DL0037

matrix_rank_injective_prefix_decidable

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 29 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

finite_prefix_collision_or_injective Alpha theorem; checked-use authorized

Direct dependents

DL003D matrix_rank_selector_decidable

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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: LtBetaAt
  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 exact command ledger · 29 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have hdecision : (exists mdr_i_collision_witness mdr_j_collision_witness mdr_a_collision_witness. ((exists mdr_gap_collision_witnessi. mdr_gap_collision_witnessi + S (mdr_i_collision_witness) = (l)) /\ ((exists mdr_gap_collision_witnessj. mdr_gap_collision_witnessj + S (mdr_j_collision_witness) = (l)) /\ ((~(mdr_i_collision_witness = mdr_j_collision_witness)) /\ ((((exists ff_h_mdr_collision_witnessfirst. ff_h_mdr_collision_witnessfirst + S (mdr_a_collision_witness) = S ((S (mdr_i_collision_witness)) * c)) /\ exists ff_q_mdr_collision_witnessfirst. b = ff_q_mdr_collision_witnessfirst * S ((S (mdr_i_collision_witness)) * c) + (mdr_a_collision_witness))) /\ (((exists ff_h_mdr_collision_witnesssecond. ff_h_mdr_collision_witnesssecond + S (mdr_a_collision_witness) = S ((S (mdr_j_collision_witness)) * c)) /\ exists ff_q_mdr_collision_witnesssecond. b = ff_q_mdr_collision_witnesssecond * S ((S (mdr_j_collision_witness)) * c) + (mdr_a_collision_witness)))))))) \/ (forall mdr_i_injectivity_witness mdr_j_injectivity_witness mdr_a_injectivity_witness. (exists mdr_gap_injectivity_witnessi. mdr_gap_injectivity_witnessi + S (mdr_i_injectivity_witness) = (l)) -> (exists mdr_gap_injectivity_witnessj. mdr_gap_injectivity_witnessj + S (mdr_j_injectivity_witness) = (l)) -> (((exists ff_h_mdr_injectivity_witnessfirst. ff_h_mdr_injectivity_witnessfirst + S (mdr_a_injectivity_witness) = S ((S (mdr_i_injectivity_witness)) * c)) /\ exists ff_q_mdr_injectivity_witnessfirst. b = ff_q_mdr_injectivity_witnessfirst * S ((S (mdr_i_injectivity_witness)) * c) + (mdr_a_injectivity_witness))) -> (((exists ff_h_mdr_injectivity_witnesssecond. ff_h_mdr_injectivity_witnesssecond + S (mdr_a_injectivity_witness) = S ((S (mdr_j_injectivity_witness)) * c)) /\ exists ff_q_mdr_injectivity_witnesssecond. b = ff_q_mdr_injectivity_witnesssecond * S ((S (mdr_j_injectivity_witness)) * c) + (mdr_a_injectivity_witness))) -> mdr_i_injectivity_witness = mdr_j_injectivity_witness)
  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