DL002D

matrix_rank_bounded_prefix_value

The finite bounded-into witness bounds every actual decoded value, by beta uniqueness.

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

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

Definition DAG

Lt(a,b) · 5BetaAt(b,c,i,x) · 3

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c l B i a. (forall fom_index_mrf_value_source. (exists fom_gap_mrf_value_source_index_bound. fom_gap_mrf_value_source_index_bound + S (fom_index_mrf_value_source) = l) -> exists fom_value_mrf_value_source. ((((exists fom_beta_height_mrf_value_source_entry. fom_beta_height_mrf_value_source_entry + S (fom_value_mrf_value_source) = S ((S (fom_index_mrf_value_source)) * c)) /\ exists fom_beta_quotient_mrf_value_source_entry. b = fom_beta_quotient_mrf_value_source_entry * S ((S (fom_index_mrf_value_source)) * c) + (fom_value_mrf_value_source))) /\ (exists fom_gap_mrf_value_source_value_bound. fom_gap_mrf_value_source_value_bound + S (fom_value_mrf_value_source) = B))) -> (exists mdr_gap_value_index. mdr_gap_value_index + S (i) = (l)) -> (((exists ff_h_mdr_value_entry. ff_h_mdr_value_entry + S (a) = S ((S (i)) * c)) /\ exists ff_q_mdr_value_entry. b = ff_q_mdr_value_entry * S ((S (i)) * c) + (a))) -> (exists mdr_gap_value_bound. mdr_gap_value_bound + S (a) = (B))

Complete tactic proof in conservative notation

All 26 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

26 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 assumptionsL1–9

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro B
  5. L5
    intro i
  6. L6
    intro a
  7. L7
    intro hbounded
  8. L8
    intro hi
  9. L9
    intro ha
02Establish hentryL10–13

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

  1. L10
    have hentry : ∃ v. BetaAt(b,c,i,v) ∧ Lt(v,B)Definitions: BetaAt(b,c,i,v)Lt(v,B)Original native command in the exact edition
  2. L11
    specialize hbounded (i)
  3. L12
    apply hbounded
  4. L13
    exact hi
03Separate the logical casesL14–15

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

  1. L14
    cases hentry
  2. L15
    cases hentry_witness
04Establish heqL16–25

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

  1. L16
    have heq : a = x
  2. L17
    specialize beta_at_unique (b)
  3. L18
    specialize beta_at_unique (c)
  4. L19
    specialize beta_at_unique (i)
  5. L20
    specialize beta_at_unique (a)
  6. L21
    specialize beta_at_unique (x)
  7. L22
    apply beta_at_unique
  8. L23
    exact ha
  9. L24
    exact hentry_witness_left
  10. L25
    rewrite heq
05Use earlier factsL26–26

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

  1. L26
    exact hentry_witness_right

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro B
  5. 0005intro i
  6. 0006intro a
  7. 0007intro hbounded
  8. 0008intro hi
  9. 0009intro ha
  10. 0010have hentry : ∃ v. BetaAt(b,c,i,v) ∧ Lt(v,B)
  11. 0011specialize hbounded (i)
  12. 0012apply hbounded
  13. 0013exact hi
  14. 0014cases hentry
  15. 0015cases hentry_witness
  16. 0016have heq : a = x
  17. 0017specialize beta_at_unique (b)
  18. 0018specialize beta_at_unique (c)
  19. 0019specialize beta_at_unique (i)
  20. 0020specialize beta_at_unique (a)
  21. 0021specialize beta_at_unique (x)
  22. 0022apply beta_at_unique
  23. 0023exact ha
  24. 0024exact hentry_witness_left
  25. 0025rewrite heq
  26. 0026exact hentry_witness_right