FS002S

four_square_euler_coordinate_triple_decompose

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

A triple-square plus a singleton-square decomposes into its four diagonal squares and three paired cross 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 w x y z. (x + y + z) * (x + y + z) + w * w = (((((x * x) + (y * y)) + (z * z))) + w * w) + ((((((x * y) + (y * x))) + (((x * z) + (z * x)))) + (((y * z) + (z * y)))))

Constructive proof overview

Generated structural guide

A triple-square plus a singleton-square decomposes into its four diagonal squares and three paired cross blocks.

The unchanged tactic script uses 2 declared prerequisites and contains 9 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

Proof neighborhood

Direct dependencies

FS002O four_square_euler_three_square_expansion FS002N four_square_euler_add_swap_last

Direct dependents

FS002U four_square_euler_left_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

9 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.

Named ingredients (2)
four square euler three square expansionfour square euler add swap last
01Fix variables and assumptionsL1–4

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

  1. L1
    intro w
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro z
02Calculate and transport equalitiesL5–6

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

  1. L5
    trans (((((x * x) + (y * y)) + (z * z))) + ((((((x * y) + (y * x))) + (((x * z) + (z * x)))) + (((y * z) + (z * y)))))) + w * w
  2. L6
    congr
03Use earlier factsL7–7

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

  1. L7
    apply four_square_euler_three_square_expansion
04Calculate and transport equalitiesL8–8

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

  1. L8
    refl
05Use earlier factsL9–9

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

  1. L9
    apply four_square_euler_add_swap_last

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro w
  2. 0002intro x
  3. 0003intro y
  4. 0004intro z
  5. 0005trans (((((x * x) + (y * y)) + (z * z))) + ((((((x * y) + (y * x))) + (((x * z) + (z * x)))) + (((y * z) + (z * y)))))) + w * w
  6. 0006congr
  7. 0007apply four_square_euler_three_square_expansion
  8. 0008refl
  9. 0009apply four_square_euler_add_swap_last