FS002X · theorem body

four_square_euler_global_compensation

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

The exact previously missing global subtraction-free Hamilton compensation equation is proved without any remaining premise.

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

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

Proof neighborhood

Direct theorem prerequisites

FS002H four_square_euler_diagonal_expansion FS002T four_square_euler_diagonal_regroup FS002V four_square_euler_cross_decomposition FS002W four_square_euler_mixed_decomposition FS002U four_square_euler_left_decomposition

Direct theorem dependents

FS002Y four_square_euler_quaternion

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

28 script commands · 11 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 (5)
four square euler diagonal expansionfour square euler diagonal regroupfour square euler cross decompositionfour square euler mixed decompositionfour square euler left 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) * (a * e) + (a * f) * (a * f) + (a * g) * (a * g) + (a * h) * (a * h)) + ((b * e) * (b * e · expand full local formula (637 characters)trans (((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))) + (((((a * e) * (b * f + c * g + d * h) + (b * f + c * g + d * h) * (a * e)) + ((a * f + b * e + c * h) * (d * g) + (d * g) * (a * f + b * e + c * h))) + (((a * g + c * e + d * f) * (b * h) + (b * h) * (a * g + c * e + d * f)) + ((a * h + b * g + d * e) * (c * f) + (c * f) * (a * h + b * g + d * e)))))
  2. L10
    congr
03Use earlier factsL11–11

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

  1. L11
    apply four_square_euler_diagonal_expansion
04Calculate and transport equalitiesL12–15

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

  1. L12
    refl
  2. L13
    trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + · expand full local formula (707 characters)trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g))))) + ((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((b * h) * (b * h)))) + (((((((a * h) * (a * h)) + ((b * g) * (b * g))) + ((d * e) * (d * e)))) + ((c * f) * (c * f))))))) + (((((a * e) * (b * f + c * g + d * h) + (b * f + c * g + d * h) * (a * e)) + ((a * f + b * e + c * h) * (d * g) + (d * g) * (a * f + b * e + c * h))) + (((a * g + c * e + d * f) * (b * h) + (b * h) * (a * g + c * e + d * f)) + ((a * h + b * g + d * e) * (c * f) + (c * f) * (a * h + b * g + d * e)))))
  3. L14
    congr
  4. L15
    symm
05Use earlier factsL16–16

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

  1. L16
    apply four_square_euler_diagonal_regroup
06Calculate and transport equalitiesL17–20

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

  1. L17
    refl
  2. L18
    trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + · expand full local formula (1,011 characters)trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g))))) + ((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((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))))) + ((((a * e) * (d * h)) + ((d * h) * (a * e)))))) + (((((((a * f) * (d * g)) + ((d * g) * (a * f)))) + ((((b * e) * (d * g)) + ((d * g) * (b * e))))) + ((((c * h) * (d * g)) + ((d * g) * (c * h))))))) + ((((((((a * g) * (b * h)) + ((b * h) * (a * g)))) + ((((b * h) * (c * e)) + ((c * e) * (b * h))))) + ((((b * h) * (d * f)) + ((d * f) * (b * h)))))) + (((((((a * h) * (c * f)) + ((c * f) * (a * h)))) + ((((b * g) * (c * f)) + ((c * f) * (b * g))))) + ((((c * f) * (d * e)) + ((d * e) * (c * f)))))))))
  3. L19
    congr
  4. L20
    refl
07Use earlier factsL21–21

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

  1. L21
    apply four_square_euler_cross_decomposition
08Calculate and transport equalitiesL22–25

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

  1. L22
    trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + · expand full local formula (1,011 characters)trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g))))) + ((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((b * h) * (b * h)))) + (((((((a * h) * (a * h)) + ((b * g) * (b * g))) + ((d * e) * (d * e)))) + ((c * f) * (c * f))))))) + ((((((((((b * f) * (c * g)) + ((c * g) * (b * f)))) + ((((b * f) * (d * h)) + ((d * h) * (b * f))))) + ((((c * g) * (d * h)) + ((d * h) * (c * g)))))) + (((((((a * f) * (b * e)) + ((b * e) * (a * f)))) + ((((a * f) * (c * h)) + ((c * h) * (a * f))))) + ((((b * e) * (c * h)) + ((c * h) * (b * e))))))) + ((((((((a * g) * (c * e)) + ((c * e) * (a * g)))) + ((((a * g) * (d * f)) + ((d * f) * (a * g))))) + ((((c * e) * (d * f)) + ((d * f) * (c * e)))))) + (((((((a * h) * (b * g)) + ((b * g) * (a * h)))) + ((((a * h) * (d * e)) + ((d * e) * (a * h))))) + ((((b * g) * (d * e)) + ((d * e) * (b * g)))))))))
  2. L23
    congr
  3. L24
    refl
  4. L25
    symm
09Use earlier factsL26–26

