DL0001

matrix_recursive_node_code_exists

Six explicit doubled-Cantor constructors code one exact matrix evaluation record with conservatively shared intermediate values.

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

∀ d. ∀ pb. ∀ pc. ∀ nb. ∀ nc. ∀ p. ∀ n. ∃ z. SignedDeterminantNodeCode(z,d,pb,pc,nb,nc,p,n)

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

Definition DAG

SignedDeterminantNodeCode(z,d,pb,pc,nb,nc,p,n) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall d pb pc nb nc p n. exists z. (exists mdr_a_code_exists mdr_b_code_exists mdr_c_code_exists mdr_e_code_exists mdr_f_code_exists. ((mdr_a_code_exists = ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) /\ ((mdr_b_code_exists = ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) /\ ((mdr_c_code_exists = ((mdr_a_code_exists) + (mdr_b_code_exists)) * S ((mdr_a_code_exists) + (mdr_b_code_exists)) + ((mdr_b_code_exists) + (mdr_b_code_exists))) /\ ((mdr_e_code_exists = ((p) + (n)) * S ((p) + (n)) + ((n) + (n))) /\ ((mdr_f_code_exists = ((nc) + (mdr_e_code_exists)) * S ((nc) + (mdr_e_code_exists)) + ((mdr_e_code_exists) + (mdr_e_code_exists))) /\ ((z) = ((mdr_c_code_exists) + (mdr_f_code_exists)) * S ((mdr_c_code_exists) + (mdr_f_code_exists)) + ((mdr_f_code_exists) + (mdr_f_code_exists)))))))))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 12 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 assumptionsL1–7

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

  1. L1
    intro d
  2. L2
    intro pb
  3. L3
    intro pc
  4. L4
    intro nb
  5. L5
    intro nc
  6. L6
    intro p
  7. L7
    intro n
02Construct an explicit witnessL8–13

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

  1. L8
    exists ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb) · expand full local formula (1,480 characters)exists ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) * S ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) + ((((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))))
  2. L9
    exists ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))
  3. L10
    exists ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))
  4. L11
    exists ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))
  5. L12
    exists ((p) + (n)) * S ((p) + (n)) + ((n) + (n))
  6. L13
    exists ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))
03Separate the logical casesL14–14

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

  1. L14
    split
04Calculate and transport equalitiesL15–15

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

  1. L15
    refl
05Separate the logical casesL16–16

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

  1. L16
    split
06Calculate and transport equalitiesL17–17

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

  1. L17
    refl
07Separate the logical casesL18–18

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

  1. L18
    split
08Calculate and transport equalitiesL19–19

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

  1. L19
    refl
09Separate the logical casesL20–20

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

  1. L20
    split
10Calculate and transport equalitiesL21–21

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

  1. L21
    refl
11Separate the logical casesL22–22

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

  1. L22
    split
12Calculate and transport equalitiesL23–24

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

  1. L23
    refl
  2. L24
    refl

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro d
  2. 0002intro pb
  3. 0003intro pc
  4. 0004intro nb
  5. 0005intro nc
  6. 0006intro p
  7. 0007intro n
  8. 0008exists ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) * S ((((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))))) + ((((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))) + (((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))))
  9. 0009exists ((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))
  10. 0010exists ((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))
  11. 0011exists ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) * S ((((d) + (pb)) * S ((d) + (pb)) + ((pb) + (pb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb)))) + ((((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))) + (((pc) + (nb)) * S ((pc) + (nb)) + ((nb) + (nb))))
  12. 0012exists ((p) + (n)) * S ((p) + (n)) + ((n) + (n))
  13. 0013exists ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) * S ((nc) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n)))) + ((((p) + (n)) * S ((p) + (n)) + ((n) + (n))) + (((p) + (n)) * S ((p) + (n)) + ((n) + (n))))
  14. 0014split
  15. 0015refl
  16. 0016split
  17. 0017refl
  18. 0018split
  19. 0019refl
  20. 0020split
  21. 0021refl
  22. 0022split
  23. 0023refl
  24. 0024refl