FS000X

four_square_conjugate_left_decomposition

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

All conjugate scalar/vector coordinate squares separate into sixteen diagonal squares and exactly twelve same-sign 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 * f + c * h) * (a * f + c * h) + (b * e + d * g) * (b * e + d * g))) + (((a * g + d * f) * (a * g + d * f) + (c * e + b * h) * (c * e + b * h)) + ((a * h + b * g) * (a * h + b * g) + (d * e + c * f) * (d * e + c * f))))) = ((((((((((a * e) * (a * e)) + ((b * f) * (b * f))) + ((c * g) * (c * g))) + ((d * h) * (d * h)))) + ((((((a * f) * (a * f)) + ((c * h) * (c * h)))) + ((((b * e) * (b * e)) + ((d * g) * (d * g))))))) + (((((((a * g) * (a * g)) + ((d * f) * (d * f)))) + ((((c * e) * (c * e)) + ((b * h) * (b * h)))))) + ((((((a * h) * (a * h)) + ((b * g) * (b * g)))) + ((((d * e) * (d * e)) + ((c * f) * (c * f))))))))) + (((((((((((((((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)))) + (((a * f) * (c * h) + (c * h) * (a * f)))) + (((b * e) * (d * g) + (d * g) * (b * e)))) + (((a * g) * (d * f) + (d * f) * (a * g)))) + (((c * e) * (b * h) + (b * h) * (c * e)))) + (((a * h) * (b * g) + (b * g) * (a * h)))) + (((d * e) * (c * f) + (c * f) * (d * e))))))

Constructive proof overview

Generated structural guide

All conjugate scalar/vector coordinate squares separate into sixteen diagonal squares and exactly twelve same-sign mixed blocks.

The unchanged tactic script uses 4 declared prerequisites and contains 21 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

FS000W four_square_conjugate_scalar_decomposition FS005H four_square_signed_pair_block_decomposition FS002D four_square_euler_four_add_shuffle add_assoc Stable theorem; checked-use authorized

Direct dependents

FS0010 four_square_conjugate_global_compensation

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

21 script commands · 8 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 (3)
four square conjugate scalar decompositionfour square signed pair block decompositionfour square euler four add shuffle
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 equalitiesL9–11

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

  1. L9
    trans ((((((((((a * e) * (a * e)) + ((b * f) * (b * f))) + ((c * g) * (c * g))) + ((d * h) * (d * h)))) + · expand full local formula (977 characters)trans ((((((((((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))))))))) + ((((((((a * f) * (a * f)) + ((c * h) * (c * h)))) + ((((b * e) * (b * e)) + ((d * g) * (d * g)))))) + (((((a * f) * (c * h) + (c * h) * (a * f))) + (((b * e) * (d * g) + (d * g) * (b * e)))))))) + (((((((((a * g) * (a * g)) + ((d * f) * (d * f)))) + ((((c * e) * (c * e)) + ((b * h) * (b * h)))))) + (((((a * g) * (d * f) + (d * f) * (a * g))) + (((c * e) * (b * h) + (b * h) * (c * e))))))) + ((((((((a * h) * (a * h)) + ((b * g) * (b * g)))) + ((((d * e) * (d * e)) + ((c * f) * (c * f)))))) + (((((a * h) * (b * g) + (b * g) * (a * h))) + (((d * e) * (c * f) + (c * f) * (d * e)))))))))
  2. L10
    congr
  3. L11
    congr
03Use earlier factsL12–13

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

  1. L12
    apply four_square_conjugate_scalar_decomposition
  2. L13
    apply four_square_signed_pair_block_decomposition
04Calculate and transport equalitiesL14–14

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

  1. L14
    congr
05Use earlier factsL15–16

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

  1. L15
    apply four_square_signed_pair_block_decomposition
  2. L16
    apply four_square_signed_pair_block_decomposition
06Calculate and transport equalitiesL17–17

