FS003J · theorem body

four_square_lagrange_from_all_primes

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

All-natural Lagrange, including zero, follows constructively from the explicit universal prime-representation premise.

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

(∀ x. Prime(x) → ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · m) → ∀ x. ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · m

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

PD0004 Prime

In local proof propositions

none
Exact expanded first-order statement
(forall fsl_prime_universal. ((~(fsl_prime_universal = 1) /\ forall frm_prime_left_fsl_all_universal frm_prime_right_fsl_all_universal. fsl_prime_universal = frm_prime_left_fsl_all_universal * frm_prime_right_fsl_all_universal -> frm_prime_left_fsl_all_universal = 1 \/ frm_prime_right_fsl_all_universal = 1)) -> (exists fsl_a_all_universal fsl_b_all_universal fsl_c_all_universal fsl_d_all_universal. (fsl_prime_universal) = fsl_a_all_universal * fsl_a_all_universal + fsl_b_all_universal * fsl_b_all_universal + fsl_c_all_universal * fsl_c_all_universal + fsl_d_all_universal * fsl_d_all_universal)) -> forall n. (exists fsl_a_value fsl_b_value fsl_c_value fsl_d_value. (n) = fsl_a_value * fsl_a_value + fsl_b_value * fsl_b_value + fsl_c_value * fsl_c_value + fsl_d_value * fsl_d_value)

Proof neighborhood

Direct theorem prerequisites

eq_decidable · Stable closed le_refl · Stable closed FS003I four_square_lagrange_bounded_from_primes

Direct theorem dependents

FS0032 four_square_lagrange_from_strict_descent FS0036 four_square_lagrange_from_bounded_strict_descent FS003K four_square_lagrange_from_three_mod_four_primes FS003N four_square_lagrange_from_odd_signed_quaternion FS003P four_square_lagrange

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

18 script commands · 6 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 lagrange bounded from primes
01Fix variables and assumptionsL1–2

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

  1. L1
    intro hprimes
  2. L2
    intro n
02Use earlier factsL3–4

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

  1. L3
    specialize eq_decidable n
  2. L4
    specialize eq_decidable 0
03Separate the logical casesL5–5

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

  1. L5
    cases eq_decidable
04Construct an explicit witnessL6–9

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

  1. L6
    exists 0
  2. L7
    exists 0
  3. L8
    exists 0
  4. L9
    exists 0
05Calculate and transport equalitiesL10–11

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

  1. L10
    rewrite eq_decidable_left
  2. L11
    norm_num
06Use earlier factsL12–18

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

  1. L12
    specialize four_square_lagrange_bounded_from_primes n
  2. L13
    specialize four_square_lagrange_bounded_from_primes n
  3. L14
    apply four_square_lagrange_bounded_from_primes
  4. L15
    specialize le_refl n
  5. L16
    exact le_refl
  6. L17
    exact eq_decidable_right
  7. L18
    exact hprimes

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro hprimes
  2. 0002intro n
  3. 0003specialize eq_decidable n
  4. 0004specialize eq_decidable 0
  5. 0005cases eq_decidable
  6. 0006exists 0
  7. 0007exists 0
  8. 0008exists 0
  9. 0009exists 0
  10. 0010rewrite eq_decidable_left
  11. 0011norm_num
  12. 0012specialize four_square_lagrange_bounded_from_primes n
  13. 0013specialize four_square_lagrange_bounded_from_primes n
  14. 0014apply four_square_lagrange_bounded_from_primes
  15. 0015specialize le_refl n
  16. 0016exact le_refl
  17. 0017exact eq_decidable_right
  18. 0018exact hprimes