CE0001

matrix_minor_four_code_exists

Four arbitrary natural minor-code components have one exact nested doubled-Cantor record.

Alpha v34 checked-use · first admitted v25 · 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.

Historical partial components only: this chapter proves genuine signed first-row minors and unique alternating folds, with supplied cofactor values. T13 is now closed in the separate Alpha-v27 integer-linear-algebra branch with actual arbitrary determinant data, rank, and integer column spans; lattice index and normal forms are not claimed. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ up. ∀ us. ∀ un. ∀ ut. ∃ z. MatrixMinorFourCode(z,up,us,un,ut)

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

Definition DAG

MatrixMinorFourCode(z,up,us,un,ut) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall up us un ut. exists z. z = ((((up) + (us)) * S ((up) + (us)) + ((us) + (us))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut)))) * S ((((up) + (us)) * S ((up) + (us)) + ((us) + (us))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut)))) + ((((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut))))

Complete unchanged native tactic proof

All 6 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

6 script commands · 3 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 assumptionsL1–4

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

  1. L1
    intro up
  2. L2
    intro us
  3. L3
    intro un
  4. L4
    intro ut
02Construct an explicit witnessL5–5

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

  1. L5
    exists ((((up) + (us)) * S ((up) + (us)) + ((us) + (us))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut)))) * S ((((up) + (us)) * S ((up) + (us)) + ((us) + (us))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut)))) + ((((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut))))
03Calculate and transport equalitiesL6–6

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

  1. L6
    refl

Library-wide reading audit

Original defined command ledger · 6 lines
  1. 0001intro up
  2. 0002intro us
  3. 0003intro un
  4. 0004intro ut
  5. 0005exists ((((up) + (us)) * S ((up) + (us)) + ((us) + (us))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut)))) * S ((((up) + (us)) * S ((up) + (us)) + ((us) + (us))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut)))) + ((((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut))) + (((un) + (ut)) * S ((un) + (ut)) + ((ut) + (ut))))
  6. 0006refl