FS000W

four_square_conjugate_scalar_decomposition

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

The all-positive conjugate scalar square decomposes into its four diagonal and six symmetric 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 a b c d e f g h. (a * e + b * f + c * g + d * h) * (a * e + b * f + c * g + d * h) + 0 * 0 = ((((((((a * e) * (a * e)) + ((b * f) * (b * f))) + ((c * g) * (c * g))) + ((d * h) * (d * h)))) + ((((((((a * e) * (b * f) + (b * f) * (a * e))) + (((a * e) * (c * g) + (c * g) * (a * e)))) + (((b * f) * (c * g) + (c * g) * (b * f))))) + ((((((d * h) * (a * e) + (a * e) * (d * h))) + (((d * h) * (b * f) + (b * f) * (d * h)))) + (((d * h) * (c * g) + (c * g) * (d * h)))))))))

Constructive proof overview

Generated structural guide

The all-positive conjugate scalar square decomposes into its four diagonal and six symmetric mixed blocks.

The unchanged tactic script uses 1 declared prerequisite and contains 11 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

FS005G four_square_signed_sum_four_decomposition

Direct dependents

FS000X four_square_conjugate_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

11 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 signed sum four decomposition
01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
02Calculate and transport equalitiesL9–10

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

  1. L9
    trans (a * e + b * f + c * g + d * h) * (a * e + b * f + c * g + d * h)
  2. L10
    simp
03Use earlier factsL11–11

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

  1. L11
    apply four_square_signed_sum_four_decomposition

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009trans (a * e + b * f + c * g + d * h) * (a * e + b * f + c * g + d * h)
  10. 0010simp
  11. 0011apply four_square_signed_sum_four_decomposition