FS000E

four_square_product_square

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

The square of a product is the product of the two natural coordinate squares.

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 a b. (a * b) * (a * b) = (a * a) * (b * b)

Constructive proof overview

Generated structural guide

The square of a product is the product of the two natural coordinate squares.

The unchanged tactic script uses 1 declared prerequisite and contains 3 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

FS000D four_square_product_shuffle

Direct dependents

FS001A four_square_descent_square_factor_norm FS001W four_square_descent_double_square_four_sum FS002G four_square_euler_diagonal_block

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

3 script commands · 2 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.

Named ingredients (1)
four square product shuffle
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–3

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

  1. L3
    apply four_square_product_shuffle

Library-wide reading audit

Original exact command ledger · 3 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003apply four_square_product_shuffle