DL00AE

matrix_lattice_identity_selector_exists

Construct an actual beta-coded identity selector of every natural length from the checked finite range construction.

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

∀ d. ∃ b. ∃ c. IdentityMatrixSelector(b,c,d)

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

Definition DAG

IdentityMatrixSelector(b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_range_exists · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall d. exists b c. (forall mdr_i_identity_exists. (exists mdr_gap_identity_existsbound. mdr_gap_identity_existsbound + S (mdr_i_identity_exists) = (d)) -> (((exists ff_h_mdr_identity_existsentry. ff_h_mdr_identity_existsentry + S (mdr_i_identity_exists) = S ((S (mdr_i_identity_exists)) * c)) /\ exists ff_q_mdr_identity_existsentry. b = ff_q_mdr_identity_existsentry * S ((S (mdr_i_identity_exists)) * c) + (mdr_i_identity_exists))))

Complete tactic proof in conservative notation

All 20 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

20 script commands · 7 reading checkpoints · 3 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 assumptionsL1–1

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

  1. L1
    intro d
02Establish hrangeL2–5

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

  1. L2
    have hrange : ∃ b. ∃ c. ∀ i. Lt(i,d) → BetaAt(b,c,i,0 + i)Definitions: Lt(i,d)BetaAt(b,c,i,0 + i)Original native command in the exact edition
  2. L3
    specialize beta_range_exists (0)
  3. L4
    specialize beta_range_exists (d)
  4. L5
    apply beta_range_exists
03Separate the logical casesL6–7

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

  1. L6
    cases hrange
  2. L7
    cases hrange_witness
04Construct an explicit witnessL8–9

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

  1. L8
    exists x
  2. L9
    exists x1
05Fix variables and assumptionsL10–11

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

  1. L10
    intro i
  2. L11
    intro hi
06Establish hentryL12–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hrange witness witness.

  1. L12
    have hentry : BetaAt(x,x1,i,0 + i)Definitions: BetaAt(x,x1,i,0 + i)Original native command in the exact edition
  2. L13
    specialize hrange_witness_witness (i)
  3. L14
    apply hrange_witness_witness
  4. L15
    exact hi
07Establish hzeroL16–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply zero add.

  1. L16
    have hzero : 0 + i = i
  2. L17
    apply zero_add
  3. L18
    rewrite hzero at hentry
  4. L19
    rewrite hzero at hentry
  5. L20
    exact hentry

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro d
  2. 0002have hrange : ∃ b. ∃ c. ∀ i. Lt(i,d) → BetaAt(b,c,i,0 + i)
  3. 0003specialize beta_range_exists (0)
  4. 0004specialize beta_range_exists (d)
  5. 0005apply beta_range_exists
  6. 0006cases hrange
  7. 0007cases hrange_witness
  8. 0008exists x
  9. 0009exists x1
  10. 0010intro i
  11. 0011intro hi
  12. 0012have hentry : BetaAt(x,x1,i,0 + i)
  13. 0013specialize hrange_witness_witness (i)
  14. 0014apply hrange_witness_witness
  15. 0015exact hi
  16. 0016have hzero : 0 + i = i
  17. 0017apply zero_add
  18. 0018rewrite hzero at hentry
  19. 0019rewrite hzero at hentry
  20. 0020exact hentry