PC001C

prime_contribution_prefix_decoded_choice

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every decoded prime contribution has its actual valuation and power witness, or is one at a nonprime index.

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.

Exact expanded first-order arithmetic statement

forall C b c l i a. (forall bpr_prefix_index_pc_contrib_entry_source. (exists bpr_gap_pc_contrib_entry_source_bound. bpr_gap_pc_contrib_entry_source_bound + S (bpr_prefix_index_pc_contrib_entry_source) = l) -> exists bpr_prefix_value_pc_contrib_entry_source. ((((exists bpr_height_pc_contrib_entry_source_decoded. bpr_height_pc_contrib_entry_source_decoded + S (bpr_prefix_value_pc_contrib_entry_source) = S ((S (bpr_prefix_index_pc_contrib_entry_source)) * c)) /\ exists bpr_quotient_pc_contrib_entry_source_decoded. b = bpr_quotient_pc_contrib_entry_source_decoded * S ((S (bpr_prefix_index_pc_contrib_entry_source)) * c) + (bpr_prefix_value_pc_contrib_entry_source))) /\ (((((~(S (bpr_prefix_index_pc_contrib_entry_source) = 1) /\ forall bpr_left_pc_contrib_entry_source_choice_prime bpr_right_pc_contrib_entry_source_choice_prime. S (bpr_prefix_index_pc_contrib_entry_source) = bpr_left_pc_contrib_entry_source_choice_prime * bpr_right_pc_contrib_entry_source_choice_prime -> bpr_left_pc_contrib_entry_source_choice_prime = 1 \/ bpr_right_pc_contrib_entry_source_choice_prime = 1)) /\ exists bpr_choice_exponent_pc_contrib_entry_source_choice. ((((exists bpr_le_gap_pc_contrib_entry_source_choice_valuation_selected_bound. bpr_le_gap_pc_contrib_entry_source_choice_valuation_selected_bound + (bpr_choice_exponent_pc_contrib_entry_source_choice) = (C)) /\ (exists bpr_power_value_pc_contrib_entry_source_choice_valuation_selected. ((exists bpr_power_code_pc_contrib_entry_source_choice_valuation_selected_power bpr_power_scale_pc_contrib_entry_source_choice_valuation_selected_power. ((forall bpr_power_index_pc_contrib_entry_source_choice_valuation_selected_power. (exists bpr_gap_pc_contrib_entry_source_choice_valuation_selected_power_repeat_bound. bpr_gap_pc_contrib_entry_source_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_source_choice_valuation_selected_power) = bpr_choice_exponent_pc_contrib_entry_source_choice) -> (((exists bpr_height_pc_contrib_entry_source_choice_valuation_selected_power_repeat_entry. bpr_height_pc_contrib_entry_source_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_pc_contrib_entry_source)) = S ((S (bpr_power_index_pc_contrib_entry_source_choice_valuation_selected_power)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_selected_power)) /\ exists bpr_quotient_pc_contrib_entry_source_choice_valuation_selected_power_repeat_entry. bpr_power_code_pc_contrib_entry_source_choice_valuation_selected_power = bpr_quotient_pc_contrib_entry_source_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_source_choice_valuation_selected_power)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_selected_power) + (S (bpr_prefix_index_pc_contrib_entry_source))))) /\ (exists ff_u_pc_contrib_entry_source_choice_valuation_selected_power_product ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product. ((((exists ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_start. ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_start. ff_u_pc_contrib_entry_source_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_terminal. ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_terminal + S (bpr_power_value_pc_contrib_entry_source_choice_valuation_selected) = S ((S (bpr_choice_exponent_pc_contrib_entry_source_choice)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_terminal. ff_u_pc_contrib_entry_source_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_pc_contrib_entry_source_choice)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product) + (bpr_power_value_pc_contrib_entry_source_choice_valuation_selected))) /\ forall ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product. (exists ff_lt_pc_contrib_entry_source_choice_valuation_selected_power_product_bound. ff_lt_pc_contrib_entry_source_choice_valuation_selected_power_product_bound + S ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product = bpr_choice_exponent_pc_contrib_entry_source_choice) -> exists ff_p_pc_contrib_entry_source_choice_valuation_selected_power_product ff_r_pc_contrib_entry_source_choice_valuation_selected_power_product ff_s_pc_contrib_entry_source_choice_valuation_selected_power_product. ((((exists ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_factor. ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_factor + S (ff_p_pc_contrib_entry_source_choice_valuation_selected_power_product) = S ((S (ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_selected_power)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_factor. bpr_power_code_pc_contrib_entry_source_choice_valuation_selected_power = ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_factor * S ((S (ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_selected_power) + (ff_p_pc_contrib_entry_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_partial. ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_partial + S (ff_r_pc_contrib_entry_source_choice_valuation_selected_power_product) = S ((S (ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_partial. ff_u_pc_contrib_entry_source_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_partial * S ((S (ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product) + (ff_r_pc_contrib_entry_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_successor. ff_h_pc_contrib_entry_source_choice_valuation_selected_power_product_successor + S (ff_s_pc_contrib_entry_source_choice_valuation_selected_power_product) = S ((S (S ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_successor. ff_u_pc_contrib_entry_source_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_source_choice_valuation_selected_power_product_successor * S ((S (S ff_i_pc_contrib_entry_source_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_selected_power_product) + (ff_s_pc_contrib_entry_source_choice_valuation_selected_power_product))) /\ ff_s_pc_contrib_entry_source_choice_valuation_selected_power_product = ff_r_pc_contrib_entry_source_choice_valuation_selected_power_product * ff_p_pc_contrib_entry_source_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_pc_contrib_entry_source_choice_valuation_selected_divides. C = (bpr_power_value_pc_contrib_entry_source_choice_valuation_selected) * bpr_divides_quotient_pc_contrib_entry_source_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation. (exists bpr_le_gap_pc_contrib_entry_source_choice_valuation_candidate_bound. bpr_le_gap_pc_contrib_entry_source_choice_valuation_candidate_bound + (bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation) = (C)) -> (exists