FS005G · theorem body

four_square_signed_sum_four_decomposition

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

A four-addend square splits constructively into four diagonal squares and its six symmetric cross pairs.

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 x y z w. (x + y + z + w) * (x + y + z + w) = ((x * x + y * y + z * z) + w * w) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w)))

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 x y z w. (x + y + z + w) * (x + y + z + w) = ((x * x + y * y + z * z) + w * w) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w)))

Proof neighborhood

Direct theorem prerequisites

FS005F four_square_signed_sum_two_decomposition FS002O four_square_euler_three_square_expansion FS002P four_square_euler_cross_triple_expansion add_comm · Stable closed add_shuffle_middle · Alpha closed add_assoc · Stable closed FS0006 four_square_add_swap_right_tail

Direct theorem dependents

FS000W four_square_conjugate_scalar_decomposition

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

36 script commands · 7 reading checkpoints · 3 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 (4)
four square signed sum two decompositionfour square euler three square expansionfour square euler cross triple expansionfour square add swap right tail
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
  4. L4
    intro w
02Establish htwoL5–9

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have htwo : ((x + y + z) + w) * ((x + y + z) + w) = ((x + y + z) * (x + y + z) + w * w) + ((x + y + z) * w + w * (x + y + z))
  2. L6
    specialize four_square_signed_sum_two_decomposition (x + y + z)
  3. L7
    specialize four_square_signed_sum_two_decomposition w
  4. L8
    exact four_square_signed_sum_two_decomposition
  5. L9
    rewrite htwo
03Establish hthreeL10–15

Establish this local claim before using it. It is not an additional assumption.

  1. L10
    have hthree : (x + y + z) * (x + y + z) = (x * x + y * y + z * z) + ((x * y + y * x) + (x * z + z * x) + (y * z + z * y))
  2. L11
    specialize four_square_euler_three_square_expansion x
  3. L12
    specialize four_square_euler_three_square_expansion y
  4. L13
    specialize four_square_euler_three_square_expansion z
  5. L14
    exact four_square_euler_three_square_expansion
  6. L15
    rewrite hthree
04Establish hcrossL16–25

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add comm.

  1. L16
    have hcross : (x + y + z) * w + w * (x + y + z) = (w * x + x * w) + (w * y + y * w) + (w * z + z * w)
  2. L17
    trans w * (x + y + z) + (x + y + z) * w
  3. L18
    apply add_comm
  4. L19
    specialize four_square_euler_cross_triple_expansion w
  5. L20
    specialize four_square_euler_cross_triple_expansion x
  6. L21
    specialize four_square_euler_cross_triple_expansion y
  7. L22
    specialize four_square_euler_cross_triple_expansion z
  8. L23
    exact four_square_euler_cross_triple_expansion
  9. L24
    rewrite hcross
  10. L25
    trans ((x * x + y * y + z * z) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * w) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w)))))
05Calculate and transport equalitiesL26–30

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

  1. L26
    simp [add_assoc]
  2. L27
    trans ((x * x + y * y + z * z) + ((w * w) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w)))))
  3. L28
    congr
  4. L29
    refl
  5. L30
    trans ((w * w) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w))))
06Use earlier factsL31–31

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

  1. L31
    apply four_square_add_swap_right_tail
07Calculate and transport equalitiesL32–36

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

  1. L32
    congr
  2. L33
    refl
  3. L34
    refl
  4. L35
    symm
  5. L36
    simp [add_assoc]

Library-wide reading audit

Original defined command ledger · 36 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro w
  5. 0005have htwo : ((x + y + z) + w) * ((x + y + z) + w) = ((x + y + z) * (x + y + z) + w * w) + ((x + y + z) * w + w * (x + y + z))
  6. 0006specialize four_square_signed_sum_two_decomposition (x + y + z)
  7. 0007specialize four_square_signed_sum_two_decomposition w
  8. 0008exact four_square_signed_sum_two_decomposition
  9. 0009rewrite htwo
  10. 0010have hthree : (x + y + z) * (x + y + z) = (x * x + y * y + z * z) + ((x * y + y * x) + (x * z + z * x) + (y * z + z * y))
  11. 0011specialize four_square_euler_three_square_expansion x
  12. 0012specialize four_square_euler_three_square_expansion y
  13. 0013specialize four_square_euler_three_square_expansion z
  14. 0014exact four_square_euler_three_square_expansion
  15. 0015rewrite hthree
  16. 0016have hcross : (x + y + z) * w + w * (x + y + z) = (w * x + x * w) + (w * y + y * w) + (w * z + z * w)
  17. 0017trans w * (x + y + z) + (x + y + z) * w
  18. 0018apply add_comm
  19. 0019specialize four_square_euler_cross_triple_expansion w
  20. 0020specialize four_square_euler_cross_triple_expansion x
  21. 0021specialize four_square_euler_cross_triple_expansion y
  22. 0022specialize four_square_euler_cross_triple_expansion z
  23. 0023exact four_square_euler_cross_triple_expansion
  24. 0024rewrite hcross
  25. 0025trans ((x * x + y * y + z * z) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * w) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w)))))
  26. 0026simp [add_assoc]
  27. 0027trans ((x * x + y * y + z * z) + ((w * w) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w)))))
  28. 0028congr
  29. 0029refl
  30. 0030trans ((w * w) + (((x * y + y * x) + (x * z + z * x) + (y * z + z * y)) + ((w * x + x * w) + (w * y + y * w) + (w * z + z * w))))
  31. 0031apply four_square_add_swap_right_tail
  32. 0032congr
  33. 0033refl
  34. 0034refl
  35. 0035symm
  36. 0036simp [add_assoc]