FS002O

four_square_euler_three_square_expansion

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

A three-term natural square expands into three diagonal squares and three explicitly paired mixed blocks.

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 x y z. (x + y + z) * (x + y + z) = ((((x * x) + (y * y)) + (z * z))) + ((((((x * y) + (y * x))) + (((x * z) + (z * x)))) + (((y * z) + (z * y)))))

Constructive proof overview

Generated structural guide

A three-term natural square expands into three diagonal squares and three explicitly paired mixed blocks.

The unchanged tactic script uses 3 declared prerequisites and contains 6 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

mul_add Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized FS002K four_square_euler_add_permute_nine

Direct dependents

FS002R four_square_euler_coordinate_single_decompose FS002S four_square_euler_coordinate_triple_decompose FS005G four_square_signed_sum_four_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

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.

Named ingredients (1)
four square euler add permute nine
01Fix variables and assumptionsL1–3

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
02Calculate and transport equalitiesL4–5

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

  1. L4
    trans (x * x + y * x + z * x) + (x * y + y * y + z * y) + (x * z + y * z + z * z)
  2. L5
    simp [mul_add, add_mul]
03Use earlier factsL6–6

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

  1. L6
    apply four_square_euler_add_permute_nine

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004trans (x * x + y * x + z * x) + (x * y + y * y + z * y) + (x * z + y * z + z * z)
  5. 0005simp [mul_add, add_mul]
  6. 0006apply four_square_euler_add_permute_nine