FS000J · theorem body

four_square_two_square_factor_total

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

Every four-square norm multiplied by any two-square norm 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. exists u v w x. (a * a + b * b + c * c + d * d) * (e * e + f * f) = u * u + v * v + w * w + x * x

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. exists u v w x. (a * a + b * b + c * c + d * d) * (e * e + f * f) = u * u + v * v + w * w + x * x

Proof neighborhood

Direct theorem prerequisites

FS000H four_square_absolute_difference_total FS000I four_square_two_square_factor_identity add_assoc · Stable closed

Direct theorem dependents

none

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

22 script commands · 10 reading checkpoints · 2 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)
four square absolute difference totalfour square two square factor identity
01Fix variables and assumptionsL1–6

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
02Establish hfirstL7–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square absolute difference total.

  1. L7
    have hfirst : exists m. (a * f = b * e + m \/ b * e = a * f + m)
  2. L8
    apply four_square_absolute_difference_total
03Separate the logical casesL9–9

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

  1. L9
    cases hfirst
04Establish hsecondL10–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square absolute difference total.

  1. L10
    have hsecond : exists n. (c * f = d * e + n \/ d * e = c * f + n)
  2. L11
    apply four_square_absolute_difference_total
05Separate the logical casesL12–12

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

  1. L12
    cases hsecond
06Construct an explicit witnessL13–16

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

  1. L13
    exists a * e + b * f
  2. L14
    exists x
  3. L15
    exists c * e + d * f
  4. L16
    exists x1
07Calculate and transport equalitiesL17–17

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

  1. L17
    trans ((a * e + b * f) * (a * e + b * f) + x * x) + ((c * e + d * f) * (c * e + d * f) + x1 * x1)
08Use earlier factsL18–20

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

  1. L18
    apply four_square_two_square_factor_identity
  2. L19
    exact hfirst_witness
  3. L20
    exact hsecond_witness
09Calculate and transport equalitiesL21–21

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

  1. L21
    symm
10Use earlier factsL22–22

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

  1. L22
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007have hfirst : exists m. (a * f = b * e + m \/ b * e = a * f + m)
  8. 0008apply four_square_absolute_difference_total
  9. 0009cases hfirst
  10. 0010have hsecond : exists n. (c * f = d * e + n \/ d * e = c * f + n)
  11. 0011apply four_square_absolute_difference_total
  12. 0012cases hsecond
  13. 0013exists a * e + b * f
  14. 0014exists x
  15. 0015exists c * e + d * f
  16. 0016exists x1
  17. 0017trans ((a * e + b * f) * (a * e + b * f) + x * x) + ((c * e + d * f) * (c * e + d * f) + x1 * x1)
  18. 0018apply four_square_two_square_factor_identity
  19. 0019exact hfirst_witness
  20. 0020exact hsecond_witness
  21. 0021symm
  22. 0022apply add_assoc