FS000Y

four_square_conjugate_cross_decomposition

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The three conjugate vector corrections decompose into the twelve opposite-sign mixed blocks; the scalar correction vanishes.

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 a b c d e f g h. (((((a * e + b * f + c * g + d * h) * (0) + (0) * (a * e + b * f + c * g + d * h)) + ((a * f + c * h) * (b * e + d * g) + (b * e + d * g) * (a * f + c * h))) + (((a * g + d * f) * (c * e + b * h) + (c * e + b * h) * (a * g + d * f)) + ((a * h + b * g) * (d * e + c * f) + (d * e + c * f) * (a * h + b * g))))) = (((((((((((((((a * f) * (b * e) + (b * e) * (a * f))) + (((a * f) * (d * g) + (d * g) * (a * f)))) + (((c * h) * (b * e) + (b * e) * (c * h)))) + (((c * h) * (d * g) + (d * g) * (c * h)))) + (((a * g) * (c * e) + (c * e) * (a * g)))) + (((a * g) * (b * h) + (b * h) * (a * g)))) + (((d * f) * (c * e) + (c * e) * (d * f)))) + (((d * f) * (b * h) + (b * h) * (d * f)))) + (((a * h) * (d * e) + (d * e) * (a * h)))) + (((a * h) * (c * f) + (c * f) * (a * h)))) + (((b * g) * (d * e) + (d * e) * (b * g)))) + (((b * g) * (c * f) + (c * f) * (b * g)))))

Constructive proof overview

Generated structural guide

The three conjugate vector corrections decompose into the twelve opposite-sign mixed blocks; the scalar correction vanishes.

The unchanged tactic script uses 4 declared prerequisites and contains 17 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

FS005I four_square_signed_pair_cross_decomposition add_assoc Stable theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

FS0010 four_square_conjugate_global_compensation

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

17 script commands · 6 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.

Named ingredients (1)
four square signed pair cross 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 equalitiesL9–12

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

  1. L9
    trans (((0) + (((((((a * f) * (b * e) + (b * e) * (a * f))) + (((a * f) * (d * g) + (d * g) * (a * f))))) + (((((c * h) * (b * e) + (b * e) * (c * h))) + (((c * h) * (d * g) + (d * g) * (c * h)))))))) + ((((((((a * g) * (c * e) + (c * e) * (a * g))) + (((a * g) * (b * h) + (b * h) * (a * g))))) + (((((d * f) * (c * e) + (c * e) * (d * f))) + (((d * f) * (b * h) + (b * h) * (d * f))))))) + (((((((a * h) * (d * e) + (d * e) * (a * h))) + (((a * h) * (c * f) + (c * f) * (a * h))))) + (((((b * g) * (d * e) + (d * e) * (b * g))) + (((b * g) * (c * f) + (c * f) * (b * g)))))))))
  2. L10
    congr
  3. L11
    congr
  4. L12
    simp [mul_zero_left, zero_add]
03Use earlier factsL13–13

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

  1. L13
    apply four_square_signed_pair_cross_decomposition
04Calculate and transport equalitiesL14–14

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

  1. L14
    congr
05Use earlier factsL15–16

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

  1. L15
    apply four_square_signed_pair_cross_decomposition
  2. L16
    apply four_square_signed_pair_cross_decomposition
06Calculate and transport equalitiesL17–17

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

  1. L17
    simp [add_assoc, zero_add]

Library-wide reading audit

Original exact command ledger · 17 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 (((0) + (((((((a * f) * (b * e) + (b * e) * (a * f))) + (((a * f) * (d * g) + (d * g) * (a * f))))) + (((((c * h) * (b * e) + (b * e) * (c * h))) + (((c * h) * (d * g) + (d * g) * (c * h)))))))) + ((((((((a * g) * (c * e) + (c * e) * (a * g))) + (((a * g) * (b * h) + (b * h) * (a * g))))) + (((((d * f) * (c * e) + (c * e) * (d * f))) + (((d * f) * (b * h) + (b * h) * (d * f))))))) + (((((((a * h) * (d * e) + (d * e) * (a * h))) + (((a * h) * (c * f) + (c * f) * (a * h))))) + (((((b * g) * (d * e) + (d * e) * (b * g))) + (((b * g) * (c * f) + (c * f) * (b * g)))))))))
  10. 0010congr
  11. 0011congr
  12. 0012simp [mul_zero_left, zero_add]
  13. 0013apply four_square_signed_pair_cross_decomposition
  14. 0014congr
  15. 0015apply four_square_signed_pair_cross_decomposition
  16. 0016apply four_square_signed_pair_cross_decomposition
  17. 0017simp [add_assoc, zero_add]