earlier factsL22–22

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

  1. L22
    exact hsecond
07Calculate and transport equalitiesL23–23

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

  1. L23
    symm
08Use earlier factsL24–24

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

  1. L24
    apply PA3
09Separate the logical casesL25–25

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

  1. L25
    split
10Use earlier factsL26–33

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

  1. L26
    specialize add_eq_zero_left (D)
  2. L27
    specialize add_eq_zero_left (E)
  3. L28
    apply add_eq_zero_left
  4. L29
    exact hsum
  5. L30
    specialize add_eq_zero_right (D)
  6. L31
    specialize add_eq_zero_right (E)
  7. L32
    apply add_eq_zero_right
  8. L33
    exact hsum

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro D
  4. 0004intro E
  5. 0005intro hfirst
  6. 0006intro hsecond
  7. 0007have hsum : D + E = 0
  8. 0008specialize add_left_cancel (b)
  9. 0009specialize add_left_cancel (D + E)
  10. 0010specialize add_left_cancel (0)
  11. 0011apply add_left_cancel
  12. 0012trans (b + D) + E
  13. 0013symm
  14. 0014apply add_assoc
  15. 0015trans a + E
  16. 0016congr
  17. 0017symm
  18. 0018exact hfirst
  19. 0019refl
  20. 0020trans b
  21. 0021symm
  22. 0022exact hsecond
  23. 0023symm
  24. 0024apply PA3
  25. 0025split
  26. 0026specialize add_eq_zero_left (D)
  27. 0027specialize add_eq_zero_left (E)
  28. 0028apply add_eq_zero_left
  29. 0029exact hsum
  30. 0030specialize add_eq_zero_right (D)
  31. 0031specialize add_eq_zero_right (E)
  32. 0032apply add_eq_zero_right
  33. 0033exact hsum