DL0071

integer_vector_equal_components_zero

Equal positive and negative beta streams represent the zero vector at every finite dimension.

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

∀ b. ∀ c. ∀ l. IntegerVectorZero(b,c,b,c,l)

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

Definition DAG

IntegerVectorZero(ab,ac,db,dc,l) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c l. (forall ics_index_zero_same ics_value0_zero_same ics_value1_zero_same. (exists ics_gap_zero_same_bound. ics_gap_zero_same_bound + S (ics_index_zero_same) = (l)) -> (((exists fs_h_ics_zero_same_at0. fs_h_ics_zero_same_at0 + S (ics_value0_zero_same) = S ((S (ics_index_zero_same)) * c)) /\ exists fs_q_ics_zero_same_at0. b = fs_q_ics_zero_same_at0 * S ((S (ics_index_zero_same)) * c) + (ics_value0_zero_same))) -> (((exists fs_h_ics_zero_same_at1. fs_h_ics_zero_same_at1 + S (ics_value1_zero_same) = S ((S (ics_index_zero_same)) * c)) /\ exists fs_q_ics_zero_same_at1. b = fs_q_ics_zero_same_at1 * S ((S (ics_index_zero_same)) * c) + (ics_value1_zero_same))) -> ics_value0_zero_same = ics_value1_zero_same)

Complete tactic proof in conservative notation

All 17 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

17 script commands · 2 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 assumptionsL1–9

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro i
  5. L5
    intro a
  6. L6
    intro d
  7. L7
    intro hi
  8. L8
    intro ha
  9. L9
    intro hd
02Use earlier factsL10–17

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

  1. L10
    specialize beta_at_unique (b)
  2. L11
    specialize beta_at_unique (c)
  3. L12
    specialize beta_at_unique (i)
  4. L13
    specialize beta_at_unique (a)
  5. L14
    specialize beta_at_unique (d)
  6. L15
    apply beta_at_unique
  7. L16
    exact ha
  8. L17
    exact hd

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro a
  6. 0006intro d
  7. 0007intro hi
  8. 0008intro ha
  9. 0009intro hd
  10. 0010specialize beta_at_unique (b)
  11. 0011specialize beta_at_unique (c)
  12. 0012specialize beta_at_unique (i)
  13. 0013specialize beta_at_unique (a)
  14. 0014specialize beta_at_unique (d)
  15. 0015apply beta_at_unique
  16. 0016exact ha
  17. 0017exact hd