FS002W · theorem body

four_square_euler_mixed_decomposition

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

All twelve same-sign Hamilton mixed blocks become the twelve opposite-sign correction blocks by explicit crossed-factor swaps.

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. ((((((((((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))))))))) = ((((((((((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)))))))))

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. ((((((((((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))))))))) = ((((((((((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)))))))))

Proof neighborhood

Direct theorem prerequisites

FS002Q four_square_euler_double_cross_swap FS002L four_square_euler_add_permute_twelve

Direct theorem dependents

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

33 script commands · 9 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 euler double cross swapfour square euler add permute twelve
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 equalitiesL9–13

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

  1. L9
    trans (((((((((b * g) * (c * f)) + ((c * f) * (b * g)))) + ((((b * h) * (d * f)) + ((d * f) * (b * h))))) · expand full local formula (609 characters)trans (((((((((b * g) * (c * f)) + ((c * f) * (b * g)))) + ((((b * h) * (d * f)) + ((d * f) * (b * h))))) + ((((c * h) * (d * g)) + ((d * g) * (c * h)))))) + (((((((a * e) * (b * f)) + ((b * f) * (a * e)))) + ((((a * h) * (c * f)) + ((c * f) * (a * h))))) + ((((b * h) * (c * e)) + ((c * e) * (b * h))))))) + ((((((((a * e) * (c * g)) + ((c * g) * (a * e)))) + ((((a * f) * (d * g)) + ((d * g) * (a * f))))) + ((((c * f) * (d * e)) + ((d * e) * (c * f)))))) + (((((((a * g) * (b * h)) + ((b * h) * (a * g)))) + ((((a * e) * (d * h)) + ((d * h) * (a * e))))) + ((((b * e) * (d * g)) + ((d * g) * (b * e))))))))
  2. L10
    congr
  3. L11
    congr
  4. L12
    congr
  5. L13
    congr
03Use earlier factsL14–16

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

  1. L14
    apply four_square_euler_double_cross_swap
  2. L15
    apply four_square_euler_double_cross_swap
  3. L16
    apply four_square_euler_double_cross_swap
04Calculate and transport equalitiesL17–18

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

  1. L17
    congr
  2. L18
    congr
05Use earlier factsL19–21

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

  1. L19
    apply four_square_euler_double_cross_swap
  2. L20
    apply four_square_euler_double_cross_swap
  3. L21
    apply four_square_euler_double_cross_swap
06Calculate and transport equalitiesL22–24

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

  1. L22
    congr
  2. L23
    congr
  3. L24
    congr
07Use earlier factsL25–27

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

  1. L25
    apply four_square_euler_double_cross_swap
  2. L26
    apply four_square_euler_double_cross_swap
  3. L27
    apply four_square_euler_double_cross_swap
08Calculate and transport equalitiesL28–29

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

  1. L28
    congr
  2. L29
    congr
09Use earlier factsL30–33

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

  1. L30
    apply four_square_euler_double_cross_swap
  2. L31
    apply four_square_euler_double_cross_swap
  3. L32
    apply four_square_euler_double_cross_swap
  4. L33
    apply four_square_euler_add_permute_twelve

Library-wide reading audit

Original defined command ledger · 33 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 (((((((((b * g) * (c * f)) + ((c * f) * (b * g)))) + ((((b * h) * (d * f)) + ((d * f) * (b * h))))) + ((((c * h) * (d * g)) + ((d * g) * (c * h)))))) + (((((((a * e) * (b * f)) + ((b * f) * (a * e)))) + ((((a * h) * (c * f)) + ((c * f) * (a * h))))) + ((((b * h) * (c * e)) + ((c * e) * (b * h))))))) + ((((((((a * e) * (c * g)) + ((c * g) * (a * e)))) + ((((a * f) * (d * g)) + ((d * g) * (a * f))))) + ((((c * f) * (d * e)) + ((d * e) * (c * f)))))) + (((((((a * g) * (b * h)) + ((b * h) * (a * g)))) + ((((a * e) * (d * h)) + ((d * h) * (a * e))))) + ((((b * e) * (d * g)) + ((d * g) * (b * e))))))))
  10. 0010congr
  11. 0011congr
  12. 0012congr
  13. 0013congr
  14. 0014apply four_square_euler_double_cross_swap
  15. 0015apply four_square_euler_double_cross_swap
  16. 0016apply four_square_euler_double_cross_swap
  17. 0017congr
  18. 0018congr
  19. 0019apply four_square_euler_double_cross_swap
  20. 0020apply four_square_euler_double_cross_swap
  21. 0021apply four_square_euler_double_cross_swap
  22. 0022congr
  23. 0023congr
  24. 0024congr
  25. 0025apply four_square_euler_double_cross_swap
  26. 0026apply four_square_euler_double_cross_swap
  27. 0027apply four_square_euler_double_cross_swap
  28. 0028congr
  29. 0029congr
  30. 0030apply four_square_euler_double_cross_swap
  31. 0031apply four_square_euler_double_cross_swap
  32. 0032apply four_square_euler_double_cross_swap
  33. 0033apply four_square_euler_add_permute_twelve