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.
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PowerValuation(p,n,e)Exact expansion
((exists bpv_gap_bertrand_defined_power_valuation_exponent_bound. bpv_gap_bertrand_defined_power_valuation_exponent_bound + e = n) /\ (exists bpv_result_bertrand_defined_power_valuation_selected. ((exists ff_b_bertrand_defined_power_valuation_selected_power ff_c_bertrand_defined_power_valuation_selected_power. ((forall ff_i_bertrand_defined_power_valuation_selected_power_repeat. (exists ff_lt_bertrand_defined_power_valuation_selected_power_repeat_bound. ff_lt_bertrand_defined_power_valuation_selected_power_repeat_bound + S ff_i_bertrand_defined_power_valuation_selected_power_repeat = e) -> (((exists ff_h_bertrand_defined_power_valuation_selected_power_repeat_decoded. ff_h_bertrand_defined_power_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bertrand_defined_power_valuation_selected_power_repeat)) * ff_c_bertrand_defined_power_valuation_selected_power)) /\ exists ff_q_bertrand_defined_power_valuation_selected_power_repeat_decoded. ff_b_bertrand_defined_power_valuation_selected_power = ff_q_bertrand_defined_power_valuation_selected_power_repeat_decoded * S ((S (ff_i_bertrand_defined_power_valuation_selected_power_repeat)) * ff_c_bertrand_defined_power_valuation_selected_power) + (p)))) /\ (exists ff_u_bertrand_defined_power_valuation_selected_power_product ff_v_bertrand_defined_power_valuation_selected_power_product. ((((exists ff_h_bertrand_defined_power_valuation_selected_power_product_start. ff_h_bertrand_defined_power_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bertrand_defined_power_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_selected_power_product_start. ff_u_bertrand_defined_power_valuation_selected_power_product = ff_q_bertrand_defined_power_valuation_selected_power_product_start * S ((S (0)) * ff_v_bertrand_defined_power_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bertrand_defined_power_valuation_selected_power_product_terminal. ff_h_bertrand_defined_power_valuation_selected_power_product_terminal + S (bpv_result_bertrand_defined_power_valuation_selected) = S ((S (e)) * ff_v_bertrand_defined_power_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_selected_power_product_terminal. ff_u_bertrand_defined_power_valuation_selected_power_product = ff_q_bertrand_defined_power_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bertrand_defined_power_valuation_selected_power_product) + (bpv_result_bertrand_defined_power_valuation_selected))) /\ forall ff_i_bertrand_defined_power_valuation_selected_power_product. (exists ff_lt_bertrand_defined_power_valuation_selected_power_product_bound. ff_lt_bertrand_defined_power_valuation_selected_power_product_bound + S ff_i_bertrand_defined_power_valuation_selected_power_product = e) -> exists ff_p_bertrand_defined_power_valuation_selected_power_product ff_r_bertrand_defined_power_valuation_selected_power_product ff_s_bertrand_defined_power_valuation_selected_power_product. ((((exists ff_h_bertrand_defined_power_valuation_selected_power_product_factor. ff_h_bertrand_defined_power_valuation_selected_power_product_factor + S (ff_p_bertrand_defined_power_valuation_selected_power_product) = S ((S (ff_i_bertrand_defined_power_valuation_selected_power_product)) * ff_c_bertrand_defined_power_valuation_selected_power)) /\ exists ff_q_bertrand_defined_power_valuation_selected_power_product_factor. ff_b_bertrand_defined_power_valuation_selected_power = ff_q_bertrand_defined_power_valuation_selected_power_product_factor * S ((S (ff_i_bertrand_defined_power_valuation_selected_power_product)) * ff_c_bertrand_defined_power_valuation_selected_power) + (ff_p_bertrand_defined_power_valuation_selected_power_product))) /\ ((((exists ff_h_bertrand_defined_power_valuation_selected_power_product_partial. ff_h_bertrand_defined_power_valuation_selected_power_product_partial + S (ff_r_bertrand_defined_power_valuation_selected_power_product) = S ((S (ff_i_bertrand_defined_power_valuation_selected_power_product)) * ff_v_bertrand_defined_power_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_selected_power_product_partial. ff_u_bertrand_defined_power_valuation_selected_power_product = ff_q_bertrand_defined_power_valuation_selected_power_product_partial * S ((S (ff_i_bertrand_defined_power_valuation_selected_power_product)) * ff_v_bertrand_defined_power_valuation_selected_power_product) + (ff_r_bertrand_defined_power_valuation_selected_power_product))) /\ ((((exists ff_h_bertrand_defined_power_valuation_selected_power_product_successor. ff_h_bertrand_defined_power_valuation_selected_power_product_successor + S (ff_s_bertrand_defined_power_valuation_selected_power_product) = S ((S (S