FS003K · theorem body

four_square_lagrange_from_three_mod_four_primes

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

The full universal four-square theorem is reduced precisely to constructive representation of primes congruent to three modulo four.

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) → Mod4Three(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 PD0012 Mod4Three

In local proof propositions

none
Exact expanded first-order statement
(forall fsl_three_prime_universal. ((~(fsl_three_prime_universal = 1) /\ forall frm_prime_left_fsl_three_universal frm_prime_right_fsl_three_universal. fsl_three_prime_universal = frm_prime_left_fsl_three_universal * frm_prime_right_fsl_three_universal -> frm_prime_left_fsl_three_universal = 1 \/ frm_prime_right_fsl_three_universal = 1)) -> (exists fsl_three_residue_universal. fsl_three_prime_universal = 4 * fsl_three_residue_universal + 3) -> (exists fsl_a_three_universal fsl_b_three_universal fsl_c_three_universal fsl_d_three_universal. (fsl_three_prime_universal) = fsl_a_three_universal * fsl_a_three_universal + fsl_b_three_universal * fsl_b_three_universal + fsl_c_three_universal * fsl_c_three_universal + fsl_d_three_universal * fsl_d_three_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

FS003H four_square_prime_case_reduction FS003J four_square_lagrange_from_all_primes

Direct theorem dependents

FS001K four_square_lagrange_from_modular_seeds_and_strict_descent FS003L four_square_lagrange_iff_three_mod_four_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

4 script commands · 2 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 (2)
four square prime case reductionfour square lagrange from all primes
01Fix variables and assumptionsL1–1

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

  1. L1
    intro hthree
02Use earlier factsL2–4

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

  1. L2
    apply four_square_lagrange_from_all_primes
  2. L3
    apply four_square_prime_case_reduction
  3. L4
    exact hthree

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001intro hthree
  2. 0002apply four_square_lagrange_from_all_primes
  3. 0003apply four_square_prime_case_reduction
  4. 0004exact hthree