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
∃ bpr_code_bertrand_defined_primorial. ∃ bpr_scale_bertrand_defined_primorial. (∀ x. Lt(x,n) → ∃ y. BetaAt(bpr_code_bertrand_defined_primorial,bpr_scale_bertrand_defined_primorial,x,y) ∧ (Prime(S x) ∧ y = S x ∨ ¬Prime(S x) ∧ y = 1)) ∧ Product(bpr_code_bertrand_defined_primorial,bpr_scale_bertrand_defined_primorial,n,z)Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists bpr_code_bertrand_defined_primorial bpr_scale_bertrand_defined_primorial. ((forall bpr_index_bertrand_defined_primorial_mask. (exists bpr_gap_bertrand_defined_primorial_mask_bound. bpr_gap_bertrand_defined_primorial_mask_bound + S (bpr_index_bertrand_defined_primorial_mask) = n) -> exists bpr_value_bertrand_defined_primorial_mask. ((((exists bpr_height_bertrand_defined_primorial_mask_decoded. bpr_height_bertrand_defined_primorial_mask_decoded + S (bpr_value_bertrand_defined_primorial_mask) = S ((S (bpr_index_bertrand_defined_primorial_mask)) * bpr_scale_bertrand_defined_primorial)) /\ exists bpr_quotient_bertrand_defined_primorial_mask_decoded. bpr_code_bertrand_defined_primorial = bpr_quotient_bertrand_defined_primorial_mask_decoded * S ((S (bpr_index_bertrand_defined_primorial_mask)) * bpr_scale_bertrand_defined_primorial) + (bpr_value_bertrand_defined_primorial_mask))) /\ (((((~(S (bpr_index_bertrand_defined_primorial_mask) = 1) /\ forall bpr_left_bertrand_defined_primorial_mask_choice_prime bpr_right_bertrand_defined_primorial_mask_choice_prime. S (bpr_index_bertrand_defined_primorial_mask) = bpr_left_bertrand_defined_primorial_mask_choice_prime * bpr_right_bertrand_defined_primorial_mask_choice_prime -> bpr_left_bertrand_defined_primorial_mask_choice_prime = 1 \/ bpr_right_bertrand_defined_primorial_mask_choice_prime = 1)) /\ bpr_value_bertrand_defined_primorial_mask = S (bpr_index_bertrand_defined_primorial_mask)) \/ (~((~(S (bpr_index_bertrand_defined_primorial_mask) = 1) /\ forall bpr_left_bertrand_defined_primorial_mask_choice_prime bpr_right_bertrand_defined_primorial_mask_choice_prime. S (bpr_index_bertrand_defined_primorial_mask) = bpr_left_bertrand_defined_primorial_mask_choice_prime * bpr_right_bertrand_defined_primorial_mask_choice_prime -> bpr_left_bertrand_defined_primorial_mask_choice_prime = 1 \/ bpr_right_bertrand_defined_primorial_mask_choice_prime = 1)) /\ bpr_value_bertrand_defined_primorial_mask = 1))))) /\ (exists ff_u_bertrand_defined_primorial_product ff_v_bertrand_defined_primorial_product. ((((exists ff_h_bertrand_defined_primorial_product_start. ff_h_bertrand_defined_primorial_product_start + S (1) = S ((S (0)) * ff_v_bertrand_defined_primorial_product)) /\ exists ff_q_bertrand_defined_primorial_product_start. ff_u_bertrand_defined_primorial_product = ff_q_bertrand_defined_primorial_product_start * S ((S (0)) * ff_v_bertrand_defined_primorial_product) + (1))) /\ ((((exists ff_h_bertrand_defined_primorial_product_terminal. ff_h_bertrand_defined_primorial_product_terminal + S (z) = S ((S (n)) * ff_v_bertrand_defined_primorial_product)) /\ exists ff_q_bertrand_defined_primorial_product_terminal. ff_u_bertrand_defined_primorial_product = ff_q_bertrand_defined_primorial_product_terminal * S ((S (n)) * ff_v_bertrand_defined_primorial_product) + (z))) /\ forall ff_i_bertrand_defined_primorial_product. (exists ff_lt_bertrand_defined_primorial_product_bound. ff_lt_bertrand_defined_primorial_product_bound + S ff_i_bertrand_defined_primorial_product = n) -> exists ff_p_bertrand_defined_primorial_product ff_r_bertrand_defined_primorial_product ff_s_bertrand_defined_primorial_product. ((((exists ff_h_bertrand_defined_primorial_product_factor. ff_h_bertrand_defined_primorial_product_factor + S (ff_p_bertrand_defined_primorial_product) = S ((S (ff_i_bertrand_defined_primorial_product)) * bpr_scale_bertrand_defined_primorial)) /\ exists ff_q_bertrand_defined_primorial_product_factor. bpr_code_bertrand_defined_primorial = ff_q_bertrand_defined_primorial_product_factor * S ((S (ff_i_bertrand_defined_primorial_product)) * bpr_scale_bertrand_defined_primorial) + (ff_p_bertrand_defined_primorial_product))) /\ ((((exists ff_h_bertrand_defined_primorial_product_partial. ff_h_bertrand_defined_primorial_product_partial + S (ff_r_bertrand_defined_primorial_product) = S ((S (ff_i_bertrand_defined_primorial_product)) * ff_v_bertrand_defined_primorial_product)) /\ exists ff_q_bertrand_defined_primorial_product_partial. ff_u_bertrand_defined_primorial_product = ff_q_bertrand_defined_primorial_product_partial * S ((S (ff_i_bertrand_defined_primorial_product)) * ff_v_bertrand_defined_primorial_product) + (ff_r_bertrand_defined_primorial_product))) /\ ((((exists ff_h_bertrand_defined_primorial_product_successor. ff_h_bertrand_defined_primorial_product_successor + S (ff_s_bertrand_defined_primorial_product) = S ((S (S ff_i_bertrand_defined_primorial_product)) * ff_v_bertrand_defined_primorial_product)) /\ exists ff_q_bertrand_defined_primorial_product_successor. ff_u_bertrand_defined_primorial_product = ff_q_bertrand_defined_primorial_product_successor * S ((S (S ff_i_bertrand_defined_primorial_product)) * ff_v_bertrand_defined_primorial_product) + (ff_s_bertrand_defined_primorial_product))) /\ ff_s_bertrand_defined_primorial_product = ff_r_bertrand_defined_primorial_product * ff_p_bertrand_defined_primorial_product)))))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
none