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

  1. L17
    trans (((((((((a * e) * (a * e)) + ((b * f) * (b * f))) + ((c * g) * (c * g))) + ((d * h) * (d * h)))) + ( · expand full local formula (971 characters)trans (((((((((a * e) * (a * e)) + ((b * f) * (b * f))) + ((c * g) * (c * g))) + ((d * h) * (d * h)))) + ((((((a * f) * (a * f)) + ((c * h) * (c * h)))) + ((((b * e) * (b * e)) + ((d * g) * (d * g))))))) + (((((((a * g) * (a * g)) + ((d * f) * (d * f)))) + ((((c * e) * (c * e)) + ((b * h) * (b * h)))))) + ((((((a * h) * (a * h)) + ((b * g) * (b * g)))) + ((((d * e) * (d * e)) + ((c * f) * (c * f))))))))) + (((((((((((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))))))) + (((((a * f) * (c * h) + (c * h) * (a * f))) + (((b * e) * (d * g) + (d * g) * (b * e)))))) + ((((((a * g) * (d * f) + (d * f) * (a * g))) + (((c * e) * (b * h) + (b * h) * (c * e))))) + (((((a * h) * (b * g) + (b * g) * (a * h))) + (((d * e) * (c * f) + (c * f) * (d * e))))))))
07Use earlier factsL18–18

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

  1. L18
    apply four_square_euler_four_add_shuffle
08Calculate and transport equalitiesL19–21

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

  1. L19
    congr
  2. L20
    refl
  3. L21
    simp [add_assoc]

Library-wide reading audit

Original exact command ledger · 21 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) * (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))))))))) + ((((((((a * f) * (a * f)) + ((c * h) * (c * h)))) + ((((b * e) * (b * e)) + ((d * g) * (d * g)))))) + (((((a * f) * (c * h) + (c * h) * (a * f))) + (((b * e) * (d * g) + (d * g) * (b * e)))))))) + (((((((((a * g) * (a * g)) + ((d * f) * (d * f)))) + ((((c * e) * (c * e)) + ((b * h) * (b * h)))))) + (((((a * g) * (d * f) + (d * f) * (a * g))) + (((c * e) * (b * h) + (b * h) * (c * e))))))) + ((((((((a * h) * (a * h)) + ((b * g) * (b * g)))) + ((((d * e) * (d * e)) + ((c * f) * (c * f)))))) + (((((a * h) * (b * g) + (b * g) * (a * h))) + (((d * e) * (c * f) + (c * f) * (d * e)))))))))
  10. 0010congr
  11. 0011congr
  12. 0012apply four_square_conjugate_scalar_decomposition
  13. 0013apply four_square_signed_pair_block_decomposition
  14. 0014congr
  15. 0015apply four_square_signed_pair_block_decomposition
  16. 0016apply four_square_signed_pair_block_decomposition
  17. 0017trans (((((((((a * e) * (a * e)) + ((b * f) * (b * f))) + ((c * g) * (c * g))) + ((d * h) * (d * h)))) + ((((((a * f) * (a * f)) + ((c * h) * (c * h)))) + ((((b * e) * (b * e)) + ((d * g) * (d * g))))))) + (((((((a * g) * (a * g)) + ((d * f) * (d * f)))) + ((((c * e) * (c * e)) + ((b * h) * (b * h)))))) + ((((((a * h) * (a * h)) + ((b * g) * (b * g)))) + ((((d * e) * (d * e)) + ((c * f) * (c * f))))))))) + (((((((((((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))))))) + (((((a * f) * (c * h) + (c * h) * (a * f))) + (((b * e) * (d * g) + (d * g) * (b * e)))))) + ((((((a * g) * (d * f) + (d * f) * (a * g))) + (((c * e) * (b * h) + (b * h) * (c * e))))) + (((((a * h) * (b * g) + (b * g) * (a * h))) + (((d * e) * (c * f) + (c * f) * (d * e))))))))
  18. 0018apply four_square_euler_four_add_shuffle
  19. 0019congr
  20. 0020refl
  21. 0021simp [add_assoc]