FS000I · theorem body

four_square_two_square_factor_identity

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

The six-variable Euler subclass with a two-square right factor is exactly the sum of two independently composed two-square norms.

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 m n. (a * f = b * e + m \/ b * e = a * f + m) -> (c * f = d * e + n \/ d * e = c * f + n) -> (a * a + b * b + c * c + d * d) * (e * e + f * f) = ((a * e + b * f) * (a * e + b * f) + m * m) + ((c * e + d * f) * (c * e + d * f) + n * n)

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 m n. (a * f = b * e + m \/ b * e = a * f + m) -> (c * f = d * e + n \/ d * e = c * f + n) -> (a * a + b * b + c * c + d * d) * (e * e + f * f) = ((a * e + b * f) * (a * e + b * f) + m * m) + ((c * e + d * f) * (c * e + d * f) + n * n)

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_mul · Stable closed brahmagupta_fibonacci_two_square_identity · Alpha closed

Direct theorem dependents

FS000J four_square_two_square_factor_total FS001R four_square_descent_double_pair_identity

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

21 script commands · 7 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.

01Fix variables and assumptionsL1–10

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 m
  8. L8
    intro n
  9. L9
    intro hfirst
  10. L10
    intro hsecond
02Calculate and transport equalitiesL11–12

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

  1. L11
    trans ((a * a + b * b) + (c * c + d * d)) * (e * e + f * f)
  2. L12
    congr
03Use earlier factsL13–13

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

  1. L13
    apply add_assoc
04Calculate and transport equalitiesL14–15

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

  1. L14
    refl
  2. L15
    trans (a * a + b * b) * (e * e + f * f) + (c * c + d * d) * (e * e + f * f)
05Use earlier factsL16–16

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

  1. L16
    apply add_mul
06Calculate and transport equalitiesL17–17

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

  1. L17
    congr
07Use earlier factsL18–21

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

  1. L18
    apply brahmagupta_fibonacci_two_square_identity
  2. L19
    exact hfirst
  3. L20
    apply brahmagupta_fibonacci_two_square_identity
  4. L21
    exact hsecond

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro m
  8. 0008intro n
  9. 0009intro hfirst
  10. 0010intro hsecond
  11. 0011trans ((a * a + b * b) + (c * c + d * d)) * (e * e + f * f)
  12. 0012congr
  13. 0013apply add_assoc
  14. 0014refl
  15. 0015trans (a * a + b * b) * (e * e + f * f) + (c * c + d * d) * (e * e + f * f)
  16. 0016apply add_mul
  17. 0017congr
  18. 0018apply brahmagupta_fibonacci_two_square_identity
  19. 0019exact hfirst
  20. 0020apply brahmagupta_fibonacci_two_square_identity
  21. 0021exact hsecond