DL006E

integer_vector_equal_reflexive

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

Any signed coded vector equals itself as an integer vector, using functionality of the actual positive and negative decoded entries.

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 ab ac db dc l. (forall ics_index_equal_refl ics_value0_equal_refl ics_value1_equal_refl ics_value2_equal_refl ics_value3_equal_refl. (exists ics_gap_equal_refl_bound. ics_gap_equal_refl_bound + S (ics_index_equal_refl) = (l)) -> (((exists fs_h_ics_equal_refl_at0. fs_h_ics_equal_refl_at0 + S (ics_value0_equal_refl) = S ((S (ics_index_equal_refl)) * ac)) /\ exists fs_q_ics_equal_refl_at0. ab = fs_q_ics_equal_refl_at0 * S ((S (ics_index_equal_refl)) * ac) + (ics_value0_equal_refl))) -> (((exists fs_h_ics_equal_refl_at1. fs_h_ics_equal_refl_at1 + S (ics_value1_equal_refl) = S ((S (ics_index_equal_refl)) * dc)) /\ exists fs_q_ics_equal_refl_at1. db = fs_q_ics_equal_refl_at1 * S ((S (ics_index_equal_refl)) * dc) + (ics_value1_equal_refl))) -> (((exists fs_h_ics_equal_refl_at2. fs_h_ics_equal_refl_at2 + S (ics_value2_equal_refl) = S ((S (ics_index_equal_refl)) * ac)) /\ exists fs_q_ics_equal_refl_at2. ab = fs_q_ics_equal_refl_at2 * S ((S (ics_index_equal_refl)) * ac) + (ics_value2_equal_refl))) -> (((exists fs_h_ics_equal_refl_at3. fs_h_ics_equal_refl_at3 + S (ics_value3_equal_refl) = S ((S (ics_index_equal_refl)) * dc)) /\ exists fs_q_ics_equal_refl_at3. db = fs_q_ics_equal_refl_at3 * S ((S (ics_index_equal_refl)) * dc) + (ics_value3_equal_refl))) -> ics_value0_equal_refl + ics_value3_equal_refl = ics_value2_equal_refl + ics_value1_equal_refl)

Constructive proof overview

Generated structural guide

Any signed coded vector equals itself as an integer vector, using functionality of the actual positive and negative decoded entries.

The unchanged tactic script uses 1 declared prerequisite and contains 36 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

DL0077 integer_matrix_vector_product_exists DL0084 integer_column_span_negate_exists

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

36 script commands · 5 reading checkpoints · 2 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 assumptionsL1–10

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

  1. L1
    intro ab
  2. L2
    intro ac
  3. L3
    intro db
  4. L4
    intro dc
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro a
  8. L8
    intro b
  9. L9
    intro c
  10. L10
    intro d
02Fix variables and assumptionsL11–15

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hb
  4. L14
    intro hc
  5. L15
    intro hd
03Establish hpositiveL16–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L16
    have hpositive : a = c
  2. L17
    specialize beta_at_unique (ab)
  3. L18
    specialize beta_at_unique (ac)
  4. L19
    specialize beta_at_unique (i)
  5. L20
    specialize beta_at_unique (a)
  6. L21
    specialize beta_at_unique (c)
  7. L22
    apply beta_at_unique
  8. L23
    exact ha
  9. L24
    exact hc
04Establish hnegativeL25–34

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L25
    have hnegative : d = b
  2. L26
    specialize beta_at_unique (db)
  3. L27
    specialize beta_at_unique (dc)
  4. L28
    specialize beta_at_unique (i)
  5. L29
    specialize beta_at_unique (d)
  6. L30
    specialize beta_at_unique (b)
  7. L31
    apply beta_at_unique
  8. L32
    exact hd
  9. L33
    exact hb
  10. L34
    congr
05Use earlier factsL35–36

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

  1. L35
    exact hpositive
  2. L36
    exact hnegative

Library-wide reading audit

Original exact command ledger · 36 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro db
  4. 0004intro dc
  5. 0005intro l
  6. 0006intro i
  7. 0007intro a
  8. 0008intro b
  9. 0009intro c
  10. 0010intro d
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hb
  14. 0014intro hc
  15. 0015intro hd
  16. 0016have hpositive : a = c
  17. 0017specialize beta_at_unique (ab)
  18. 0018specialize beta_at_unique (ac)
  19. 0019specialize beta_at_unique (i)
  20. 0020specialize beta_at_unique (a)
  21. 0021specialize beta_at_unique (c)
  22. 0022apply beta_at_unique
  23. 0023exact ha
  24. 0024exact hc
  25. 0025have hnegative : d = b
  26. 0026specialize beta_at_unique (db)
  27. 0027specialize beta_at_unique (dc)
  28. 0028specialize beta_at_unique (i)
  29. 0029specialize beta_at_unique (d)
  30. 0030specialize beta_at_unique (b)
  31. 0031apply beta_at_unique
  32. 0032exact hd
  33. 0033exact hb
  34. 0034congr
  35. 0035exact hpositive
  36. 0036exact hnegative