FS0008

four_square_sum_expansion

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

A natural square of a sum expands to its four ordered diagonal and cross contributions.

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 + a * b) + (b * a + b * b)

Constructive proof overview

Generated structural guide

A natural square of a sum expands to its four ordered diagonal and cross contributions.

The unchanged tactic script uses 2 declared prerequisites 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

add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized

Direct dependents

FS0009 four_square_gap_balance_right FS005F four_square_signed_sum_two_decomposition

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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Calculate and transport equalitiesL3–3

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

  1. L3
    simp [add_mul, mul_add]

Library-wide reading audit

Original exact command ledger · 3 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003simp [add_mul, mul_add]