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. Prime(x) → ¬y = 0 → ¬y = 1 → Lt(y,x) → (∃ 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
Exact expanded first-order statement
(forall fslb_bounded_prime_bridge fslb_bounded_multiplier_bridge. ((~(fslb_bounded_prime_bridge = 1) /\ forall frm_prime_left_fslb_bounded_prime_bridge frm_prime_right_fslb_bounded_prime_bridge. fslb_bounded_prime_bridge = frm_prime_left_fslb_bounded_prime_bridge * frm_prime_right_fslb_bounded_prime_bridge -> frm_prime_left_fslb_bounded_prime_bridge = 1 \/ frm_prime_right_fslb_bounded_prime_bridge = 1)) -> ~(fslb_bounded_multiplier_bridge = 0) -> ~(fslb_bounded_multiplier_bridge = 1) -> (exists fslb_bounded_upper_gap_bridge. fslb_bounded_upper_gap_bridge + S fslb_bounded_multiplier_bridge = fslb_bounded_prime_bridge) -> (exists fsl_a_fslb_bounded_source_bridge fsl_b_fslb_bounded_source_bridge fsl_c_fslb_bounded_source_bridge fsl_d_fslb_bounded_source_bridge. (fslb_bounded_prime_bridge * fslb_bounded_multiplier_bridge) = fsl_a_fslb_bounded_source_bridge * fsl_a_fslb_bounded_source_bridge + fsl_b_fslb_bounded_source_bridge * fsl_b_fslb_bounded_source_bridge + fsl_c_fslb_bounded_source_bridge * fsl_c_fslb_bounded_source_bridge + fsl_d_fslb_bounded_source_bridge * fsl_d_fslb_bounded_source_bridge) -> exists fslb_bounded_smaller_bridge. (~(fslb_bounded_smaller_bridge = 0) /\ ((exists fslb_bounded_lower_gap_bridge. fslb_bounded_lower_gap_bridge + S fslb_bounded_smaller_bridge = fslb_bounded_multiplier_bridge) /\ (exists fsl_a_fslb_bounded_target_bridge fsl_b_fslb_bounded_target_bridge fsl_c_fslb_bounded_target_bridge fsl_d_fslb_bounded_target_bridge. (fslb_bounded_prime_bridge * fslb_bounded_smaller_bridge) = fsl_a_fslb_bounded_target_bridge * fsl_a_fslb_bounded_target_bridge + fsl_b_fslb_bounded_target_bridge * fsl_b_fslb_bounded_target_bridge + fsl_c_fslb_bounded_target_bridge * fsl_c_fslb_bounded_target_bridge + fsl_d_fslb_bounded_target_bridge * fsl_d_fslb_bounded_target_bridge)))) -> forall n. (exists fsl_a_fslb_natural_result fsl_b_fslb_natural_result fsl_c_fslb_natural_result fsl_d_fslb_natural_result. (n) = fsl_a_fslb_natural_result * fsl_a_fslb_natural_result + fsl_b_fslb_natural_result * fsl_b_fslb_natural_result + fsl_c_fslb_natural_result * fsl_c_fslb_natural_result + fsl_d_fslb_natural_result * fsl_d_fslb_natural_result)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 (2)
01Fix variables and assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro hstep