DL0036

matrix_rank_injective_prefix_transport

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

A recoded finite selector remains genuinely injective; code equality is not assumed.

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 u v l. (forall mdr_i_injective_transport mdr_a_injective_transport. (exists mdr_gap_injective_transportb. mdr_gap_injective_transportb + S (mdr_i_injective_transport) = (l)) -> (((exists ff_h_mdr_injective_transporto. ff_h_mdr_injective_transporto + S (mdr_a_injective_transport) = S ((S (mdr_i_injective_transport)) * c)) /\ exists ff_q_mdr_injective_transporto. b = ff_q_mdr_injective_transporto * S ((S (mdr_i_injective_transport)) * c) + (mdr_a_injective_transport))) -> (((exists ff_h_mdr_injective_transportn. ff_h_mdr_injective_transportn + S (mdr_a_injective_transport) = S ((S (mdr_i_injective_transport)) * v)) /\ exists ff_q_mdr_injective_transportn. u = ff_q_mdr_injective_transportn * S ((S (mdr_i_injective_transport)) * v) + (mdr_a_injective_transport)))) -> (forall mdr_i_injective_source mdr_j_injective_source mdr_a_injective_source. (exists mdr_gap_injective_sourcei. mdr_gap_injective_sourcei + S (mdr_i_injective_source) = (l)) -> (exists mdr_gap_injective_sourcej. mdr_gap_injective_sourcej + S (mdr_j_injective_source) = (l)) -> (((exists ff_h_mdr_injective_sourcefirst. ff_h_mdr_injective_sourcefirst + S (mdr_a_injective_source) = S ((S (mdr_i_injective_source)) * c)) /\ exists ff_q_mdr_injective_sourcefirst. b = ff_q_mdr_injective_sourcefirst * S ((S (mdr_i_injective_source)) * c) + (mdr_a_injective_source))) -> (((exists ff_h_mdr_injective_sourcesecond. ff_h_mdr_injective_sourcesecond + S (mdr_a_injective_source) = S ((S (mdr_j_injective_source)) * c)) /\ exists ff_q_mdr_injective_sourcesecond. b = ff_q_mdr_injective_sourcesecond * S ((S (mdr_j_injective_source)) * c) + (mdr_a_injective_source))) -> mdr_i_injective_source = mdr_j_injective_source) -> (forall mdr_i_injective_target mdr_j_injective_target mdr_a_injective_target. (exists mdr_gap_injective_targeti. mdr_gap_injective_targeti + S (mdr_i_injective_target) = (l)) -> (exists mdr_gap_injective_targetj. mdr_gap_injective_targetj + S (mdr_j_injective_target) = (l)) -> (((exists ff_h_mdr_injective_targetfirst. ff_h_mdr_injective_targetfirst + S (mdr_a_injective_target) = S ((S (mdr_i_injective_target)) * v)) /\ exists ff_q_mdr_injective_targetfirst. u = ff_q_mdr_injective_targetfirst * S ((S (mdr_i_injective_target)) * v) + (mdr_a_injective_target))) -> (((exists ff_h_mdr_injective_targetsecond. ff_h_mdr_injective_targetsecond + S (mdr_a_injective_target) = S ((S (mdr_j_injective_target)) * v)) /\ exists ff_q_mdr_injective_targetsecond. u = ff_q_mdr_injective_targetsecond * S ((S (mdr_j_injective_target)) * v) + (mdr_a_injective_target))) -> mdr_i_injective_target = mdr_j_injective_target)

Constructive proof overview

Generated structural guide

A recoded finite selector remains genuinely injective; code equality is not assumed.

The unchanged tactic script uses 1 declared prerequisite and contains 38 exact native proof lines.

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

Proof neighborhood

Direct dependencies

DL0034 matrix_rank_prefix_equality_symmetric

Direct dependents

DL003C matrix_rank_selector_transport

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

38 script commands · 5 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.

Named ingredients (1)
matrix rank prefix equality symmetric

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 assumptionsL1–7

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 hprefix
  7. L7
    intro hinjective
02Establish hreverseL8–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix rank prefix equality symmetric.

