FS002R

four_square_euler_coordinate_single_decompose

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

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

The unchanged tactic script uses 2 declared prerequisites and contains 10 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 add_assoc Stable theorem; checked-use authorized

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

10 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 (1)
four square euler three square expansion
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 equalitiesL5–7

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

  1. L5
    trans w * w + (((((x * x) + (y * y)) + (z * z))) + ((((((x * y) + (y * x))) + (((x * z) + (z * x)))) + (((y * z) + (z * y))))))
  2. L6
    congr
  3. L7
    refl
03Use earlier factsL8–8

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

  1. L8
    apply four_square_euler_three_square_expansion
04Calculate and transport equalitiesL9–9

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

  1. L9
    symm
05Use earlier factsL10–10

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

  1. L10
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro w
  2. 0002intro x
  3. 0003intro y
  4. 0004intro z
  5. 0005trans w * w + (((((x * x) + (y * y)) + (z * z))) + ((((((x * y) + (y * x))) + (((x * z) + (z * x)))) + (((y * z) + (z * y))))))
  6. 0006congr
  7. 0007refl
  8. 0008apply four_square_euler_three_square_expansion
  9. 0009symm
  10. 0010apply add_assoc