CE0002 — matrix_minor_four_code_output_functional

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. ∀ w. MatrixMinorFourCode(z,up,us,un,ut) → MatrixMinorFourCode(w,up,us,un,ut) → z = w

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

Definition DAG

MatrixMinorFourCode(z,up,us,un,ut) · 2

Actual proof prerequisites

none

Original expanded first-order statement

forall up us un ut z w. 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)))) -> w = ((((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)))) -> z = w

Complete unchanged native tactic proof

All 12 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

12 script commands · 5 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 assumptionsL1–8

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

L1
intro up
L2
intro us
L3
intro un
L4
intro ut
L5
intro z
L6
intro w
L7
intro hz
L8
intro hw

02Calculate and transport equalitiesL9–9

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

L9
trans ((((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))))

03Use earlier factsL10–10

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

L10
exact hz

04Calculate and transport equalitiesL11–11

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

L11
symm

05Use earlier factsL12–12

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

L12
exact hw

Library-wide reading audit

Original defined command ledger · 12 lines

0001intro up
0002intro us
0003intro un
0004intro ut
0005intro z
0006intro w
0007intro hz
0008intro hw
0009trans ((((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))))
0010exact hz
0011symm
0012exact hw