FS005G

four_square_signed_sum_four_decomposition

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 7 declared prerequisites and contains 36 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

FS005F four_square_signed_sum_two_decomposition FS002O four_square_euler_three_square_expansion FS002P four_square_euler_cross_triple_expansion add_comm Stable theorem; checked-use authorized add_shuffle_middle Alpha theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized FS0006 four_square_add_swap_right_tail

Direct dependents

FS000W four_square_conjugate_scalar_decomposition

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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]