FS000I

four_square_two_square_factor_identity

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 21 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

add_assoc Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized brahmagupta_fibonacci_two_square_identity Alpha theorem; checked-use authorized

Direct dependents

FS000J four_square_two_square_factor_total FS001R four_square_descent_double_pair_identity

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

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.

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 exact 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