DL003A

matrix_rank_bounded_prefix_extend

A checked final decoded value extends an existing bounded selector prefix.

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. ∀ a. (∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B)) → BetaAt(b,c,l,a) → Lt(a,B) → ∀ x. Lt(x,S l) → ∃ y. BetaAt(b,c,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) · 3

Actual proof prerequisites

finite_lt_succ_eq_or_lt · checked external prerequisite
Original expanded first-order statement 
forall b c l B a. (forall fom_index_mrf_extend_source. (exists fom_gap_mrf_extend_source_index_bound. fom_gap_mrf_extend_source_index_bound + S (fom_index_mrf_extend_source) = l) -> exists fom_value_mrf_extend_source. ((((exists fom_beta_height_mrf_extend_source_entry. fom_beta_height_mrf_extend_source_entry + S (fom_value_mrf_extend_source) = S ((S (fom_index_mrf_extend_source)) * c)) /\ exists fom_beta_quotient_mrf_extend_source_entry. b = fom_beta_quotient_mrf_extend_source_entry * S ((S (fom_index_mrf_extend_source)) * c) + (fom_value_mrf_extend_source))) /\ (exists fom_gap_mrf_extend_source_value_bound. fom_gap_mrf_extend_source_value_bound + S (fom_value_mrf_extend_source) = B))) -> (((exists ff_h_mdr_extend_entry. ff_h_mdr_extend_entry + S (a) = S ((S (l)) * c)) /\ exists ff_q_mdr_extend_entry. b = ff_q_mdr_extend_entry * S ((S (l)) * c) + (a))) -> (exists mdr_gap_extend_bound. mdr_gap_extend_bound + S (a) = (B)) -> (forall fom_index_mrf_extend_result. (exists fom_gap_mrf_extend_result_index_bound. fom_gap_mrf_extend_result_index_bound + S (fom_index_mrf_extend_result) = S l) -> exists fom_value_mrf_extend_result. ((((exists fom_beta_height_mrf_extend_result_entry. fom_beta_height_mrf_extend_result_entry + S (fom_value_mrf_extend_result) = S ((S (fom_index_mrf_extend_result)) * c)) /\ exists fom_beta_quotient_mrf_extend_result_entry. b = fom_beta_quotient_mrf_extend_result_entry * S ((S (fom_index_mrf_extend_result)) * c) + (fom_value_mrf_extend_result))) /\ (exists fom_gap_mrf_extend_result_value_bound. fom_gap_mrf_extend_result_value_bound + S (fom_value_mrf_extend_result) = B)))

Complete tactic proof in conservative notation

All 25 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

25 script commands · 7 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 l
  4. L4
    intro B
  5. L5
    intro a
  6. L6
    intro hbounded
  7. L7
    intro ha
  8. L8
    intro habound
  9. L9
    intro i
  10. L10
    intro hi
02Establish hcaseL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite lt succ eq or lt.

  1. L11
    have hcase : i = l ∨ Lt(i,l)Definitions: Lt(i,l)Original native command in the exact edition
  2. L12
    specialize finite_lt_succ_eq_or_lt (l)
  3. L13
    specialize finite_lt_succ_eq_or_lt (i)
  4. L14
    apply finite_lt_succ_eq_or_lt
  5. L15
    exact hi
03Separate the logical casesL16–16

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

  1. L16
    cases hcase
04Construct an explicit witnessL17–17

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

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

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

  1. L18
    split
06Calculate and transport equalitiesL19–20

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    rewrite hcase_left
  2. L20
    rewrite hcase_left
07Use earlier factsL21–25

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

  1. L21
    exact ha
  2. L22
    exact habound
  3. L23
    specialize hbounded (i)
  4. L24
    apply hbounded
  5. L25
    exact hcase_right

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro B
  5. 0005intro a
  6. 0006intro hbounded
  7. 0007intro ha
  8. 0008intro habound
  9. 0009intro i
  10. 0010intro hi
  11. 0011have hcase : i = l ∨ Lt(i,l)
  12. 0012specialize finite_lt_succ_eq_or_lt (l)
  13. 0013specialize finite_lt_succ_eq_or_lt (i)
  14. 0014apply finite_lt_succ_eq_or_lt
  15. 0015exact hi
  16. 0016cases hcase
  17. 0017exists a
  18. 0018split
  19. 0019rewrite hcase_left
  20. 0020rewrite hcase_left
  21. 0021exact ha
  22. 0022exact habound
  23. 0023specialize hbounded (i)
  24. 0024apply hbounded
  25. 0025exact hcase_right