DL006E

integer_vector_equal_reflexive

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

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

∀ ab. ∀ ac. ∀ db. ∀ dc. ∀ l. IntegerVectorEqual(ab,ac,db,dc,ab,ac,db,dc,l)

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

Definition DAG

IntegerVectorEqual(ab,ac,db,dc,eb,ec,fb,fc,l) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order 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)

Complete tactic proof in conservative notation

All 36 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

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.

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 assumptionsL1–10

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 defined 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