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.
Definition in prerequisite notation
∃ ff_u_defined_product. ∃ ff_v_defined_product. BetaAt(ff_u_defined_product,ff_v_defined_product,0,1) ∧ (BetaAt(ff_u_defined_product,ff_v_defined_product,l,z) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ n. ∃ m. BetaAt(b,c,x,y) ∧ (BetaAt(ff_u_defined_product,ff_v_defined_product,x,n) ∧ (BetaAt(ff_u_defined_product,ff_v_defined_product,S x,m) ∧ m = n · y))))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists ff_u_defined_product ff_v_defined_product. ((((exists ff_h_defined_product_start. ff_h_defined_product_start + S (1) = S ((S (0)) * ff_v_defined_product)) /\ exists ff_q_defined_product_start. ff_u_defined_product = ff_q_defined_product_start * S ((S (0)) * ff_v_defined_product) + (1))) /\ ((((exists ff_h_defined_product_terminal. ff_h_defined_product_terminal + S (z) = S ((S (l)) * ff_v_defined_product)) /\ exists ff_q_defined_product_terminal. ff_u_defined_product = ff_q_defined_product_terminal * S ((S (l)) * ff_v_defined_product) + (z))) /\ forall ff_i_defined_product. (exists ff_lt_defined_product_bound. ff_lt_defined_product_bound + S ff_i_defined_product = l) -> exists ff_p_defined_product ff_r_defined_product ff_s_defined_product. ((((exists ff_h_defined_product_factor. ff_h_defined_product_factor + S (ff_p_defined_product) = S ((S (ff_i_defined_product)) * c)) /\ exists ff_q_defined_product_factor. b = ff_q_defined_product_factor * S ((S (ff_i_defined_product)) * c) + (ff_p_defined_product))) /\ ((((exists ff_h_defined_product_partial. ff_h_defined_product_partial + S (ff_r_defined_product) = S ((S (ff_i_defined_product)) * ff_v_defined_product)) /\ exists ff_q_defined_product_partial. ff_u_defined_product = ff_q_defined_product_partial * S ((S (ff_i_defined_product)) * ff_v_defined_product) + (ff_r_defined_product))) /\ ((((exists ff_h_defined_product_successor. ff_h_defined_product_successor + S (ff_s_defined_product) = S ((S (S ff_i_defined_product)) * ff_v_defined_product)) /\ exists ff_q_defined_product_successor. ff_u_defined_product = ff_q_defined_product_successor * S ((S (S ff_i_defined_product)) * ff_v_defined_product) + (ff_s_defined_product))) /\ ff_s_defined_product = ff_r_defined_product * ff_p_defined_product)))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.