FS002Z · theorem body

four_square_euler_four_square_product_total

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

Every product of two arbitrary four-square natural norms has four explicitly constructed natural-square witnesses.

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. exists m0 m1 m2 m3. ((a * a + b * b + c * c + d * d) * (e * e + f * f + g * g + h * h)) = m0 * m0 + m1 * m1 + m2 * m2 + m3 * m3

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. exists m0 m1 m2 m3. ((a * a + b * b + c * c + d * d) * (e * e + f * f + g * g + h * h)) = m0 * m0 + m1 * m1 + m2 * m2 + m3 * m3

Proof neighborhood

Direct theorem prerequisites

FS0005 quaternion_coordinate_absolute_total FS002Y four_square_euler_quaternion

Direct theorem dependents

FS0030 four_square_euler_representations_closed_under_multiplication

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

26 script commands · 5 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)
quaternion coordinate absolute totalfour square euler quaternion
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
02Use earlier factsL9–16

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

  1. L9
    specialize quaternion_coordinate_absolute_total a
  2. L10
    specialize quaternion_coordinate_absolute_total b
  3. L11
    specialize quaternion_coordinate_absolute_total c
  4. L12
    specialize quaternion_coordinate_absolute_total d
  5. L13
    specialize quaternion_coordinate_absolute_total e
  6. L14
    specialize quaternion_coordinate_absolute_total f
  7. L15
    specialize quaternion_coordinate_absolute_total g
  8. L16
    specialize quaternion_coordinate_absolute_total h
03Separate the logical casesL17–20

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L17
    cases quaternion_coordinate_absolute_total
  2. L18
    cases quaternion_coordinate_absolute_total_witness
  3. L19
    cases quaternion_coordinate_absolute_total_witness_witness
  4. L20
    cases quaternion_coordinate_absolute_total_witness_witness_witness
04Construct an explicit witnessL21–24

Supply the displayed value, then prove that it has the required property.

  1. L21
    exists x
  2. L22
    exists x1
  3. L23
    exists x2
  4. L24
    exists x3
05Use earlier factsL25–26

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

  1. L25
    apply four_square_euler_quaternion
  2. L26
    exact quaternion_coordinate_absolute_total_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 26 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. 0009specialize quaternion_coordinate_absolute_total a
  10. 0010specialize quaternion_coordinate_absolute_total b
  11. 0011specialize quaternion_coordinate_absolute_total c
  12. 0012specialize quaternion_coordinate_absolute_total d
  13. 0013specialize quaternion_coordinate_absolute_total e
  14. 0014specialize quaternion_coordinate_absolute_total f
  15. 0015specialize quaternion_coordinate_absolute_total g
  16. 0016specialize quaternion_coordinate_absolute_total h
  17. 0017cases quaternion_coordinate_absolute_total
  18. 0018cases quaternion_coordinate_absolute_total_witness
  19. 0019cases quaternion_coordinate_absolute_total_witness_witness
  20. 0020cases quaternion_coordinate_absolute_total_witness_witness_witness
  21. 0021exists x
  22. 0022exists x1
  23. 0023exists x2
  24. 0024exists x3
  25. 0025apply four_square_euler_quaternion
  26. 0026exact quaternion_coordinate_absolute_total_witness_witness_witness_witness