DL00AF

matrix_lattice_identity_is_selector

The actual identity selector is in range and genuinely injective, not merely a supplied permutation label.

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. ∀ d. IdentityMatrixSelector(b,c,d) → FiniteMatrixSelector(b,c,d,d)

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

Definition DAG

FiniteMatrixSelector(b,c,l,B) · 1IdentityMatrixSelector(b,c,l) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisiteeq_trans · checked external prerequisite
Original expanded first-order statement 
forall b c d. (forall mdr_i_identity_source. (exists mdr_gap_identity_sourcebound. mdr_gap_identity_sourcebound + S (mdr_i_identity_source) = (d)) -> (((exists ff_h_mdr_identity_sourceentry. ff_h_mdr_identity_sourceentry + S (mdr_i_identity_source) = S ((S (mdr_i_identity_source)) * c)) /\ exists ff_q_mdr_identity_sourceentry. b = ff_q_mdr_identity_sourceentry * S ((S (mdr_i_identity_source)) * c) + (mdr_i_identity_source)))) -> (((forall fom_index_mrf_identity_selectorbound. (exists fom_gap_mrf_identity_selectorbound_index_bound. fom_gap_mrf_identity_selectorbound_index_bound + S (fom_index_mrf_identity_selectorbound) = d) -> exists fom_value_mrf_identity_selectorbound. ((((exists fom_beta_height_mrf_identity_selectorbound_entry. fom_beta_height_mrf_identity_selectorbound_entry + S (fom_value_mrf_identity_selectorbound) = S ((S (fom_index_mrf_identity_selectorbound)) * c)) /\ exists fom_beta_quotient_mrf_identity_selectorbound_entry. b = fom_beta_quotient_mrf_identity_selectorbound_entry * S ((S (fom_index_mrf_identity_selectorbound)) * c) + (fom_value_mrf_identity_selectorbound))) /\ (exists fom_gap_mrf_identity_selectorbound_value_bound. fom_gap_mrf_identity_selectorbound_value_bound + S (fom_value_mrf_identity_selectorbound) = d))) /\ (forall mdr_i_identity_selectordistinct mdr_j_identity_selectordistinct mdr_a_identity_selectordistinct. (exists mdr_gap_identity_selectordistincti. mdr_gap_identity_selectordistincti + S (mdr_i_identity_selectordistinct) = (d)) -> (exists mdr_gap_identity_selectordistinctj. mdr_gap_identity_selectordistinctj + S (mdr_j_identity_selectordistinct) = (d)) -> (((exists ff_h_mdr_identity_selectordistinctfirst. ff_h_mdr_identity_selectordistinctfirst + S (mdr_a_identity_selectordistinct) = S ((S (mdr_i_identity_selectordistinct)) * c)) /\ exists ff_q_mdr_identity_selectordistinctfirst. b = ff_q_mdr_identity_selectordistinctfirst * S ((S (mdr_i_identity_selectordistinct)) * c) + (mdr_a_identity_selectordistinct))) -> (((exists ff_h_mdr_identity_selectordistinctsecond. ff_h_mdr_identity_selectordistinctsecond + S (mdr_a_identity_selectordistinct) = S ((S (mdr_j_identity_selectordistinct)) * c)) /\ exists ff_q_mdr_identity_selectordistinctsecond. b = ff_q_mdr_identity_selectordistinctsecond * S ((S (mdr_j_identity_selectordistinct)) * c) + (mdr_a_identity_selectordistinct))) -> mdr_i_identity_selectordistinct = mdr_j_identity_selectordistinct)))

Complete tactic proof in conservative notation

All 44 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

44 script commands · 10 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 assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro hidentity
02Separate the logical casesL5–5

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

  1. L5
    split
03Fix variables and assumptionsL6–7

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

  1. L6
    intro i
  2. L7
    intro hi
04Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists i
05Separate the logical casesL9–9

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

  1. L9
    split
06Use earlier factsL10–13

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

  1. L10
    specialize hidentity (i)
  2. L11
    apply hidentity
  3. L12
    exact hi
  4. L13
    exact hi
07Fix variables and assumptionsL14–20

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

  1. L14
    intro i
  2. L15
    intro j
  3. L16
    intro a
  4. L17
    intro hi
  5. L18
    intro hj
  6. L19
    intro ha
  7. L20
    intro hb
08Use earlier factsL21–30

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

  1. L21
    specialize eq_trans (i)
  2. L22
    specialize eq_trans (a)
  3. L23
    specialize eq_trans (j)
  4. L24
    apply eq_trans
  5. L25
    specialize beta_at_unique (b)
  6. L26
    specialize beta_at_unique (c)
  7. L27
    specialize beta_at_unique (i)
  8. L28
    specialize beta_at_unique (i)
  9. L29
    specialize beta_at_unique (a)
  10. L30
    apply beta_at_unique
09Use earlier factsL31–40

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

  1. L31
    specialize hidentity (i)
  2. L32
    apply hidentity
  3. L33
    exact hi
  4. L34
    exact ha
  5. L35
    specialize beta_at_unique (b)
  6. L36
    specialize beta_at_unique (c)
  7. L37
    specialize beta_at_unique (j)
  8. L38
    specialize beta_at_unique (a)
  9. L39
    specialize beta_at_unique (j)
  10. L40
    apply beta_at_unique
10Use earlier factsL41–44

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

  1. L41
    exact hb
  2. L42
    specialize hidentity (j)
  3. L43
    apply hidentity
  4. L44
    exact hj

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro hidentity
  5. 0005split
  6. 0006intro i
  7. 0007intro hi
  8. 0008exists i
  9. 0009split
  10. 0010specialize hidentity (i)
  11. 0011apply hidentity
  12. 0012exact hi
  13. 0013exact hi
  14. 0014intro i
  15. 0015intro j
  16. 0016intro a
  17. 0017intro hi
  18. 0018intro hj
  19. 0019intro ha
  20. 0020intro hb
  21. 0021specialize eq_trans (i)
  22. 0022specialize eq_trans (a)
  23. 0023specialize eq_trans (j)
  24. 0024apply eq_trans
  25. 0025specialize beta_at_unique (b)
  26. 0026specialize beta_at_unique (c)
  27. 0027specialize beta_at_unique (i)
  28. 0028specialize beta_at_unique (i)
  29. 0029specialize beta_at_unique (a)
  30. 0030apply beta_at_unique
  31. 0031specialize hidentity (i)
  32. 0032apply hidentity
  33. 0033exact hi
  34. 0034exact ha
  35. 0035specialize beta_at_unique (b)
  36. 0036specialize beta_at_unique (c)
  37. 0037specialize beta_at_unique (j)
  38. 0038specialize beta_at_unique (a)
  39. 0039specialize beta_at_unique (j)
  40. 0040apply beta_at_unique
  41. 0041exact hb
  42. 0042specialize hidentity (j)
  43. 0043apply hidentity
  44. 0044exact hj