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. Dvd(x,y · y + z · z + 1)) → (∀ x. ∀ y. Prime(x) → ¬y = 0 → ¬y = 1 → (∃ z. ∃ n. ∃ m. ∃ k. x · y = z · z + n · n + m · m + k · k) → ∃ z. ¬z = 0 ∧ (Lt(z,y) ∧ (∃ n. ∃ m. ∃ k. ∃ i. x · z = n · n + m · m + k · k + i · i))) → ∀ x. ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · mEvery 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
In local proof propositions
none
Exact expanded first-order statement
(forall fsd_seed_prime_universal. ((~(fsd_seed_prime_universal = 1) /\ forall frm_prime_left_fsd_seed_universal frm_prime_right_fsd_seed_universal. fsd_seed_prime_universal = frm_prime_left_fsd_seed_universal * frm_prime_right_fsd_seed_universal -> frm_prime_left_fsd_seed_universal = 1 \/ frm_prime_right_fsd_seed_universal = 1)) -> (exists fsd_seed_residue_universal. fsd_seed_prime_universal = 4 * fsd_seed_residue_universal + 3) -> exists fsd_seed_first_universal fsd_seed_second_universal fsd_seed_multiplier_universal. fsd_seed_first_universal * fsd_seed_first_universal + fsd_seed_second_universal * fsd_seed_second_universal + 1 = fsd_seed_prime_universal * fsd_seed_multiplier_universal) -> (forall fsd_prime_universal fsd_multiplier_universal. ((~(fsd_prime_universal = 1) /\ forall frm_prime_left_fsd_universal_prime frm_prime_right_fsd_universal_prime. fsd_prime_universal = frm_prime_left_fsd_universal_prime * frm_prime_right_fsd_universal_prime -> frm_prime_left_fsd_universal_prime = 1 \/ frm_prime_right_fsd_universal_prime = 1)) -> ~(fsd_multiplier_universal = 0) -> ~(fsd_multiplier_universal = 1) -> (exists fsl_a_fsd_universal_source fsl_b_fsd_universal_source fsl_c_fsd_universal_source fsl_d_fsd_universal_source. (fsd_prime_universal * fsd_multiplier_universal) = fsl_a_fsd_universal_source * fsl_a_fsd_universal_source + fsl_b_fsd_universal_source * fsl_b_fsd_universal_source + fsl_c_fsd_universal_source * fsl_c_fsd_universal_source + fsl_d_fsd_universal_source * fsl_d_fsd_universal_source) -> exists fsd_smaller_universal. (~(fsd_smaller_universal = 0) /\ ((exists fsd_gap_universal. fsd_gap_universal + S fsd_smaller_universal = fsd_multiplier_universal) /\ (exists fsl_a_fsd_universal_target fsl_b_fsd_universal_target fsl_c_fsd_universal_target fsl_d_fsd_universal_target. (fsd_prime_universal * fsd_smaller_universal) = fsl_a_fsd_universal_target * fsl_a_fsd_universal_target + fsl_b_fsd_universal_target * fsl_b_fsd_universal_target + fsl_c_fsd_universal_target * fsl_c_fsd_universal_target + fsl_d_fsd_universal_target * fsl_d_fsd_universal_target)))) -> forall n. (exists fsl_a_fsd_lagrange_result fsl_b_fsd_lagrange_result fsl_c_fsd_lagrange_result fsl_d_fsd_lagrange_result. (n) = fsl_a_fsd_lagrange_result * fsl_a_fsd_lagrange_result + fsl_b_fsd_lagrange_result * fsl_b_fsd_lagrange_result + fsl_c_fsd_lagrange_result * fsl_c_fsd_lagrange_result + fsl_d_fsd_lagrange_result * fsl_d_fsd_lagrange_result)Proof neighborhood
Direct theorem prerequisites
FS001J four_square_descent_three_mod_four_primes_from_seed_and_step FS003K four_square_lagrange_from_three_mod_four_primesDirect theorem dependents
none
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
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.