DL0005

matrix_recursive_record_transport

Preserved beta entries carry the same actual seven-field evaluation record without expanding nested pairing polynomials.

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. ∀ u. ∀ v. ∀ l. ∀ i. ∀ d. ∀ pb. ∀ pc. ∀ nb. ∀ nc. ∀ p. ∀ n. (∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(u,v,x,y)) → Lt(i,l) → SignedDeterminantNodeAt(b,c,i,d,pb,pc,nb,nc,p,n) → SignedDeterminantNodeAt(u,v,i,d,pb,pc,nb,nc,p,n)

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

Definition DAG

SignedDeterminantNodeAt(b,c,i,d,pb,pc,nb,nc,p,n) · 2Lt(a,b) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c u v l i d pb pc nb nc p n. (forall mdr_i_source mdr_a_source. (exists mdr_gap_sourceb. mdr_gap_sourceb + S (mdr_i_source) = (l)) -> (((exists ff_h_mdr_sourceo. ff_h_mdr_sourceo + S (mdr_a_source) = S ((S (mdr_i_source)) * c)) /\ exists ff_q_mdr_sourceo. b = ff_q_mdr_sourceo * S ((S (mdr_i_source)) * c) + (mdr_a_source))) -> (((exists ff_h_mdr_sourcen. ff_h_mdr_sourcen + S (mdr_a_source) = S ((S (mdr_i_source)) * v)) /\ exists ff_q_mdr_sourcen. u = ff_q_mdr_sourcen * S ((S (mdr_i_source)) * v) + (mdr_a_source)))) -> (exists mdr_gap_record_bound. mdr_gap_record_bound + S (i) = (l)) -> (exists mdr_z_record_old. ((exists mdr_a_record_oldc mdr_b_record_oldc mdr_c_record_oldc mdr_e_record_oldc mdr_f_record_oldc. ((mdr_a_record_oldc = ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) /\ ((mdr_b_record_oldc = ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) /\ ((mdr_c_record_oldc = ((mdr_a_record_oldc) + (mdr_b_record_oldc)) * S ((mdr_a_record_oldc) + (mdr_b_record_oldc)) + ((mdr_b_record_oldc) + (mdr_b_record_oldc))) /\ ((mdr_e_record_oldc = ((p) + (n)) * S ((p) + (n)) + ((n) + (n))) /\ ((mdr_f_record_oldc = ((nc) + (mdr_e_record_oldc)) * S ((nc) + (mdr_e_record_oldc)) + ((mdr_e_record_oldc) + (mdr_e_record_oldc))) /\ ((mdr_z_record_old) = ((mdr_c_record_oldc) + (mdr_f_record_oldc)) * S ((mdr_c_record_oldc) + (mdr_f_record_oldc)) + ((mdr_f_record_oldc) + (mdr_f_record_oldc))))))))) /\ (((exists ff_h_mdr_record_oldb. ff_h_mdr_record_oldb + S (mdr_z_record_old) = S ((S (i)) * c)) /\ exists ff_q_mdr_record_oldb. b = ff_q_mdr_record_oldb * S ((S (i)) * c) + (mdr_z_record_old))))) -> (exists mdr_z_record_new. ((exists mdr_a_record_newc mdr_b_record_newc mdr_c_record_newc mdr_e_record_newc mdr_f_record_newc. ((mdr_a_record_newc = ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) /\ ((mdr_b_record_newc = ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) /\ ((mdr_c_record_newc = ((mdr_a_record_newc) + (mdr_b_record_newc)) * S ((mdr_a_record_newc) + (mdr_b_record_newc)) + ((mdr_b_record_newc) + (mdr_b_record_newc))) /\ ((mdr_e_record_newc = ((p) + (n)) * S ((p) + (n)) + ((n) + (n))) /\ ((mdr_f_record_newc = ((nc) + (mdr_e_record_newc)) * S ((nc) + (mdr_e_record_newc)) + ((mdr_e_record_newc) + (mdr_e_record_newc))) /\ ((mdr_z_record_new) = ((mdr_c_record_newc) + (mdr_f_record_newc)) * S ((mdr_c_record_newc) + (mdr_f_record_newc)) + ((mdr_f_record_newc) + (mdr_f_record_newc))))))))) /\ (((exists ff_h_mdr_record_newb. ff_h_mdr_record_newb + S (mdr_z_record_new) = S ((S (i)) * v)) /\ exists ff_q_mdr_record_newb. u = ff_q_mdr_record_newb * S ((S (i)) * v) + (mdr_z_record_new)))))

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 · 6 reading checkpoints · 0 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 u
  4. L4
    intro v
  5. L5
    intro l
  6. L6
    intro i
  7. L7
    intro d
  8. L8
    intro pb
  9. L9
    intro pc
  10. L10
    intro nb
02Fix variables and assumptionsL11–16

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

  1. L11
    intro nc
  2. L12
    intro p
  3. L13
    intro n
  4. L14
    intro hprefix
  5. L15
    intro hi
  6. L16
    intro hrecord
03Separate the logical casesL17–18

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

  1. L17
    cases hrecord
  2. L18
    cases hrecord_witness
04Construct an explicit witnessL19–19

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

  1. L19
    exists x
05Separate the logical casesL20–20

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

  1. L20
    split
06Use earlier factsL21–26

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

  1. L21
    exact hrecord_witness_left
  2. L22
    specialize hprefix (i)
  3. L23
    specialize hprefix (x)
  4. L24
    apply hprefix
  5. L25
    exact hi
  6. L26
    exact hrecord_witness_right

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro u
  4. 0004intro v
  5. 0005intro l
  6. 0006intro i
  7. 0007intro d
  8. 0008intro pb
  9. 0009intro pc
  10. 0010intro nb
  11. 0011intro nc
  12. 0012intro p
  13. 0013intro n
  14. 0014intro hprefix
  15. 0015intro hi
  16. 0016intro hrecord
  17. 0017cases hrecord
  18. 0018cases hrecord_witness
  19. 0019exists x
  20. 0020split
  21. 0021exact hrecord_witness_left
  22. 0022specialize hprefix (i)
  23. 0023specialize hprefix (x)
  24. 0024apply hprefix
  25. 0025exact hi
  26. 0026exact hrecord_witness_right