FS005I · theorem body

four_square_signed_pair_cross_decomposition

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

The symmetric cross product of two two-addend blocks decomposes into four independent symmetric coordinate pairs.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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

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

Proof neighborhood

Direct theorem prerequisites

mul_add · Stable closed add_mul · Stable closed add_assoc · Stable closed add_comm · Stable closed FS0006 four_square_add_swap_right_tail

Direct theorem dependents

FS000Y four_square_conjugate_cross_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

43 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 (1)
four 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
02Calculate and transport equalitiesL5–14

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

  1. L5
    simp [mul_add, add_mul]
  2. L6
    trans ((x * z) + ((y * z) + ((x * w) + ((y * w) + ((z * x) + ((w * x) + ((z * y) + (w * y))))))))
  3. L7
    simp [add_assoc]
  4. L8
    trans ((x * z) + ((z * x) + ((x * w) + ((w * x) + ((y * z) + ((z * y) + ((y * w) + (w * y))))))))
  5. L9
    congr
  6. L10
    refl
  7. L11
    trans ((z * x) + ((y * z) + ((x * w) + ((y * w) + ((w * x) + ((z * y) + (w * y)))))))
  8. L12
    trans ((y * z) + ((z * x) + ((x * w) + ((y * w) + ((w * x) + ((z * y) + (w * y)))))))
  9. L13
    congr
  10. L14
    refl
03Calculate and transport equalitiesL15–17

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

  1. L15
    trans ((x * w) + ((z * x) + ((y * w) + ((w * x) + ((z * y) + (w * y))))))
  2. L16
    congr
  3. L17
    refl
04Use earlier factsL18–20

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

  1. L18
    apply four_square_add_swap_right_tail
  2. L19
    apply four_square_add_swap_right_tail
  3. L20
    apply four_square_add_swap_right_tail
05Calculate and transport equalitiesL21–23

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

  1. L21
    congr
  2. L22
    refl
  3. L23
    trans ((x * w) + ((y * z) + ((y * w) + ((w * x) + ((z * y) + (w * y))))))
06Use earlier factsL24–24

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

  1. L24
    apply four_square_add_swap_right_tail
07Calculate and transport equalitiesL25–30

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

  1. L25
    congr
  2. L26
    refl
  3. L27
    trans ((w * x) + ((y * z) + ((y * w) + ((z * y) + (w * y)))))
  4. L28
    trans ((y * z) + ((w * x) + ((y * w) + ((z * y) + (w * y)))))
  5. L29
    congr
  6. L30
    refl
08Use earlier factsL31–32

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

  1. L31
    apply four_square_add_swap_right_tail
  2. L32
    apply four_square_add_swap_right_tail
09Calculate and transport equalitiesL33–37

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

  1. L33
    congr
  2. L34
    refl
  3. L35
    congr
  4. L36
    refl
  5. L37
    trans ((z * y) + ((y * w) + (w * y)))
10Use earlier factsL38–38

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

  1. L38
    apply four_square_add_swap_right_tail
11Calculate and transport equalitiesL39–43

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

  1. L39
    congr
  2. L40
    refl
  3. L41
    refl
  4. L42
    symm
  5. L43
    simp [add_assoc]

Library-wide reading audit

Original defined command ledger · 43 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro w
  5. 0005simp [mul_add, add_mul]
  6. 0006trans ((x * z) + ((y * z) + ((x * w) + ((y * w) + ((z * x) + ((w * x) + ((z * y) + (w * y))))))))
  7. 0007simp [add_assoc]
  8. 0008trans ((x * z) + ((z * x) + ((x * w) + ((w * x) + ((y * z) + ((z * y) + ((y * w) + (w * y))))))))
  9. 0009congr
  10. 0010refl
  11. 0011trans ((z * x) + ((y * z) + ((x * w) + ((y * w) + ((w * x) + ((z * y) + (w * y)))))))
  12. 0012trans ((y * z) + ((z * x) + ((x * w) + ((y * w) + ((w * x) + ((z * y) + (w * y)))))))
  13. 0013congr
  14. 0014refl
  15. 0015trans ((x * w) + ((z * x) + ((y * w) + ((w * x) + ((z * y) + (w * y))))))
  16. 0016congr
  17. 0017refl
  18. 0018apply four_square_add_swap_right_tail
  19. 0019apply four_square_add_swap_right_tail
  20. 0020apply four_square_add_swap_right_tail
  21. 0021congr
  22. 0022refl
  23. 0023trans ((x * w) + ((y * z) + ((y * w) + ((w * x) + ((z * y) + (w * y))))))
  24. 0024apply four_square_add_swap_right_tail
  25. 0025congr
  26. 0026refl
  27. 0027trans ((w * x) + ((y * z) + ((y * w) + ((z * y) + (w * y)))))
  28. 0028trans ((y * z) + ((w * x) + ((y * w) + ((z * y) + (w * y)))))
  29. 0029congr
  30. 0030refl
  31. 0031apply four_square_add_swap_right_tail
  32. 0032apply four_square_add_swap_right_tail
  33. 0033congr
  34. 0034refl
  35. 0035congr
  36. 0036refl
  37. 0037trans ((z * y) + ((y * w) + (w * y)))
  38. 0038apply four_square_add_swap_right_tail
  39. 0039congr
  40. 0040refl
  41. 0041refl
  42. 0042symm
  43. 0043simp [add_assoc]