CE0001

matrix_minor_four_code_exists

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

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

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 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))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 0 declared prerequisites and contains 6 exact native proof lines.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

none

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

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.

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 exact 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

Separate complete second-wave branches: Full T13 proof · Alpha v27.