bpr_power_value_pc_contrib_entry_source_choice_valuation_candidate. ((exists bpr_power_code_pc_contrib_entry_source_choice_valuation_candidate_power bpr_power_scale_pc_contrib_entry_source_choice_valuation_candidate_power. ((forall bpr_power_index_pc_contrib_entry_source_choice_valuation_candidate_power. (exists bpr_gap_pc_contrib_entry_source_choice_valuation_candidate_power_repeat_bound. bpr_gap_pc_contrib_entry_source_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_source_choice_valuation_candidate_power) = bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation) -> (((exists bpr_height_pc_contrib_entry_source_choice_valuation_candidate_power_repeat_entry. bpr_height_pc_contrib_entry_source_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_pc_contrib_entry_source)) = S ((S (bpr_power_index_pc_contrib_entry_source_choice_valuation_candidate_power)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_candidate_power)) /\ exists bpr_quotient_pc_contrib_entry_source_choice_valuation_candidate_power_repeat_entry. bpr_power_code_pc_contrib_entry_source_choice_valuation_candidate_power = bpr_quotient_pc_contrib_entry_source_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_source_choice_valuation_candidate_power)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_candidate_power) + (S (bpr_prefix_index_pc_contrib_entry_source))))) /\ (exists ff_u_pc_contrib_entry_source_choice_valuation_candidate_power_product ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product. ((((exists ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_start. ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_start. ff_u_pc_contrib_entry_source_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_terminal. ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_pc_contrib_entry_source_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_terminal. ff_u_pc_contrib_entry_source_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product) + (bpr_power_value_pc_contrib_entry_source_choice_valuation_candidate))) /\ forall ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product. (exists ff_lt_pc_contrib_entry_source_choice_valuation_candidate_power_product_bound. ff_lt_pc_contrib_entry_source_choice_valuation_candidate_power_product_bound + S ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product = bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation) -> exists ff_p_pc_contrib_entry_source_choice_valuation_candidate_power_product ff_r_pc_contrib_entry_source_choice_valuation_candidate_power_product ff_s_pc_contrib_entry_source_choice_valuation_candidate_power_product. ((((exists ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_factor. ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_factor + S (ff_p_pc_contrib_entry_source_choice_valuation_candidate_power_product) = S ((S (ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_candidate_power)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_factor. bpr_power_code_pc_contrib_entry_source_choice_valuation_candidate_power = ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_factor * S ((S (ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product)) * bpr_power_scale_pc_contrib_entry_source_choice_valuation_candidate_power) + (ff_p_pc_contrib_entry_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_partial. ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_partial + S (ff_r_pc_contrib_entry_source_choice_valuation_candidate_power_product) = S ((S (ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_partial. ff_u_pc_contrib_entry_source_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_partial * S ((S (ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product) + (ff_r_pc_contrib_entry_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_successor. ff_h_pc_contrib_entry_source_choice_valuation_candidate_power_product_successor + S (ff_s_pc_contrib_entry_source_choice_valuation_candidate_power_product) = S ((S (S ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_successor. ff_u_pc_contrib_entry_source_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_source_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_pc_contrib_entry_source_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_source_choice_valuation_candidate_power_product) + (ff_s_pc_contrib_entry_source_choice_valuation_candidate_power_product))) /\ ff_s_pc_contrib_entry_source_choice_valuation_candidate_power_product = ff_r_pc_contrib_entry_source_choice_valuation_candidate_power_product * ff_p_pc_contrib_entry_source_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_pc_contrib_entry_source_choice_valuation_candidate_divides. C = (bpr_power_value_pc_contrib_entry_source_choice_valuation_candidate) * bpr_divides_quotient_pc_contrib_entry_source_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_pc_contrib_entry_source_choice_valuation_candidate_below. bpr_le_gap_pc_contrib_entry_source_choice_valuation_candidate_below + (bpr_valuation_candidate_pc_contrib_entry_source_choice_valuation) = (bpr_choice_exponent_pc_contrib_entry_source_choice))) /\ (exists bpr_power_code_pc_contrib_entry_source_choice_power bpr_power_scale_pc_contrib_entry_source_choice_power. ((forall bpr_power_index_pc_contrib_entry_source_choice_power. (exists bpr_gap_pc_contrib_entry_source_choice_power_repeat_bound. bpr_gap_pc_contrib_entry_source_choice_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_source_choice_power) = bpr_choice_exponent_pc_contrib_entry_source_choice) -> (((exists bpr_height_pc_contrib_entry_source_choice_power_repeat_entry. bpr_height_pc_contrib_entry_source_choice_power_repeat_entry + S (S (bpr_prefix_index_pc_contrib_entry_source)) = S ((S (bpr_power_index_pc_contrib_entry_source_choice_power)) * bpr_power_scale_pc_contrib_entry_source_choice_power)) /\ exists bpr_quotient_pc_contrib_entry_source_choice_power_repeat_entry. bpr_power_code_pc_contrib_entry_source_choice_power = bpr_quotient_pc_contrib_entry_source_choice_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_source_choice_power)) * bpr_power_scale_pc_contrib_entry_source_choice_power) + (S (bpr_prefix_index_pc_contrib_entry_source))))) /\ (exists ff_u_pc_contrib_entry_source_choice_power_product ff_v_pc_contrib_entry_source_choice_power_product. ((((exists ff_h_pc_contrib_entry_source_choice_power_product_start. ff_h_pc_contrib_entry_source_choice_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_source_choice_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_power_product_start. ff_u_pc_contrib_entry_source_choice_power_product = ff_q_pc_contrib_entry_source_choice_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_source_choice_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_power_product_terminal. ff_h_pc_contrib_entry_source_choice_power_product_terminal + S (bpr_prefix_value_pc_contrib_entry_source) = S ((S (bpr_choice_exponent_pc_contrib_entry_source_choice)) * ff_v_pc_contrib_entry_source_choice_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_power_product_terminal. ff_u_pc_contrib_entry_source_choice_power_product = ff_q_pc_contrib_entry_source_choice_power_product_terminal * S ((S (bpr_choice_exponent_pc_contrib_entry_source_choice)) * ff_v_pc_contrib_entry_source_choice_power_product) + (bpr_prefix_value_pc_contrib_entry_source))) /\ forall ff_i_pc_contrib_entry_source_choice_power_product. (exists ff_lt_pc_contrib_entry_source_choice_power_product_bound. ff_lt_pc_contrib_entry_source_choice_power_product_bound + S ff_i_pc_contrib_entry_source_choice_power_product = bpr_choice_exponent_pc_contrib_entry_source_choice) -> exists ff_p_pc_contrib_entry_source_choice_power_product ff_r_pc_contrib_entry_source_choice_power_product ff_s_pc_contrib_entry_source_choice_power_product. ((((exists ff_h_pc_contrib_entry_source_choice_power_product_factor. ff_h_pc_contrib_entry_source_choice_power_product_factor + S (ff_p_pc_contrib_entry_source_choice_power_product) = S ((S (ff_i_pc_contrib_entry_source_choice_power_product)) * bpr_power_scale_pc_contrib_entry_source_choice_power)) /\ exists ff_q_pc_contrib_entry_source_choice_power_product_factor. bpr_power_code_pc_contrib_entry_source_choice_power = ff_q_pc_contrib_entry_source_choice_power_product_factor * S ((S (ff_i_pc_contrib_entry_source_choice_power_product)) * bpr_power_scale_pc_contrib_entry_source_choice_power) + (ff_p_pc_contrib_entry_source_choice_power_product))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_power_product_partial. ff_h_pc_contrib_entry_source_choice_power_product_partial + S (ff_r_pc_contrib_entry_source_choice_power_product) = S ((S (ff_i_pc_contrib_entry_source_choice_power_product)) * ff_v_pc_contrib_entry_source_choice_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_power_product_partial. ff_u_pc_contrib_entry_source_choice_power_product = ff_q_pc_contrib_entry_source_choice_power_product_partial * S ((S (ff_i_pc_contrib_entry_source_choice_power_product)) * ff_v_pc_contrib_entry_source_choice_power_product) + (ff_r_pc_contrib_entry_source_choice_power_product))) /\ ((((exists ff_h_pc_contrib_entry_source_choice_power_product_successor. ff_h_pc_contrib_entry_source_choice_power_product_successor + S (ff_s_pc_contrib_entry_source_choice_power_product) = S ((S (S ff_i_pc_contrib_entry_source_choice_power_product)) * ff_v_pc_contrib_entry_source_choice_power_product)) /\ exists ff_q_pc_contrib_entry_source_choice_power_product_successor. ff_u_pc_contrib_entry_source_choice_power_product = ff_q_pc_contrib_entry_source_choice_power_product_successor * S ((S (S ff_i_pc_contrib_entry_source_choice_power_product)) * ff_v_pc_contrib_entry_source_choice_power_product) + (ff_s_pc_contrib_entry_source_choice_power_product))) /\ ff_s_pc_contrib_entry_source_choice_power_product = ff_r_pc_contrib_entry_source_choice_power_product * ff_p_pc_contrib_entry_source_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_pc_contrib_entry_source) = 1) /\ forall bpr_left_pc_contrib_entry_source_choice_prime bpr_right_pc_contrib_entry_source_choice_prime. S (bpr_prefix_index_pc_contrib_entry_source) = bpr_left_pc_contrib_entry_source_choice_prime * bpr_right_pc_contrib_entry_source_choice_prime -> bpr_left_pc_contrib_entry_source_choice_prime = 1 \/ bpr_right_pc_contrib_entry_source_choice_prime = 1)) /\ bpr_prefix_value_pc_contrib_entry_source = 1))))) -> (exists pc_lt_contrib_entry_bound. pc_lt_contrib_entry_bound + S (i) = (l)) -> (((exists fs_h_pc_contrib_entry_given. fs_h_pc_contrib_entry_given + S (a) = S ((S (i)) * c)) /\ exists fs_q_pc_contrib_entry_given. b = fs_q_pc_contrib_entry_given * S ((S (i)) * c) + (a))) -> (((((~(S (i) = 1) /\ forall bpr_left_pc_contrib_entry_result_prime bpr_right_pc_contrib_entry_result_prime. S (i) = bpr_left_pc_contrib_entry_result_prime * bpr_right_pc_contrib_entry_result_prime -> bpr_left_pc_contrib_entry_result_prime = 1 \/ bpr_right_pc_contrib_entry_result_prime = 1)) /\ exists bpr_choice_exponent_pc_contrib_entry_result. ((((exists bpr_le_gap_pc_contrib_entry_result_valuation_selected_bound. bpr_le_gap_pc_contrib_entry_result_valuation_selected_bound + (bpr_choice_exponent_pc_contrib_entry_result) = (C)) /\ (exists bpr_power_value_pc_contrib_entry_result_valuation_selected. ((exists bpr_power_code_pc_contrib_entry_result_valuation_selected_power bpr_power_scale_pc_contrib_entry_result_valuation_selected_power. ((forall bpr_power_index_pc_contrib_entry_result_valuation_selected_power. (exists bpr_gap_pc_contrib_entry_result_valuation_selected_power_repeat_bound. bpr_gap_pc_contrib_entry_result_valuation_selected_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_result_valuation_selected_power) = bpr_choice_exponent_pc_contrib_entry_result) -> (((exists bpr_height_pc_contrib_entry_result_valuation_selected_power_repeat_entry. bpr_height_pc_contrib_entry_result_valuation_selected_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_pc_contrib_entry_result_valuation_selected_power)) * bpr_power_scale_pc_contrib_entry_result_valuation_selected_power)) /\ exists bpr_quotient_pc_contrib_entry_result_valuation_selected_power_repeat_entry. bpr_power_code_pc_contrib_entry_result_valuation_selected_power = bpr_quotient_pc_contrib_entry_result_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_result_valuation_selected_power)) * bpr_power_scale_pc_contrib_entry_result_valuation_selected_power) + (S (i))))) /\ (exists ff_u_pc_contrib_entry_result_valuation_selected_power_product ff_v_pc_contrib_entry_result_valuation_selected_power_product. ((((exists ff_h_pc_contrib_entry_result_valuation_selected_power_product_start. ff_h_pc_contrib_entry_result_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_selected_power_product_start. ff_u_pc_contrib_entry_result_valuation_selected_power_product = ff_q_pc_contrib_entry_result_valuation_selected_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_result_valuation_selected_power_product_terminal. ff_h_pc_contrib_entry_result_valuation_selected_power_product_terminal + S (bpr_power_value_pc_contrib_entry_result_valuation_selected) = S ((S (bpr_choice_exponent_pc_contrib_entry_result)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_selected_power_product_terminal. ff_u_pc_contrib_entry_result_valuation_selected_power_product = ff_q_pc_contrib_entry_result_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_pc_contrib_entry_result)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product) + (bpr_power_value_pc_contrib_entry_result_valuation_selected))) /\ forall ff_i_pc_contrib_entry_result_valuation_selected_power_product. (exists