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. ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · m) → ∀ x. Prime(x) → Mod4Three(x) → ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · m) ∧ ((∀ 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
In local proof propositions
Exact expanded first-order statement
(((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)) -> (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 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
Direct theorem dependents
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.
Named ingredients (1)
01Separate the logical casesL1–1
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L1
split