  1. L8
    have hreverse : ∀ mdr_i_reverse_injective. ∀ mdr_a_reverse_injective. Lt(mdr_i_reverse_injective,l) → BetaAt(u,v,mdr_i_reverse_injective,mdr_a_reverse_injective) → BetaAt(b,c,mdr_i_reverse_injective,mdr_a_reverse_injective)Definitions: LtBetaAt
  2. L9
    specialize matrix_rank_prefix_equality_symmetric (b)
  3. L10
    specialize matrix_rank_prefix_equality_symmetric (c)
  4. L11
    specialize matrix_rank_prefix_equality_symmetric (u)
  5. L12
    specialize matrix_rank_prefix_equality_symmetric (v)
  6. L13
    specialize matrix_rank_prefix_equality_symmetric (l)
  7. L14
    apply matrix_rank_prefix_equality_symmetric
  8. L15
    exact hprefix
  9. L16
    intro i
  10. L17
    intro j
03Fix variables and assumptionsL18–22

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

  1. L18
    intro a
  2. L19
    intro hi
  3. L20
    intro hj
  4. L21
    intro ha
  5. L22
    intro hb
04Use earlier factsL23–32

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

  1. L23
    specialize hinjective (i)
  2. L24
    specialize hinjective (j)
  3. L25
    specialize hinjective (a)
  4. L26
    apply hinjective
  5. L27
    exact hi
  6. L28
    exact hj
  7. L29
    specialize hreverse (i)
  8. L30
    specialize hreverse (a)
  9. L31
    apply hreverse
  10. L32
    exact hi
05Use earlier factsL33–38

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

  1. L33
    exact ha
  2. L34
    specialize hreverse (j)
  3. L35
    specialize hreverse (a)
  4. L36
    apply hreverse
  5. L37
    exact hj
  6. L38
    exact hb

Library-wide reading audit

Original exact command ledger · 38 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro u
  4. 0004intro v
  5. 0005intro l
  6. 0006intro hprefix
  7. 0007intro hinjective
  8. 0008have hreverse : forall mdr_i_reverse_injective mdr_a_reverse_injective. (exists mdr_gap_reverse_injectiveb. mdr_gap_reverse_injectiveb + S (mdr_i_reverse_injective) = (l)) -> (((exists ff_h_mdr_reverse_injectiveo. ff_h_mdr_reverse_injectiveo + S (mdr_a_reverse_injective) = S ((S (mdr_i_reverse_injective)) * v)) /\ exists ff_q_mdr_reverse_injectiveo. u = ff_q_mdr_reverse_injectiveo * S ((S (mdr_i_reverse_injective)) * v) + (mdr_a_reverse_injective))) -> (((exists ff_h_mdr_reverse_injectiven. ff_h_mdr_reverse_injectiven + S (mdr_a_reverse_injective) = S ((S (mdr_i_reverse_injective)) * c)) /\ exists ff_q_mdr_reverse_injectiven. b = ff_q_mdr_reverse_injectiven * S ((S (mdr_i_reverse_injective)) * c) + (mdr_a_reverse_injective)))
  9. 0009specialize matrix_rank_prefix_equality_symmetric (b)
  10. 0010specialize matrix_rank_prefix_equality_symmetric (c)
  11. 0011specialize matrix_rank_prefix_equality_symmetric (u)
  12. 0012specialize matrix_rank_prefix_equality_symmetric (v)
  13. 0013specialize matrix_rank_prefix_equality_symmetric (l)
  14. 0014apply matrix_rank_prefix_equality_symmetric
  15. 0015exact hprefix
  16. 0016intro i
  17. 0017intro j
  18. 0018intro a
  19. 0019intro hi
  20. 0020intro hj
  21. 0021intro ha
  22. 0022intro hb
  23. 0023specialize hinjective (i)
  24. 0024specialize hinjective (j)
  25. 0025specialize hinjective (a)
  26. 0026apply hinjective
  27. 0027exact hi
  28. 0028exact hj
  29. 0029specialize hreverse (i)
  30. 0030specialize hreverse (a)
  31. 0031apply hreverse
  32. 0032exact hi
  33. 0033exact ha
  34. 0034specialize hreverse (j)
  35. 0035specialize hreverse (a)
  36. 0036apply hreverse
  37. 0037exact hj
  38. 0038exact hb