ff_lt_pc_contrib_entry_result_valuation_selected_power_product_bound. ff_lt_pc_contrib_entry_result_valuation_selected_power_product_bound + S ff_i_pc_contrib_entry_result_valuation_selected_power_product = bpr_choice_exponent_pc_contrib_entry_result) -> exists ff_p_pc_contrib_entry_result_valuation_selected_power_product ff_r_pc_contrib_entry_result_valuation_selected_power_product ff_s_pc_contrib_entry_result_valuation_selected_power_product. ((((exists ff_h_pc_contrib_entry_result_valuation_selected_power_product_factor. ff_h_pc_contrib_entry_result_valuation_selected_power_product_factor + S (ff_p_pc_contrib_entry_result_valuation_selected_power_product) = S ((S (ff_i_pc_contrib_entry_result_valuation_selected_power_product)) * bpr_power_scale_pc_contrib_entry_result_valuation_selected_power)) /\ exists ff_q_pc_contrib_entry_result_valuation_selected_power_product_factor. bpr_power_code_pc_contrib_entry_result_valuation_selected_power = ff_q_pc_contrib_entry_result_valuation_selected_power_product_factor * S ((S (ff_i_pc_contrib_entry_result_valuation_selected_power_product)) * bpr_power_scale_pc_contrib_entry_result_valuation_selected_power) + (ff_p_pc_contrib_entry_result_valuation_selected_power_product))) /\ ((((exists ff_h_pc_contrib_entry_result_valuation_selected_power_product_partial. ff_h_pc_contrib_entry_result_valuation_selected_power_product_partial + S (ff_r_pc_contrib_entry_result_valuation_selected_power_product) = S ((S (ff_i_pc_contrib_entry_result_valuation_selected_power_product)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_selected_power_product_partial. ff_u_pc_contrib_entry_result_valuation_selected_power_product = ff_q_pc_contrib_entry_result_valuation_selected_power_product_partial * S ((S (ff_i_pc_contrib_entry_result_valuation_selected_power_product)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product) + (ff_r_pc_contrib_entry_result_valuation_selected_power_product))) /\ ((((exists ff_h_pc_contrib_entry_result_valuation_selected_power_product_successor. ff_h_pc_contrib_entry_result_valuation_selected_power_product_successor + S (ff_s_pc_contrib_entry_result_valuation_selected_power_product) = S ((S (S ff_i_pc_contrib_entry_result_valuation_selected_power_product)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_selected_power_product_successor. ff_u_pc_contrib_entry_result_valuation_selected_power_product = ff_q_pc_contrib_entry_result_valuation_selected_power_product_successor * S ((S (S ff_i_pc_contrib_entry_result_valuation_selected_power_product)) * ff_v_pc_contrib_entry_result_valuation_selected_power_product) + (ff_s_pc_contrib_entry_result_valuation_selected_power_product))) /\ ff_s_pc_contrib_entry_result_valuation_selected_power_product = ff_r_pc_contrib_entry_result_valuation_selected_power_product * ff_p_pc_contrib_entry_result_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_pc_contrib_entry_result_valuation_selected_divides. C = (bpr_power_value_pc_contrib_entry_result_valuation_selected) * bpr_divides_quotient_pc_contrib_entry_result_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_pc_contrib_entry_result_valuation. (exists bpr_le_gap_pc_contrib_entry_result_valuation_candidate_bound. bpr_le_gap_pc_contrib_entry_result_valuation_candidate_bound + (bpr_valuation_candidate_pc_contrib_entry_result_valuation) = (C)) -> (exists bpr_power_value_pc_contrib_entry_result_valuation_candidate. ((exists bpr_power_code_pc_contrib_entry_result_valuation_candidate_power bpr_power_scale_pc_contrib_entry_result_valuation_candidate_power. ((forall bpr_power_index_pc_contrib_entry_result_valuation_candidate_power. (exists bpr_gap_pc_contrib_entry_result_valuation_candidate_power_repeat_bound. bpr_gap_pc_contrib_entry_result_valuation_candidate_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_result_valuation_candidate_power) = bpr_valuation_candidate_pc_contrib_entry_result_valuation) -> (((exists bpr_height_pc_contrib_entry_result_valuation_candidate_power_repeat_entry. bpr_height_pc_contrib_entry_result_valuation_candidate_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_pc_contrib_entry_result_valuation_candidate_power)) * bpr_power_scale_pc_contrib_entry_result_valuation_candidate_power)) /\ exists bpr_quotient_pc_contrib_entry_result_valuation_candidate_power_repeat_entry. bpr_power_code_pc_contrib_entry_result_valuation_candidate_power = bpr_quotient_pc_contrib_entry_result_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_result_valuation_candidate_power)) * bpr_power_scale_pc_contrib_entry_result_valuation_candidate_power) + (S (i))))) /\ (exists ff_u_pc_contrib_entry_result_valuation_candidate_power_product ff_v_pc_contrib_entry_result_valuation_candidate_power_product. ((((exists ff_h_pc_contrib_entry_result_valuation_candidate_power_product_start. ff_h_pc_contrib_entry_result_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_candidate_power_product_start. ff_u_pc_contrib_entry_result_valuation_candidate_power_product = ff_q_pc_contrib_entry_result_valuation_candidate_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_result_valuation_candidate_power_product_terminal. ff_h_pc_contrib_entry_result_valuation_candidate_power_product_terminal + S (bpr_power_value_pc_contrib_entry_result_valuation_candidate) = S ((S (bpr_valuation_candidate_pc_contrib_entry_result_valuation)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_candidate_power_product_terminal. ff_u_pc_contrib_entry_result_valuation_candidate_power_product = ff_q_pc_contrib_entry_result_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_pc_contrib_entry_result_valuation)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product) + (bpr_power_value_pc_contrib_entry_result_valuation_candidate))) /\ forall ff_i_pc_contrib_entry_result_valuation_candidate_power_product. (exists ff_lt_pc_contrib_entry_result_valuation_candidate_power_product_bound. ff_lt_pc_contrib_entry_result_valuation_candidate_power_product_bound + S ff_i_pc_contrib_entry_result_valuation_candidate_power_product = bpr_valuation_candidate_pc_contrib_entry_result_valuation) -> exists ff_p_pc_contrib_entry_result_valuation_candidate_power_product ff_r_pc_contrib_entry_result_valuation_candidate_power_product ff_s_pc_contrib_entry_result_valuation_candidate_power_product. ((((exists ff_h_pc_contrib_entry_result_valuation_candidate_power_product_factor. ff_h_pc_contrib_entry_result_valuation_candidate_power_product_factor + S (ff_p_pc_contrib_entry_result_valuation_candidate_power_product) = S ((S (ff_i_pc_contrib_entry_result_valuation_candidate_power_product)) * bpr_power_scale_pc_contrib_entry_result_valuation_candidate_power)) /\ exists ff_q_pc_contrib_entry_result_valuation_candidate_power_product_factor. bpr_power_code_pc_contrib_entry_result_valuation_candidate_power = ff_q_pc_contrib_entry_result_valuation_candidate_power_product_factor * S ((S (ff_i_pc_contrib_entry_result_valuation_candidate_power_product)) * bpr_power_scale_pc_contrib_entry_result_valuation_candidate_power) + (ff_p_pc_contrib_entry_result_valuation_candidate_power_product))) /\ ((((exists ff_h_pc_contrib_entry_result_valuation_candidate_power_product_partial. ff_h_pc_contrib_entry_result_valuation_candidate_power_product_partial + S (ff_r_pc_contrib_entry_result_valuation_candidate_power_product) = S ((S (ff_i_pc_contrib_entry_result_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_candidate_power_product_partial. ff_u_pc_contrib_entry_result_valuation_candidate_power_product = ff_q_pc_contrib_entry_result_valuation_candidate_power_product_partial * S ((S (ff_i_pc_contrib_entry_result_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product) + (ff_r_pc_contrib_entry_result_valuation_candidate_power_product))) /\ ((((exists ff_h_pc_contrib_entry_result_valuation_candidate_power_product_successor. ff_h_pc_contrib_entry_result_valuation_candidate_power_product_successor + S (ff_s_pc_contrib_entry_result_valuation_candidate_power_product) = S ((S (S ff_i_pc_contrib_entry_result_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_result_valuation_candidate_power_product_successor. ff_u_pc_contrib_entry_result_valuation_candidate_power_product = ff_q_pc_contrib_entry_result_valuation_candidate_power_product_successor * S ((S (S ff_i_pc_contrib_entry_result_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_result_valuation_candidate_power_product) + (ff_s_pc_contrib_entry_result_valuation_candidate_power_product))) /\ ff_s_pc_contrib_entry_result_valuation_candidate_power_product = ff_r_pc_contrib_entry_result_valuation_candidate_power_product * ff_p_pc_contrib_entry_result_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_pc_contrib_entry_result_valuation_candidate_divides. C = (bpr_power_value_pc_contrib_entry_result_valuation_candidate) * bpr_divides_quotient_pc_contrib_entry_result_valuation_candidate_divides))) -> (exists bpr_le_gap_pc_contrib_entry_result_valuation_candidate_below. bpr_le_gap_pc_contrib_entry_result_valuation_candidate_below + (bpr_valuation_candidate_pc_contrib_entry_result_valuation) = (bpr_choice_exponent_pc_contrib_entry_result))) /\ (exists bpr_power_code_pc_contrib_entry_result_power bpr_power_scale_pc_contrib_entry_result_power. ((forall bpr_power_index_pc_contrib_entry_result_power. (exists bpr_gap_pc_contrib_entry_result_power_repeat_bound. bpr_gap_pc_contrib_entry_result_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_result_power) = bpr_choice_exponent_pc_contrib_entry_result) -> (((exists bpr_height_pc_contrib_entry_result_power_repeat_entry. bpr_height_pc_contrib_entry_result_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_pc_contrib_entry_result_power)) * bpr_power_scale_pc_contrib_entry_result_power)) /\ exists bpr_quotient_pc_contrib_entry_result_power_repeat_entry. bpr_power_code_pc_contrib_entry_result_power = bpr_quotient_pc_contrib_entry_result_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_result_power)) * bpr_power_scale_pc_contrib_entry_result_power) + (S (i))))) /\ (exists ff_u_pc_contrib_entry_result_power_product ff_v_pc_contrib_entry_result_power_product. ((((exists ff_h_pc_contrib_entry_result_power_product_start. ff_h_pc_contrib_entry_result_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_result_power_product)) /\ exists ff_q_pc_contrib_entry_result_power_product_start. ff_u_pc_contrib_entry_result_power_product = ff_q_pc_contrib_entry_result_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_result_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_result_power_product_terminal. ff_h_pc_contrib_entry_result_power_product_terminal + S (a) = S ((S (bpr_choice_exponent_pc_contrib_entry_result)) * ff_v_pc_contrib_entry_result_power_product)) /\ exists ff_q_pc_contrib_entry_result_power_product_terminal. ff_u_pc_contrib_entry_result_power_product = ff_q_pc_contrib_entry_result_power_product_terminal * S ((S (bpr_choice_exponent_pc_contrib_entry_result)) * ff_v_pc_contrib_entry_result_power_product) + (a))) /\ forall ff_i_pc_contrib_entry_result_power_product. (exists ff_lt_pc_contrib_entry_result_power_product_bound. ff_lt_pc_contrib_entry_result_power_product_bound + S ff_i_pc_contrib_entry_result_power_product = bpr_choice_exponent_pc_contrib_entry_result) -> exists ff_p_pc_contrib_entry_result_power_product ff_r_pc_contrib_entry_result_power_product ff_s_pc_contrib_entry_result_power_product. ((((exists ff_h_pc_contrib_entry_result_power_product_factor. ff_h_pc_contrib_entry_result_power_product_factor + S (ff_p_pc_contrib_entry_result_power_product) = S ((S (ff_i_pc_contrib_entry_result_power_product)) * bpr_power_scale_pc_contrib_entry_result_power)) /\ exists ff_q_pc_contrib_entry_result_power_product_factor. bpr_power_code_pc_contrib_entry_result_power = ff_q_pc_contrib_entry_result_power_product_factor * S ((S (ff_i_pc_contrib_entry_result_power_product)) * bpr_power_scale_pc_contrib_entry_result_power) + (ff_p_pc_contrib_entry_result_power_product))) /\ ((((exists ff_h_pc_contrib_entry_result_power_product_partial. ff_h_pc_contrib_entry_result_power_product_partial + S (ff_r_pc_contrib_entry_result_power_product) = S ((S (ff_i_pc_contrib_entry_result_power_product)) * ff_v_pc_contrib_entry_result_power_product)) /\ exists ff_q_pc_contrib_entry_result_power_product_partial. ff_u_pc_contrib_entry_result_power_product = ff_q_pc_contrib_entry_result_power_product_partial * S ((S (ff_i_pc_contrib_entry_result_power_product)) * ff_v_pc_contrib_entry_result_power_product) + (ff_r_pc_contrib_entry_result_power_product))) /\ ((((exists ff_h_pc_contrib_entry_result_power_product_successor. ff_h_pc_contrib_entry_result_power_product_successor + S (ff_s_pc_contrib_entry_result_power_product) = S ((S (S ff_i_pc_contrib_entry_result_power_product)) * ff_v_pc_contrib_entry_result_power_product)) /\ exists ff_q_pc_contrib_entry_result_power_product_successor. ff_u_pc_contrib_entry_result_power_product = ff_q_pc_contrib_entry_result_power_product_successor * S ((S (S ff_i_pc_contrib_entry_result_power_product)) * ff_v_pc_contrib_entry_result_power_product) + (ff_s_pc_contrib_entry_result_power_product))) /\ ff_s_pc_contrib_entry_result_power_product = ff_r_pc_contrib_entry_result_power_product * ff_p_pc_contrib_entry_result_power_product)))))))))) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_contrib_entry_result_prime bpr_right_pc_contrib_entry_result_prime. S (i) = bpr_left_pc_contrib_entry_result_prime * bpr_right_pc_contrib_entry_result_prime -> bpr_left_pc_contrib_entry_result_prime = 1 \/ bpr_right_pc_contrib_entry_result_prime = 1)) /\ a = 1)))

