FS002H · theorem body

four_square_euler_diagonal_expansion

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

The complete eight-variable quaternion norm product expands into exactly its sixteen squared coordinate products.

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 * a + b * b + c * c + d * d) * (e * e + f * f + g * g + h * h) = ((a * e) * (a * e) + (a * f) * (a * f) + (a * g) * (a * g) + (a * h) * (a * h)) + ((b * e) * (b * e) + (b * f) * (b * f) + (b * g) * (b * g) + (b * h) * (b * h)) + ((c * e) * (c * e) + (c * f) * (c * f) + (c * g) * (c * g) + (c * h) * (c * h)) + ((d * e) * (d * e) + (d * f) * (d * f) + (d * g) * (d * g) + (d * h) * (d * h))

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 * a + b * b + c * c + d * d) * (e * e + f * f + g * g + h * h) = ((a * e) * (a * e) + (a * f) * (a * f) + (a * g) * (a * g) + (a * h) * (a * h)) + ((b * e) * (b * e) + (b * f) * (b * f) + (b * g) * (b * g) + (b * h) * (b * h)) + ((c * e) * (c * e) + (c * f) * (c * f) + (c * g) * (c * g) + (c * h) * (c * h)) + ((d * e) * (d * e) + (d * f) * (d * f) + (d * g) * (d * g) + (d * h) * (d * h))

Proof neighborhood

Direct theorem prerequisites

FS0003 four_square_norm_distributes FS002G four_square_euler_diagonal_block

Direct theorem dependents

FS0010 four_square_conjugate_global_compensation FS002X four_square_euler_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

24 script commands · 4 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 (2)
four square norm distributesfour square euler diagonal block
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
02Use earlier factsL9–16

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

  1. L9
    specialize four_square_norm_distributes a
  2. L10
    specialize four_square_norm_distributes b
  3. L11
    specialize four_square_norm_distributes c
  4. L12
    specialize four_square_norm_distributes d
  5. L13
    specialize four_square_norm_distributes e
  6. L14
    specialize four_square_norm_distributes f
  7. L15
    specialize four_square_norm_distributes g
  8. L16
    specialize four_square_norm_distributes h
03Calculate and transport equalitiesL17–20

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

  1. L17
    rewrite four_square_norm_distributes
  2. L18
    congr
  3. L19
    congr
  4. L20
    congr
04Use earlier factsL21–24

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

  1. L21
    apply four_square_euler_diagonal_block
  2. L22
    apply four_square_euler_diagonal_block
  3. L23
    apply four_square_euler_diagonal_block
  4. L24
    apply four_square_euler_diagonal_block

Library-wide reading audit

Original defined command ledger · 24 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. 0009specialize four_square_norm_distributes a
  10. 0010specialize four_square_norm_distributes b
  11. 0011specialize four_square_norm_distributes c
  12. 0012specialize four_square_norm_distributes d
  13. 0013specialize four_square_norm_distributes e
  14. 0014specialize four_square_norm_distributes f
  15. 0015specialize four_square_norm_distributes g
  16. 0016specialize four_square_norm_distributes h
  17. 0017rewrite four_square_norm_distributes
  18. 0018congr
  19. 0019congr
  20. 0020congr
  21. 0021apply four_square_euler_diagonal_block
  22. 0022apply four_square_euler_diagonal_block
  23. 0023apply four_square_euler_diagonal_block
  24. 0024apply four_square_euler_diagonal_block