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
∀ b. ∀ c. ∀ l. ∀ a. (∀ x. ∀ y. Lt(x,S l) → BetaAt(b,c,x,y) → ∃ z. ∃ n. y = z · z + n · n) → BetaAt(b,c,l,a) → ∃ x. ∃ y. a = x · x + y · yEvery 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 b c l a. (forall ftsf_index_drop_next ftsf_factor_drop_next. (exists ftsf_gap_drop_next_bound. ftsf_gap_drop_next_bound + S ftsf_index_drop_next = (S l)) -> (((exists ff_h_ftsf_drop_next_entry. ff_h_ftsf_drop_next_entry + S (ftsf_factor_drop_next) = S ((S (ftsf_index_drop_next)) * c)) /\ exists ff_q_ftsf_drop_next_entry. b = ff_q_ftsf_drop_next_entry * S ((S (ftsf_index_drop_next)) * c) + (ftsf_factor_drop_next))) -> (exists ftsf_first_drop_next_representation ftsf_second_drop_next_representation. (ftsf_factor_drop_next) = ftsf_first_drop_next_representation * ftsf_first_drop_next_representation + ftsf_second_drop_next_representation * ftsf_second_drop_next_representation)) -> (((exists ff_h_ftsf_last_entry. ff_h_ftsf_last_entry + S (a) = S ((S (l)) * c)) /\ exists ff_q_ftsf_last_entry. b = ff_q_ftsf_last_entry * S ((S (l)) * c) + (a))) -> (exists ftsf_first_last_result ftsf_second_last_result. (a) = ftsf_first_last_result * ftsf_first_last_result + ftsf_second_last_result * ftsf_second_last_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.