FS000X · theorem body

four_square_conjugate_left_decomposition

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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))))))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order 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))))))

Proof neighborhood

Direct theorem prerequisites

FS000W four_square_conjugate_scalar_decomposition FS005H four_square_signed_pair_block_decomposition FS002D four_square_euler_four_add_shuffle add_assoc · Stable closed

Direct theorem dependents

FS0010 four_square_conjugate_global_compensation

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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]