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

  1. L26
    apply four_square_euler_mixed_decomposition
10Calculate and transport equalitiesL27–27

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

  1. L27
    symm
11Use earlier factsL28–28

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

  1. L28
    apply four_square_euler_left_decomposition

Library-wide reading audit

Original defined command ledger · 28 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) + (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))) + (((((a * e) * (b * f + c * g + d * h) + (b * f + c * g + d * h) * (a * e)) + ((a * f + b * e + c * h) * (d * g) + (d * g) * (a * f + b * e + c * h))) + (((a * g + c * e + d * f) * (b * h) + (b * h) * (a * g + c * e + d * f)) + ((a * h + b * g + d * e) * (c * f) + (c * f) * (a * h + b * g + d * e)))))
  10. 0010congr
  11. 0011apply four_square_euler_diagonal_expansion
  12. 0012refl
  13. 0013trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g))))) + ((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((b * h) * (b * h)))) + (((((((a * h) * (a * h)) + ((b * g) * (b * g))) + ((d * e) * (d * e)))) + ((c * f) * (c * f))))))) + (((((a * e) * (b * f + c * g + d * h) + (b * f + c * g + d * h) * (a * e)) + ((a * f + b * e + c * h) * (d * g) + (d * g) * (a * f + b * e + c * h))) + (((a * g + c * e + d * f) * (b * h) + (b * h) * (a * g + c * e + d * f)) + ((a * h + b * g + d * e) * (c * f) + (c * f) * (a * h + b * g + d * e)))))
  14. 0014congr
  15. 0015symm
  16. 0016apply four_square_euler_diagonal_regroup
  17. 0017refl
  18. 0018trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g))))) + ((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((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))))) + ((((a * e) * (d * h)) + ((d * h) * (a * e)))))) + (((((((a * f) * (d * g)) + ((d * g) * (a * f)))) + ((((b * e) * (d * g)) + ((d * g) * (b * e))))) + ((((c * h) * (d * g)) + ((d * g) * (c * h))))))) + ((((((((a * g) * (b * h)) + ((b * h) * (a * g)))) + ((((b * h) * (c * e)) + ((c * e) * (b * h))))) + ((((b * h) * (d * f)) + ((d * f) * (b * h)))))) + (((((((a * h) * (c * f)) + ((c * f) * (a * h)))) + ((((b * g) * (c * f)) + ((c * f) * (b * g))))) + ((((c * f) * (d * e)) + ((d * e) * (c * f)))))))))
  19. 0019congr
  20. 0020refl
  21. 0021apply four_square_euler_cross_decomposition
  22. 0022trans (((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g))))) + ((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((b * h) * (b * h)))) + (((((((a * h) * (a * h)) + ((b * g) * (b * g))) + ((d * e) * (d * e)))) + ((c * f) * (c * f))))))) + ((((((((((b * f) * (c * g)) + ((c * g) * (b * f)))) + ((((b * f) * (d * h)) + ((d * h) * (b * f))))) + ((((c * g) * (d * h)) + ((d * h) * (c * g)))))) + (((((((a * f) * (b * e)) + ((b * e) * (a * f)))) + ((((a * f) * (c * h)) + ((c * h) * (a * f))))) + ((((b * e) * (c * h)) + ((c * h) * (b * e))))))) + ((((((((a * g) * (c * e)) + ((c * e) * (a * g)))) + ((((a * g) * (d * f)) + ((d * f) * (a * g))))) + ((((c * e) * (d * f)) + ((d * f) * (c * e)))))) + (((((((a * h) * (b * g)) + ((b * g) * (a * h)))) + ((((a * h) * (d * e)) + ((d * e) * (a * h))))) + ((((b * g) * (d * e)) + ((d * e) * (b * g)))))))))
  23. 0023congr
  24. 0024refl
  25. 0025symm
  26. 0026apply four_square_euler_mixed_decomposition
  27. 0027symm
  28. 0028apply four_square_euler_left_decomposition