DL0037

matrix_rank_injective_prefix_decidable

Selector injectivity is constructively decidable using the existing witnessed-collision theorem.

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. (∀ x. ∀ y. ∀ z. Lt(x,l) → Lt(y,l) → BetaAt(b,c,x,z) → BetaAt(b,c,y,z) → x = y) ∨ ¬(∀ x. ∀ y. ∀ z. Lt(x,l) → Lt(y,l) → BetaAt(b,c,x,z) → BetaAt(b,c,y,z) → x = y)

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

Definition DAG

Lt(a,b) · 8BetaAt(b,c,i,x) · 8

Actual proof prerequisites

finite_prefix_collision_or_injective · checked external prerequisite
Original expanded first-order statement 
forall b c l. (forall mdr_i_injective_yes mdr_j_injective_yes mdr_a_injective_yes. (exists mdr_gap_injective_yesi. mdr_gap_injective_yesi + S (mdr_i_injective_yes) = (l)) -> (exists mdr_gap_injective_yesj. mdr_gap_injective_yesj + S (mdr_j_injective_yes) = (l)) -> (((exists ff_h_mdr_injective_yesfirst. ff_h_mdr_injective_yesfirst + S (mdr_a_injective_yes) = S ((S (mdr_i_injective_yes)) * c)) /\ exists ff_q_mdr_injective_yesfirst. b = ff_q_mdr_injective_yesfirst * S ((S (mdr_i_injective_yes)) * c) + (mdr_a_injective_yes))) -> (((exists ff_h_mdr_injective_yessecond. ff_h_mdr_injective_yessecond + S (mdr_a_injective_yes) = S ((S (mdr_j_injective_yes)) * c)) /\ exists ff_q_mdr_injective_yessecond. b = ff_q_mdr_injective_yessecond * S ((S (mdr_j_injective_yes)) * c) + (mdr_a_injective_yes))) -> mdr_i_injective_yes = mdr_j_injective_yes) \/ ~(forall mdr_i_injective_no mdr_j_injective_no mdr_a_injective_no. (exists mdr_gap_injective_noi. mdr_gap_injective_noi + S (mdr_i_injective_no) = (l)) -> (exists mdr_gap_injective_noj. mdr_gap_injective_noj + S (mdr_j_injective_no) = (l)) -> (((exists ff_h_mdr_injective_nofirst. ff_h_mdr_injective_nofirst + S (mdr_a_injective_no) = S ((S (mdr_i_injective_no)) * c)) /\ exists ff_q_mdr_injective_nofirst. b = ff_q_mdr_injective_nofirst * S ((S (mdr_i_injective_no)) * c) + (mdr_a_injective_no))) -> (((exists ff_h_mdr_injective_nosecond. ff_h_mdr_injective_nosecond + S (mdr_a_injective_no) = S ((S (mdr_j_injective_no)) * c)) /\ exists ff_q_mdr_injective_nosecond. b = ff_q_mdr_injective_nosecond * S ((S (mdr_j_injective_no)) * c) + (mdr_a_injective_no))) -> mdr_i_injective_no = mdr_j_injective_no)

Complete tactic proof in conservative notation

All 29 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

29 script commands · 8 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 assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
02Establish hdecisionL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite prefix collision or injective.

  1. L4
    have hdecision : (∃ x. ∃ y. ∃ z. Lt(x,l) ∧ (Lt(y,l) ∧ (¬x = y ∧ (BetaAt(b,c,x,z) ∧ BetaAt(b,c,y,z))))) ∨ (∀ x. ∀ y. ∀ z. Lt(x,l) → Lt(y,l) → BetaAt(b,c,x,z) → BetaAt(b,c,y,z) → x = y)Definitions: Lt(x,l)Lt(y,l)BetaAt(b,c,x,z)BetaAt(b,c,y,z)Original native command in the exact edition
  2. L5
    specialize finite_prefix_collision_or_injective (b)
  3. L6
    specialize finite_prefix_collision_or_injective (c)
  4. L7
    specialize finite_prefix_collision_or_injective (l)
  5. L8
    apply finite_prefix_collision_or_injective
03Separate the logical casesL9–10

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

  1. L9
    cases hdecision
  2. L10
    right
04Fix variables and assumptionsL11–11

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

  1. L11
    intro hinjective
05Separate the logical casesL12–18

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

  1. L12
    cases hdecision_left
  2. L13
    cases hdecision_left_witness
  3. L14
    cases hdecision_left_witness_witness
  4. L15
    cases hdecision_left_witness_witness_witness
  5. L16
    cases hdecision_left_witness_witness_witness_right
  6. L17
    cases hdecision_left_witness_witness_witness_right_right
  7. L18
    cases hdecision_left_witness_witness_witness_right_right_right
06Use earlier factsL19–27

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

  1. L19
    apply hdecision_left_witness_witness_witness_right_right_left
  2. L20
    specialize hinjective (x)
  3. L21
    specialize hinjective (x1)
  4. L22
    specialize hinjective (x2)
  5. L23
    apply hinjective
  6. L24
    exact hdecision_left_witness_witness_witness_left
  7. L25
    exact hdecision_left_witness_witness_witness_right_left
  8. L26
    exact hdecision_left_witness_witness_witness_right_right_right_left
  9. L27
    exact hdecision_left_witness_witness_witness_right_right_right_right
07Separate the logical casesL28–28

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

  1. L28
    left
08Use earlier factsL29–29

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

  1. L29
    exact hdecision_right

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have hdecision : (∃ x. ∃ y. ∃ z. Lt(x,l) ∧ (Lt(y,l) ∧ (¬x = y ∧ (BetaAt(b,c,x,z) ∧ BetaAt(b,c,y,z))))) ∨ (∀ x. ∀ y. ∀ z. Lt(x,l) → Lt(y,l) → BetaAt(b,c,x,z) → BetaAt(b,c,y,z) → x = y)
  5. 0005specialize finite_prefix_collision_or_injective (b)
  6. 0006specialize finite_prefix_collision_or_injective (c)
  7. 0007specialize finite_prefix_collision_or_injective (l)
  8. 0008apply finite_prefix_collision_or_injective
  9. 0009cases hdecision
  10. 0010right
  11. 0011intro hinjective
  12. 0012cases hdecision_left
  13. 0013cases hdecision_left_witness
  14. 0014cases hdecision_left_witness_witness
  15. 0015cases hdecision_left_witness_witness_witness
  16. 0016cases hdecision_left_witness_witness_witness_right
  17. 0017cases hdecision_left_witness_witness_witness_right_right
  18. 0018cases hdecision_left_witness_witness_witness_right_right_right
  19. 0019apply hdecision_left_witness_witness_witness_right_right_left
  20. 0020specialize hinjective (x)
  21. 0021specialize hinjective (x1)
  22. 0022specialize hinjective (x2)
  23. 0023apply hinjective
  24. 0024exact hdecision_left_witness_witness_witness_left
  25. 0025exact hdecision_left_witness_witness_witness_right_left
  26. 0026exact hdecision_left_witness_witness_witness_right_right_right_left
  27. 0027exact hdecision_left_witness_witness_witness_right_right_right_right
  28. 0028left
  29. 0029exact hdecision_right