Constructive proof overview

Generated structural guide

Every decoded prime contribution has its actual valuation and power witness, or is one at a nonprime index.

The unchanged tactic script uses 1 declared prerequisite and contains 28 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

PC001D central_binom_prime_mask_weighted_upper

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

28 script commands · 6 reading checkpoints · 2 local claims

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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–9

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro C
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro i
  6. L6
    intro a
  7. L7
    intro h
  8. L8
    intro hi
  9. L9
    intro ha
02Establish hpL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply h.

  1. L10
    have hp : ∃ v. BetaAt(b,c,i,v) ∧ (Prime(S i) ∧ (∃ x. Le(x,C) ∧ (∃ y. Pow(S i,x,y) ∧ (∃ z. C = y · z)) ∧ (∀ y. Le(y,C) → (∃ z. Pow(S i,y,z) ∧ (∃ n. C = z · n)) → Le(y,x)) ∧ Pow(S i,x,v)) ∨ ¬Prime(S i) ∧ v = 1)Definitions: LePrimeBetaAtPow
  2. L11
    specialize h i
  3. L12
    apply h
  4. L13
    exact hi
03Separate the logical casesL14–15

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    cases hp
  2. L15
    cases hp_witness
04Establish heqL16–25

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L16
    have heq : x = a
  2. L17
    specialize beta_at_unique b
  3. L18
    specialize beta_at_unique c
  4. L19
    specialize beta_at_unique i
  5. L20
    specialize beta_at_unique x
  6. L21
    specialize beta_at_unique a
  7. L22
    apply beta_at_unique
  8. L23
    exact hp_witness_left
  9. L24
    exact ha
  10. L25
    rewrite heq at hp_witness_right
05Calculate and transport equalitiesL26–27

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L26
    rewrite heq at hp_witness_right
  2. L27
    rewrite heq at hp_witness_right
06Use earlier factsL28–28

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L28
    exact hp_witness_right

