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.
Readable signature
Product(b,c,l,z)Exact expansion
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)))))This node is conservative notation, not a theorem, new axiom, predicate constant, or kernel rule. Its expansion is checked for exact first-order AST equivalence.
Definition neighborhood
Expands using
Used by definitions
Used by theorem statements or local proof propositions
BT005F beta_product_exists BT005I beta_product_zero BT005J beta_product_succ_decompose BT005K beta_product_succ_append BT005L beta_product_transport_prefix BT0069 beta_factor_divides_product BT0082 pow_functional BT0083 pow_successor_decompose BT0090 factorial_functional BT0092 factorial_succ_decompose BT00DH beta_product_pointwise_coprime BT00UA primorial_exists BT00UD primorial_succ_decompose BT00UQ beta_product_prefix_suffix_split BT00UY primorial_prefix_interval_split BT00VD beta_pairwise_coprime_product_divides_common_multiple BT00VF primorial_interval_divides_choose_between BT00VG primorial_even_interval_divides_central BT00VH primorial_odd_interval_divides_middle BT00VI primorial_even_interval_le_central BT00VJ primorial_odd_interval_le_middle BT00VU primorial_four_power_support_package BT00VV primorial_le_four_pow_bounded BT00X9 beta_product_pointwise_le BT00XA beta_product_uniform_le_pow BT00YS prime_contribution_product_exists BT00YY prime_contribution_product_divides BT0100 prime_contribution_selected_entry BT0102 prime_contribution_cofactor_prime_contradiction BT0103 prime_contribution_cofactor_eq_one BT0104 prime_contribution_reverse_divides BT0105 prime_contribution_product_eq BT0106 prime_contribution_complete_exists BT0107 central_binom_prime_contribution_product_exists BT010R prime_contribution_prefix_interval_split BT010S prime_contribution_product_length_eq_transport BT010U beta_product_all_one_exact BT010Y no_bertrand_small_contribution_product_le_power BT0110 no_bertrand_middle_contribution_interval_le_primorial_interval BT0111 no_bertrand_middle_contribution_interval_le_four_pow BT0112 no_bertrand_high_contribution_interval_eq_one BT0113 central_binom_factorization_small BT0114 central_binom_le_of_no_bertrand_prime