DL0071

integer_vector_equal_components_zero

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

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

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 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 17 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

DL007F integer_column_span_contains_zero

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

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.

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 exact 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