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro C
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro i
  6. 0006intro a
  7. 0007intro h
  8. 0008intro hi
  9. 0009intro ha
  10. 0010have hp : exists v. (((exists fs_h_pc_contrib_entry_actual. fs_h_pc_contrib_entry_actual + S (v) = S ((S (i)) * c)) /\ exists fs_q_pc_contrib_entry_actual. b = fs_q_pc_contrib_entry_actual * S ((S (i)) * c) + (v))) /\ (((((~(S (i) = 1) /\ forall bpr_left_pc_contrib_entry_choice_prime bpr_right_pc_contrib_entry_choice_prime. S (i) = bpr_left_pc_contrib_entry_choice_prime * bpr_right_pc_contrib_entry_choice_prime -> bpr_left_pc_contrib_entry_choice_prime = 1 \/ bpr_right_pc_contrib_entry_choice_prime = 1)) /\ exists bpr_choice_exponent_pc_contrib_entry_choice. ((((exists bpr_le_gap_pc_contrib_entry_choice_valuation_selected_bound. bpr_le_gap_pc_contrib_entry_choice_valuation_selected_bound + (bpr_choice_exponent_pc_contrib_entry_choice) = (C)) /\ (exists bpr_power_value_pc_contrib_entry_choice_valuation_selected. ((exists bpr_power_code_pc_contrib_entry_choice_valuation_selected_power bpr_power_scale_pc_contrib_entry_choice_valuation_selected_power. ((forall bpr_power_index_pc_contrib_entry_choice_valuation_selected_power. (exists bpr_gap_pc_contrib_entry_choice_valuation_selected_power_repeat_bound. bpr_gap_pc_contrib_entry_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_choice_valuation_selected_power) = bpr_choice_exponent_pc_contrib_entry_choice) -> (((exists bpr_height_pc_contrib_entry_choice_valuation_selected_power_repeat_entry. bpr_height_pc_contrib_entry_choice_valuation_selected_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_pc_contrib_entry_choice_valuation_selected_power)) * bpr_power_scale_pc_contrib_entry_choice_valuation_selected_power)) /\ exists bpr_quotient_pc_contrib_entry_choice_valuation_selected_power_repeat_entry. bpr_power_code_pc_contrib_entry_choice_valuation_selected_power = bpr_quotient_pc_contrib_entry_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_choice_valuation_selected_power)) * bpr_power_scale_pc_contrib_entry_choice_valuation_selected_power) + (S (i))))) /\ (exists ff_u_pc_contrib_entry_choice_valuation_selected_power_product ff_v_pc_contrib_entry_choice_valuation_selected_power_product. ((((exists ff_h_pc_contrib_entry_choice_valuation_selected_power_product_start. ff_h_pc_contrib_entry_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_selected_power_product_start. ff_u_pc_contrib_entry_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_choice_valuation_selected_power_product_terminal. ff_h_pc_contrib_entry_choice_valuation_selected_power_product_terminal + S (bpr_power_value_pc_contrib_entry_choice_valuation_selected) = S ((S (bpr_choice_exponent_pc_contrib_entry_choice)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_selected_power_product_terminal. ff_u_pc_contrib_entry_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_pc_contrib_entry_choice)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product) + (bpr_power_value_pc_contrib_entry_choice_valuation_selected))) /\ forall ff_i_pc_contrib_entry_choice_valuation_selected_power_product. (exists ff_lt_pc_contrib_entry_choice_valuation_selected_power_product_bound. ff_lt_pc_contrib_entry_choice_valuation_selected_power_product_bound + S ff_i_pc_contrib_entry_choice_valuation_selected_power_product = bpr_choice_exponent_pc_contrib_entry_choice) -> exists ff_p_pc_contrib_entry_choice_valuation_selected_power_product ff_r_pc_contrib_entry_choice_valuation_selected_power_product ff_s_pc_contrib_entry_choice_valuation_selected_power_product. ((((exists ff_h_pc_contrib_entry_choice_valuation_selected_power_product_factor. ff_h_pc_contrib_entry_choice_valuation_selected_power_product_factor + S (ff_p_pc_contrib_entry_choice_valuation_selected_power_product) = S ((S (ff_i_pc_contrib_entry_choice_valuation_selected_power_product)) * bpr_power_scale_pc_contrib_entry_choice_valuation_selected_power)) /\ exists ff_q_pc_contrib_entry_choice_valuation_selected_power_product_factor. bpr_power_code_pc_contrib_entry_choice_valuation_selected_power = ff_q_pc_contrib_entry_choice_valuation_selected_power_product_factor * S ((S (ff_i_pc_contrib_entry_choice_valuation_selected_power_product)) * bpr_power_scale_pc_contrib_entry_choice_valuation_selected_power) + (ff_p_pc_contrib_entry_choice_valuation_selected_power_product))) /\ ((((exists ff_h_pc_contrib_entry_choice_valuation_selected_power_product_partial. ff_h_pc_contrib_entry_choice_valuation_selected_power_product_partial + S (ff_r_pc_contrib_entry_choice_valuation_selected_power_product) = S ((S (ff_i_pc_contrib_entry_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_selected_power_product_partial. ff_u_pc_contrib_entry_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_choice_valuation_selected_power_product_partial * S ((S (ff_i_pc_contrib_entry_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product) + (ff_r_pc_contrib_entry_choice_valuation_selected_power_product))) /\ ((((exists ff_h_pc_contrib_entry_choice_valuation_selected_power_product_successor. ff_h_pc_contrib_entry_choice_valuation_selected_power_product_successor + S (ff_s_pc_contrib_entry_choice_valuation_selected_power_product) = S ((S (S ff_i_pc_contrib_entry_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_selected_power_product_successor. ff_u_pc_contrib_entry_choice_valuation_selected_power_product = ff_q_pc_contrib_entry_choice_valuation_selected_power_product_successor * S ((S (S ff_i_pc_contrib_entry_choice_valuation_selected_power_product)) * ff_v_pc_contrib_entry_choice_valuation_selected_power_product) + (ff_s_pc_contrib_entry_choice_valuation_selected_power_product))) /\ ff_s_pc_contrib_entry_choice_valuation_selected_power_product = ff_r_pc_contrib_entry_choice_valuation_selected_power_product * ff_p_pc_contrib_entry_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_pc_contrib_entry_choice_valuation_selected_divides. C = (bpr_power_value_pc_contrib_entry_choice_valuation_selected) * bpr_divides_quotient_pc_contrib_entry_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_pc_contrib_entry_choice_valuation. (exists bpr_le_gap_pc_contrib_entry_choice_valuation_candidate_bound. bpr_le_gap_pc_contrib_entry_choice_valuation_candidate_bound + (bpr_valuation_candidate_pc_contrib_entry_choice_valuation) = (C)) -> (exists bpr_power_value_pc_contrib_entry_choice_valuation_candidate. ((exists bpr_power_code_pc_contrib_entry_choice_valuation_candidate_power bpr_power_scale_pc_contrib_entry_choice_valuation_candidate_power. ((forall bpr_power_index_pc_contrib_entry_choice_valuation_candidate_power. (exists bpr_gap_pc_contrib_entry_choice_valuation_candidate_power_repeat_bound. bpr_gap_pc_contrib_entry_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_choice_valuation_candidate_power) = bpr_valuation_candidate_pc_contrib_entry_choice_valuation) -> (((exists bpr_height_pc_contrib_entry_choice_valuation_candidate_power_repeat_entry. bpr_height_pc_contrib_entry_choice_valuation_candidate_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_pc_contrib_entry_choice_valuation_candidate_power)) * bpr_power_scale_pc_contrib_entry_choice_valuation_candidate_power)) /\ exists bpr_quotient_pc_contrib_entry_choice_valuation_candidate_power_repeat_entry. bpr_power_code_pc_contrib_entry_choice_valuation_candidate_power = bpr_quotient_pc_contrib_entry_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_choice_valuation_candidate_power)) * bpr_power_scale_pc_contrib_entry_choice_valuation_candidate_power) + (S (i))))) /\ (exists ff_u_pc_contrib_entry_choice_valuation_candidate_power_product ff_v_pc_contrib_entry_choice_valuation_candidate_power_product. ((((exists ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_start. ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_start. ff_u_pc_contrib_entry_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_terminal. ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_pc_contrib_entry_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_pc_contrib_entry_choice_valuation)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_terminal. ff_u_pc_contrib_entry_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_pc_contrib_entry_choice_valuation)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product) + (bpr_power_value_pc_contrib_entry_choice_valuation_candidate))) /\ forall ff_i_pc_contrib_entry_choice_valuation_candidate_power_product. (exists ff_lt_pc_contrib_entry_choice_valuation_candidate_power_product_bound. ff_lt_pc_contrib_entry_choice_valuation_candidate_power_product_bound + S ff_i_pc_contrib_entry_choice_valuation_candidate_power_product = bpr_valuation_candidate_pc_contrib_entry_choice_valuation) -> exists ff_p_pc_contrib_entry_choice_valuation_candidate_power_product ff_r_pc_contrib_entry_choice_valuation_candidate_power_product ff_s_pc_contrib_entry_choice_valuation_candidate_power_product. ((((exists ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_factor. ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_factor + S (ff_p_pc_contrib_entry_choice_valuation_candidate_power_product) = S ((S (ff_i_pc_contrib_entry_choice_valuation_candidate_power_product)) * bpr_power_scale_pc_contrib_entry_choice_valuation_candidate_power)) /\ exists ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_factor. bpr_power_code_pc_contrib_entry_choice_valuation_candidate_power = ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_factor * S ((S (ff_i_pc_contrib_entry_choice_valuation_candidate_power_product)) * bpr_power_scale_pc_contrib_entry_choice_valuation_candidate_power) + (ff_p_pc_contrib_entry_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_partial. ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_partial + S (ff_r_pc_contrib_entry_choice_valuation_candidate_power_product) = S ((S (ff_i_pc_contrib_entry_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_partial. ff_u_pc_contrib_entry_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_partial * S ((S (ff_i_pc_contrib_entry_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product) + (ff_r_pc_contrib_entry_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_successor. ff_h_pc_contrib_entry_choice_valuation_candidate_power_product_successor + S (ff_s_pc_contrib_entry_choice_valuation_candidate_power_product) = S ((S (S ff_i_pc_contrib_entry_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product)) /\ exists ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_successor. ff_u_pc_contrib_entry_choice_valuation_candidate_power_product = ff_q_pc_contrib_entry_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_pc_contrib_entry_choice_valuation_candidate_power_product)) * ff_v_pc_contrib_entry_choice_valuation_candidate_power_product) + (ff_s_pc_contrib_entry_choice_valuation_candidate_power_product))) /\ ff_s_pc_contrib_entry_choice_valuation_candidate_power_product = ff_r_pc_contrib_entry_choice_valuation_candidate_power_product * ff_p_pc_contrib_entry_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_pc_contrib_entry_choice_valuation_candidate_divides. C = (bpr_power_value_pc_contrib_entry_choice_valuation_candidate) * bpr_divides_quotient_pc_contrib_entry_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_pc_contrib_entry_choice_valuation_candidate_below. bpr_le_gap_pc_contrib_entry_choice_valuation_candidate_below + (bpr_valuation_candidate_pc_contrib_entry_choice_valuation) = (bpr_choice_exponent_pc_contrib_entry_choice))) /\ (exists bpr_power_code_pc_contrib_entry_choice_power bpr_power_scale_pc_contrib_entry_choice_power. ((forall bpr_power_index_pc_contrib_entry_choice_power. (exists bpr_gap_pc_contrib_entry_choice_power_repeat_bound. bpr_gap_pc_contrib_entry_choice_power_repeat_bound + S (bpr_power_index_pc_contrib_entry_choice_power) = bpr_choice_exponent_pc_contrib_entry_choice) -> (((exists bpr_height_pc_contrib_entry_choice_power_repeat_entry. bpr_height_pc_contrib_entry_choice_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_pc_contrib_entry_choice_power)) * bpr_power_scale_pc_contrib_entry_choice_power)) /\ exists bpr_quotient_pc_contrib_entry_choice_power_repeat_entry. bpr_power_code_pc_contrib_entry_choice_power = bpr_quotient_pc_contrib_entry_choice_power_repeat_entry * S ((S (bpr_power_index_pc_contrib_entry_choice_power)) * bpr_power_scale_pc_contrib_entry_choice_power) + (S (i))))) /\ (exists ff_u_pc_contrib_entry_choice_power_product ff_v_pc_contrib_entry_choice_power_product. ((((exists ff_h_pc_contrib_entry_choice_power_product_start. ff_h_pc_contrib_entry_choice_power_product_start + S (1) = S ((S (0)) * ff_v_pc_contrib_entry_choice_power_product)) /\ exists ff_q_pc_contrib_entry_choice_power_product_start. ff_u_pc_contrib_entry_choice_power_product = ff_q_pc_contrib_entry_choice_power_product_start * S ((S (0)) * ff_v_pc_contrib_entry_choice_power_product) + (1))) /\ ((((exists ff_h_pc_contrib_entry_choice_power_product_terminal. ff_h_pc_contrib_entry_choice_power_product_terminal + S (v) = S ((S (bpr_choice_exponent_pc_contrib_entry_choice)) * ff_v_pc_contrib_entry_choice_power_product)) /\ exists ff_q_pc_contrib_entry_choice_power_product_terminal. ff_u_pc_contrib_entry_choice_power_product = ff_q_pc_contrib_entry_choice_power_product_terminal * S ((S (bpr_choice_exponent_pc_contrib_entry_choice)) * ff_v_pc_contrib_entry_choice_power_product) + (v))) /\ forall ff_i_pc_contrib_entry_choice_power_product. (exists ff_lt_pc_contrib_entry_choice_power_product_bound. ff_lt_pc_contrib_entry_choice_power_product_bound + S ff_i_pc_contrib_entry_choice_power_product = bpr_choice_exponent_pc_contrib_entry_choice) -> exists ff_p_pc_contrib_entry_choice_power_product ff_r_pc_contrib_entry_choice_power_product ff_s_pc_contrib_entry_choice_power_product. ((((exists ff_h_pc_contrib_entry_choice_power_product_factor. ff_h_pc_contrib_entry_choice_power_product_factor + S (ff_p_pc_contrib_entry_choice_power_product) = S ((S (ff_i_pc_contrib_entry_choice_power_product)) * bpr_power_scale_pc_contrib_entry_choice_power)) /\ exists ff_q_pc_contrib_entry_choice_power_product_factor. bpr_power_code_pc_contrib_entry_choice_power = ff_q_pc_contrib_entry_choice_power_product_factor * S ((S (ff_i_pc_contrib_entry_choice_power_product)) * bpr_power_scale_pc_contrib_entry_choice_power) + (ff_p_pc_contrib_entry_choice_power_product))) /\ ((((exists ff_h_pc_contrib_entry_choice_power_product_partial. ff_h_pc_contrib_entry_choice_power_product_partial + S (ff_r_pc_contrib_entry_choice_power_product) = S ((S (ff_i_pc_contrib_entry_choice_power_product)) * ff_v_pc_contrib_entry_choice_power_product)) /\ exists ff_q_pc_contrib_entry_choice_power_product_partial. ff_u_pc_contrib_entry_choice_power_product = ff_q_pc_contrib_entry_choice_power_product_partial * S ((S (ff_i_pc_contrib_entry_choice_power_product)) * ff_v_pc_contrib_entry_choice_power_product) + (ff_r_pc_contrib_entry_choice_power_product))) /\ ((((exists ff_h_pc_contrib_entry_choice_power_product_successor. ff_h_pc_contrib_entry_choice_power_product_successor + S (ff_s_pc_contrib_entry_choice_power_product) = S ((S (S ff_i_pc_contrib_entry_choice_power_product)) * ff_v_pc_contrib_entry_choice_power_product)) /\ exists ff_q_pc_contrib_entry_choice_power_product_successor. ff_u_pc_contrib_entry_choice_power_product = ff_q_pc_contrib_entry_choice_power_product_successor * S ((S (S ff_i_pc_contrib_entry_choice_power_product)) * ff_v_pc_contrib_entry_choice_power_product) + (ff_s_pc_contrib_entry_choice_power_product))) /\ ff_s_pc_contrib_entry_choice_power_product = ff_r_pc_contrib_entry_choice_power_product * ff_p_pc_contrib_entry_choice_power_product)))))))))) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_contrib_entry_choice_prime bpr_right_pc_contrib_entry_choice_prime. S (i) = bpr_left_pc_contrib_entry_choice_prime * bpr_right_pc_contrib_entry_choice_prime -> bpr_left_pc_contrib_entry_choice_prime = 1 \/ bpr_right_pc_contrib_entry_choice_prime = 1)) /\ v = 1)))
  11. 0011specialize h i
  12. 0012apply h
  13. 0013exact hi
  14. 0014cases hp
  15. 0015cases hp_witness
  16. 0016have heq : x = a
  17. 0017specialize beta_at_unique b
  18. 0018specialize beta_at_unique c
  19. 0019specialize beta_at_unique i
  20. 0020specialize beta_at_unique x
  21. 0021specialize beta_at_unique a
  22. 0022apply beta_at_unique
  23. 0023exact hp_witness_left
  24. 0024exact ha
  25. 0025rewrite heq at hp_witness_right
  26. 0026rewrite heq at hp_witness_right
  27. 0027rewrite heq at hp_witness_right
  28. 0028exact hp_witness_right