DL0035

matrix_rank_bounded_prefix_transport

Finite selector value bounds transport across actual pointwise equality of their code prefixes.

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. ∀ u. ∀ v. ∀ l. ∀ B. (∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(u,v,x,y)) → (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B)) → ∀ x. Lt(x,l) → ∃ y. BetaAt(u,v,x,y) ∧ Lt(y,B)

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

Definition DAG

Lt(a,b) · 6BetaAt(b,c,i,x) · 5

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c u v l B. (forall mdr_i_bounded_transport mdr_a_bounded_transport. (exists mdr_gap_bounded_transportb. mdr_gap_bounded_transportb + S (mdr_i_bounded_transport) = (l)) -> (((exists ff_h_mdr_bounded_transporto. ff_h_mdr_bounded_transporto + S (mdr_a_bounded_transport) = S ((S (mdr_i_bounded_transport)) * c)) /\ exists ff_q_mdr_bounded_transporto. b = ff_q_mdr_bounded_transporto * S ((S (mdr_i_bounded_transport)) * c) + (mdr_a_bounded_transport))) -> (((exists ff_h_mdr_bounded_transportn. ff_h_mdr_bounded_transportn + S (mdr_a_bounded_transport) = S ((S (mdr_i_bounded_transport)) * v)) /\ exists ff_q_mdr_bounded_transportn. u = ff_q_mdr_bounded_transportn * S ((S (mdr_i_bounded_transport)) * v) + (mdr_a_bounded_transport)))) -> (forall fom_index_mrf_bounded_source. (exists fom_gap_mrf_bounded_source_index_bound. fom_gap_mrf_bounded_source_index_bound + S (fom_index_mrf_bounded_source) = l) -> exists fom_value_mrf_bounded_source. ((((exists fom_beta_height_mrf_bounded_source_entry. fom_beta_height_mrf_bounded_source_entry + S (fom_value_mrf_bounded_source) = S ((S (fom_index_mrf_bounded_source)) * c)) /\ exists fom_beta_quotient_mrf_bounded_source_entry. b = fom_beta_quotient_mrf_bounded_source_entry * S ((S (fom_index_mrf_bounded_source)) * c) + (fom_value_mrf_bounded_source))) /\ (exists fom_gap_mrf_bounded_source_value_bound. fom_gap_mrf_bounded_source_value_bound + S (fom_value_mrf_bounded_source) = B))) -> (forall fom_index_mrf_bounded_target. (exists fom_gap_mrf_bounded_target_index_bound. fom_gap_mrf_bounded_target_index_bound + S (fom_index_mrf_bounded_target) = l) -> exists fom_value_mrf_bounded_target. ((((exists fom_beta_height_mrf_bounded_target_entry. fom_beta_height_mrf_bounded_target_entry + S (fom_value_mrf_bounded_target) = S ((S (fom_index_mrf_bounded_target)) * v)) /\ exists fom_beta_quotient_mrf_bounded_target_entry. u = fom_beta_quotient_mrf_bounded_target_entry * S ((S (fom_index_mrf_bounded_target)) * v) + (fom_value_mrf_bounded_target))) /\ (exists fom_gap_mrf_bounded_target_value_bound. fom_gap_mrf_bounded_target_value_bound + S (fom_value_mrf_bounded_target) = B)))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 6 reading checkpoints · 1 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 b
  2. L2
    intro c
  3. L3
    intro u
  4. L4
    intro v
  5. L5
    intro l
  6. L6
    intro B
  7. L7
    intro hprefix
  8. L8
    intro hbounded
  9. L9
    intro i
  10. L10
    intro hi
02Establish hentryL11–14

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

  1. L11
    have hentry : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,B)Definitions: BetaAt(b,c,i,a)Lt(a,B)Original native command in the exact edition
  2. L12
    specialize hbounded (i)
  3. L13
    apply hbounded
  4. L14
    exact hi
03Separate the logical casesL15–16

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

  1. L15
    cases hentry
  2. L16
    cases hentry_witness
04Construct an explicit witnessL17–17

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

  1. L17
    exists x
05Separate the logical casesL18–18

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

  1. L18
    split
06Use earlier factsL19–24

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

  1. L19
    specialize hprefix (i)
  2. L20
    specialize hprefix (x)
  3. L21
    apply hprefix
  4. L22
    exact hi
  5. L23
    exact hentry_witness_left
  6. L24
    exact hentry_witness_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro u
  4. 0004intro v
  5. 0005intro l
  6. 0006intro B
  7. 0007intro hprefix
  8. 0008intro hbounded
  9. 0009intro i
  10. 0010intro hi
  11. 0011have hentry : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,B)
  12. 0012specialize hbounded (i)
  13. 0013apply hbounded
  14. 0014exact hi
  15. 0015cases hentry
  16. 0016cases hentry_witness
  17. 0017exists x
  18. 0018split
  19. 0019specialize hprefix (i)
  20. 0020specialize hprefix (x)
  21. 0021apply hprefix
  22. 0022exact hi
  23. 0023exact hentry_witness_left
  24. 0024exact hentry_witness_right