ff_i_bertrand_defined_power_valuation_selected_power_product)) * ff_v_bertrand_defined_power_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_selected_power_product_successor. ff_u_bertrand_defined_power_valuation_selected_power_product = ff_q_bertrand_defined_power_valuation_selected_power_product_successor * S ((S (S ff_i_bertrand_defined_power_valuation_selected_power_product)) * ff_v_bertrand_defined_power_valuation_selected_power_product) + (ff_s_bertrand_defined_power_valuation_selected_power_product))) /\ ff_s_bertrand_defined_power_valuation_selected_power_product = ff_r_bertrand_defined_power_valuation_selected_power_product * ff_p_bertrand_defined_power_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bertrand_defined_power_valuation_selected_divides. n = bpv_result_bertrand_defined_power_valuation_selected * bpv_factor_bertrand_defined_power_valuation_selected_divides)))) /\ forall bpv_candidate_bertrand_defined_power_valuation. (exists bpv_gap_bertrand_defined_power_valuation_candidate_bound. bpv_gap_bertrand_defined_power_valuation_candidate_bound + bpv_candidate_bertrand_defined_power_valuation = n) -> (exists bpv_result_bertrand_defined_power_valuation_candidate. ((exists ff_b_bertrand_defined_power_valuation_candidate_power ff_c_bertrand_defined_power_valuation_candidate_power. ((forall ff_i_bertrand_defined_power_valuation_candidate_power_repeat. (exists ff_lt_bertrand_defined_power_valuation_candidate_power_repeat_bound. ff_lt_bertrand_defined_power_valuation_candidate_power_repeat_bound + S ff_i_bertrand_defined_power_valuation_candidate_power_repeat = bpv_candidate_bertrand_defined_power_valuation) -> (((exists ff_h_bertrand_defined_power_valuation_candidate_power_repeat_decoded. ff_h_bertrand_defined_power_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bertrand_defined_power_valuation_candidate_power_repeat)) * ff_c_bertrand_defined_power_valuation_candidate_power)) /\ exists ff_q_bertrand_defined_power_valuation_candidate_power_repeat_decoded. ff_b_bertrand_defined_power_valuation_candidate_power = ff_q_bertrand_defined_power_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bertrand_defined_power_valuation_candidate_power_repeat)) * ff_c_bertrand_defined_power_valuation_candidate_power) + (p)))) /\ (exists ff_u_bertrand_defined_power_valuation_candidate_power_product ff_v_bertrand_defined_power_valuation_candidate_power_product. ((((exists ff_h_bertrand_defined_power_valuation_candidate_power_product_start. ff_h_bertrand_defined_power_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bertrand_defined_power_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_candidate_power_product_start. ff_u_bertrand_defined_power_valuation_candidate_power_product = ff_q_bertrand_defined_power_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bertrand_defined_power_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bertrand_defined_power_valuation_candidate_power_product_terminal. ff_h_bertrand_defined_power_valuation_candidate_power_product_terminal + S (bpv_result_bertrand_defined_power_valuation_candidate) = S ((S (bpv_candidate_bertrand_defined_power_valuation)) * ff_v_bertrand_defined_power_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_candidate_power_product_terminal. ff_u_bertrand_defined_power_valuation_candidate_power_product = ff_q_bertrand_defined_power_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bertrand_defined_power_valuation)) * ff_v_bertrand_defined_power_valuation_candidate_power_product) + (bpv_result_bertrand_defined_power_valuation_candidate))) /\ forall ff_i_bertrand_defined_power_valuation_candidate_power_product. (exists ff_lt_bertrand_defined_power_valuation_candidate_power_product_bound. ff_lt_bertrand_defined_power_valuation_candidate_power_product_bound + S ff_i_bertrand_defined_power_valuation_candidate_power_product = bpv_candidate_bertrand_defined_power_valuation) -> exists ff_p_bertrand_defined_power_valuation_candidate_power_product ff_r_bertrand_defined_power_valuation_candidate_power_product ff_s_bertrand_defined_power_valuation_candidate_power_product. ((((exists ff_h_bertrand_defined_power_valuation_candidate_power_product_factor. ff_h_bertrand_defined_power_valuation_candidate_power_product_factor + S (ff_p_bertrand_defined_power_valuation_candidate_power_product) = S ((S (ff_i_bertrand_defined_power_valuation_candidate_power_product)) * ff_c_bertrand_defined_power_valuation_candidate_power)) /\ exists ff_q_bertrand_defined_power_valuation_candidate_power_product_factor. ff_b_bertrand_defined_power_valuation_candidate_power = ff_q_bertrand_defined_power_valuation_candidate_power_product_factor * S ((S (ff_i_bertrand_defined_power_valuation_candidate_power_product)) * ff_c_bertrand_defined_power_valuation_candidate_power) + (ff_p_bertrand_defined_power_valuation_candidate_power_product))) /\ ((((exists ff_h_bertrand_defined_power_valuation_candidate_power_product_partial. ff_h_bertrand_defined_power_valuation_candidate_power_product_partial + S (ff_r_bertrand_defined_power_valuation_candidate_power_product) = S ((S (ff_i_bertrand_defined_power_valuation_candidate_power_product)) * ff_v_bertrand_defined_power_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_candidate_power_product_partial. ff_u_bertrand_defined_power_valuation_candidate_power_product = ff_q_bertrand_defined_power_valuation_candidate_power_product_partial * S ((S (ff_i_bertrand_defined_power_valuation_candidate_power_product)) * ff_v_bertrand_defined_power_valuation_candidate_power_product) + (ff_r_bertrand_defined_power_valuation_candidate_power_product))) /\ ((((exists ff_h_bertrand_defined_power_valuation_candidate_power_product_successor. ff_h_bertrand_defined_power_valuation_candidate_power_product_successor + S (ff_s_bertrand_defined_power_valuation_candidate_power_product) = S ((S (S ff_i_bertrand_defined_power_valuation_candidate_power_product)) * ff_v_bertrand_defined_power_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_power_valuation_candidate_power_product_successor. ff_u_bertrand_defined_power_valuation_candidate_power_product = ff_q_bertrand_defined_power_valuation_candidate_power_product_successor * S ((S (S ff_i_bertrand_defined_power_valuation_candidate_power_product)) * ff_v_bertrand_defined_power_valuation_candidate_power_product) + (ff_s_bertrand_defined_power_valuation_candidate_power_product))) /\ ff_s_bertrand_defined_power_valuation_candidate_power_product = ff_r_bertrand_defined_power_valuation_candidate_power_product * ff_p_bertrand_defined_power_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bertrand_defined_power_valuation_candidate_divides. n = bpv_result_bertrand_defined_power_valuation_candidate * bpv_factor_bertrand_defined_power_valuation_candidate_divides))) -> (exists bpv_gap_bertrand_defined_power_valuation_maximal. bpv_gap_bertrand_defined_power_valuation_maximal + bpv_candidate_bertrand_defined_power_valuation = e)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
BT00Q7 power_valuation_exists BT00Q8 power_valuation_functional BT00Q9 power_valuation_power_divides BT00QA power_valuation_dominates BT00QH power_valuation_successor_not_divides BT00QI power_valuation_selected_and_successor_not_divides BT00QP power_valuation_exact_cofactor BT00QQ power_valuation_mul_successor_not_divides BT00QR power_valuation_mul_lower BT00QS power_valuation_mul_upper BT00QT prime_power_valuation_mul BT00RL prime_power_valuation_one_zero BT00RM factorial_valuation_exists BT00RP prime_factorial_valuation_succ BT00SB prime_power_divides_exponent_le_valuation BT00SC power_divides_of_exponent_le_valuation BT00SI valuation_threshold_bit_decides_power_divides BT00SK power_quotient_successor_pointwise_add BT00SV prime_legendre_sum_succ BT00T0 factorial_legendre_successor_agreement BT00T1 prime_factorial_valuation_eq_legendre_sum BT00W0 power_valuation_nonzero_exponent_divides_base BT00XL power_valuation_value_eq_transport BT00XM central_binom_factorial_valuation_balance BT00XN central_binom_legendre_valuation_balance BT00Y4 central_binom_carry_bit_count BT00Y5 central_binom_prime_power_contribution_le_double BT00Y6 central_binom_prime_square_tail_exponent_not_two_le BT00Y7 central_binom_prime_square_tail_valuation_le_one BT00YD central_binom_prime_valuation_zero_of_exact_double_quotients BT00YE central_binom_prime_valuation_zero_two_thirds_range BT00YG central_binom_prime_valuation_zero_above_third_quotient BT00YI central_binom_prime_above_floor_sqrt_valuation_le_one BT00YJ no_bertrand_central_nonzero_valuation_live_ranges BT00YK no_bertrand_central_nonzero_valuation_factor_ranges BT00YL no_bertrand_central_nonzero_contribution_factor_ranges BT00YM no_bertrand_central_prime_contribution_ranges BT00YN prime_contribution_choice_exists BT00YO prime_contribution_choice_functional BT00YP prime_contribution_prefix_extend BT00YQ prime_contribution_prefix_exists BT00YS prime_contribution_product_exists BT00YW prime_contribution_prefix_pairwise_coprime BT00YX prime_contribution_factor_divides 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 BT0108 no_bertrand_central_contribution_choice_ranges BT010K prime_contribution_interval_prefix_extend BT010L prime_contribution_interval_prefix_exists BT010P prime_contribution_interval_prefix_shift BT010Q prime_contribution_prefix_restrict_add BT010R prime_contribution_prefix_interval_split BT010S prime_contribution_product_length_eq_transport BT010V no_bertrand_small_contribution_choice_le_double BT010W no_bertrand_middle_contribution_choice_le_selector BT010X no_bertrand_high_contribution_choice_eq_one 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