FS0030 · theorem body

four_square_euler_representations_closed_under_multiplication

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

The class of constructively represented natural sums of four squares is closed under multiplication without any sign or compensation hypothesis.

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 n m. (exists a b c d. n = a * a + b * b + c * c + d * d) -> (exists e f g h. m = e * e + f * f + g * g + h * h) -> exists u v w x. n * m = 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 n m. (exists a b c d. n = a * a + b * b + c * c + d * d) -> (exists e f g h. m = e * e + f * f + g * g + h * h) -> exists u v w x. n * m = u * u + v * v + w * w + x * x

Proof neighborhood

Direct theorem prerequisites

FS002Z four_square_euler_four_square_product_total

Direct theorem dependents

FS003I four_square_lagrange_bounded_from_primes

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

15 script commands · 4 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 (1)
four square euler four square product total
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
  2. L2
    intro m
  3. L3
    intro hn
  4. L4
    intro hm
02Separate the logical casesL5–12

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

  1. L5
    cases hn
  2. L6
    cases hn_witness
  3. L7
    cases hn_witness_witness
  4. L8
    cases hn_witness_witness_witness
  5. L9
    cases hm
  6. L10
    cases hm_witness
  7. L11
    cases hm_witness_witness
  8. L12
    cases hm_witness_witness_witness
03Calculate and transport equalitiesL13–14

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

  1. L13
    rewrite hn_witness_witness_witness_witness
  2. L14
    rewrite hm_witness_witness_witness_witness
04Use earlier factsL15–15

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

  1. L15
    apply four_square_euler_four_square_product_total

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro n
  2. 0002intro m
  3. 0003intro hn
  4. 0004intro hm
  5. 0005cases hn
  6. 0006cases hn_witness
  7. 0007cases hn_witness_witness
  8. 0008cases hn_witness_witness_witness
  9. 0009cases hm
  10. 0010cases hm_witness
  11. 0011cases hm_witness_witness
  12. 0012cases hm_witness_witness_witness
  13. 0013rewrite hn_witness_witness_witness_witness
  14. 0014rewrite hm_witness_witness_witness_witness
  15. 0015apply four_square_euler_four_square_product_total