BT00YN — prime_contribution_choice

Exact expanded PA statement

forall n i. exists a. (((((~(S (i) = 1) /\ forall bpr_left_bpcce_result_prime bpr_right_bpcce_result_prime. S (i) = bpr_left_bpcce_result_prime * bpr_right_bpcce_result_prime -> bpr_left_bpcce_result_prime = 1 \/ bpr_right_bpcce_result_prime = 1)) /\ exists bpr_choice_exponent_bpcce_result. ((((exists bpr_le_gap_bpcce_result_valuation_selected_bound. bpr_le_gap_bpcce_result_valuation_selected_bound + (bpr_choice_exponent_bpcce_result) = (n)) /\ (exists bpr_power_value_bpcce_result_valuation_selected. ((exists bpr_power_code_bpcce_result_valuation_selected_power bpr_power_scale_bpcce_result_valuation_selected_power. ((forall bpr_power_index_bpcce_result_valuation_selected_power. (exists bpr_gap_bpcce_result_valuation_selected_power_repeat_bound. bpr_gap_bpcce_result_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcce_result_valuation_selected_power) = bpr_choice_exponent_bpcce_result) -> (((exists bpr_height_bpcce_result_valuation_selected_power_repeat_entry. bpr_height_bpcce_result_valuation_selected_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_bpcce_result_valuation_selected_power)) * bpr_power_scale_bpcce_result_valuation_selected_power)) /\ exists bpr_quotient_bpcce_result_valuation_selected_power_repeat_entry. bpr_power_code_bpcce_result_valuation_selected_power = bpr_quotient_bpcce_result_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcce_result_valuation_selected_power)) * bpr_power_scale_bpcce_result_valuation_selected_power) + (S (i))))) /\ (exists ff_u_bpcce_result_valuation_selected_power_product ff_v_bpcce_result_valuation_selected_power_product. ((((exists ff_h_bpcce_result_valuation_selected_power_product_start. ff_h_bpcce_result_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcce_result_valuation_selected_power_product)) /\ exists ff_q_bpcce_result_valuation_selected_power_product_start. ff_u_bpcce_result_valuation_selected_power_product = ff_q_bpcce_result_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcce_result_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcce_result_valuation_selected_power_product_terminal. ff_h_bpcce_result_valuation_selected_power_product_terminal + S (bpr_power_value_bpcce_result_valuation_selected) = S ((S (bpr_choice_exponent_bpcce_result)) * ff_v_bpcce_result_valuation_selected_power_product)) /\ exists ff_q_bpcce_result_valuation_selected_power_product_terminal. ff_u_bpcce_result_valuation_selected_power_product = ff_q_bpcce_result_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcce_result)) * ff_v_bpcce_result_valuation_selected_power_product) + (bpr_power_value_bpcce_result_valuation_selected))) /\ forall ff_i_bpcce_result_valuation_selected_power_product. (exists ff_lt_bpcce_result_valuation_selected_power_product_bound. ff_lt_bpcce_result_valuation_selected_power_product_bound + S ff_i_bpcce_result_valuation_selected_power_product = bpr_choice_exponent_bpcce_result) -> exists ff_p_bpcce_result_valuation_selected_power_product ff_r_bpcce_result_valuation_selected_power_product ff_s_bpcce_result_valuation_selected_power_product. ((((exists ff_h_bpcce_result_valuation_selected_power_product_factor. ff_h_bpcce_result_valuation_selected_power_product_factor + S (ff_p_bpcce_result_valuation_selected_power_product) = S ((S (ff_i_bpcce_result_valuation_selected_power_product)) * bpr_power_scale_bpcce_result_valuation_selected_power)) /\ exists ff_q_bpcce_result_valuation_selected_power_product_factor. bpr_power_code_bpcce_result_valuation_selected_power = ff_q_bpcce_result_valuation_selected_power_product_factor * S ((S (ff_i_bpcce_result_valuation_selected_power_product)) * bpr_power_scale_bpcce_result_valuation_selected_power) + (ff_p_bpcce_result_valuation_selected_power_product))) /\ ((((exists ff_h_bpcce_result_valuation_selected_power_product_partial. ff_h_bpcce_result_valuation_selected_power_product_partial + S (ff_r_bpcce_result_valuation_selected_power_product) = S ((S (ff_i_bpcce_result_valuation_selected_power_product)) * ff_v_bpcce_result_valuation_selected_power_product)) /\ exists ff_q_bpcce_result_valuation_selected_power_product_partial. ff_u_bpcce_result_valuation_selected_power_product = ff_q_bpcce_result_valuation_selected_power_product_partial * S ((S (ff_i_bpcce_result_valuation_selected_power_product)) * ff_v_bpcce_result_valuation_selected_power_product) + (ff_r_bpcce_result_valuation_selected_power_product))) /\ ((((exists ff_h_bpcce_result_valuation_selected_power_product_successor. ff_h_bpcce_result_valuation_selected_power_product_successor + S (ff_s_bpcce_result_valuation_selected_power_product) = S ((S (S ff_i_bpcce_result_valuation_selected_power_product)) * ff_v_bpcce_result_valuation_selected_power_product)) /\ exists ff_q_bpcce_result_valuation_selected_power_product_successor. ff_u_bpcce_result_valuation_selected_power_product = ff_q_bpcce_result_valuation_selected_power_product_successor * S ((S (S ff_i_bpcce_result_valuation_selected_power_product)) * ff_v_bpcce_result_valuation_selected_power_product) + (ff_s_bpcce_result_valuation_selected_power_product))) /\ ff_s_bpcce_result_valuation_selected_power_product = ff_r_bpcce_result_valuation_selected_power_product * ff_p_bpcce_result_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcce_result_valuation_selected_divides. n = (bpr_power_value_bpcce_result_valuation_selected) * bpr_divides_quotient_bpcce_result_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcce_result_valuation. (exists bpr_le_gap_bpcce_result_valuation_candidate_bound. bpr_le_gap_bpcce_result_valuation_candidate_bound + (bpr_valuation_candidate_bpcce_result_valuation) = (n)) -> (exists bpr_power_value_bpcce_result_valuation_candidate. ((exists bpr_power_code_bpcce_result_valuation_candidate_power bpr_power_scale_bpcce_result_valuation_candidate_power. ((forall bpr_power_index_bpcce_result_valuation_candidate_power. (exists bpr_gap_bpcce_result_valuation_candidate_power_repeat_bound. bpr_gap_bpcce_result_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcce_result_valuation_candidate_power) = bpr_valuation_candidate_bpcce_result_valuation) -> (((exists bpr_height_bpcce_result_valuation_candidate_power_repeat_entry. bpr_height_bpcce_result_valuation_candidate_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_bpcce_result_valuation_candidate_power)) * bpr_power_scale_bpcce_result_valuation_candidate_power)) /\ exists bpr_quotient_bpcce_result_valuation_candidate_power_repeat_entry. bpr_power_code_bpcce_result_valuation_candidate_power = bpr_quotient_bpcce_result_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcce_result_valuation_candidate_power)) * bpr_power_scale_bpcce_result_valuation_candidate_power) + (S (i))))) /\ (exists ff_u_bpcce_result_valuation_candidate_power_product ff_v_bpcce_result_valuation_candidate_power_product. ((((exists ff_h_bpcce_result_valuation_candidate_power_product_start. ff_h_bpcce_result_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcce_result_valuation_candidate_power_product)) /\ exists ff_q_bpcce_result_valuation_candidate_power_product_start. ff_u_bpcce_result_valuation_candidate_power_product = ff_q_bpcce_result_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcce_result_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcce_result_valuation_candidate_power_product_terminal. ff_h_bpcce_result_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcce_result_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcce_result_valuation)) * ff_v_bpcce_result_valuation_candidate_power_product)) /\ exists ff_q_bpcce_result_valuation_candidate_power_product_terminal. ff_u_bpcce_result_valuation_candidate_power_product = ff_q_bpcce_result_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcce_result_valuation)) * ff_v_bpcce_result_valuation_candidate_power_product) + (bpr_power_value_bpcce_result_valuation_candidate))) /\ forall ff_i_bpcce_result_valuation_candidate_power_product. (exists ff_lt_bpcce_result_valuation_candidate_power_product_bound. ff_lt_bpcce_result_valuation_candidate_power_product_bound + S ff_i_bpcce_result_valuation_candidate_power_product = bpr_valuation_candidate_bpcce_result_valuation) -> exists ff_p_bpcce_result_valuation_candidate_power_product ff_r_bpcce_result_valuation_candidate_power_product ff_s_bpcce_result_valuation_candidate_power_product. ((((exists ff_h_bpcce_result_valuation_candidate_power_product_factor. ff_h_bpcce_result_valuation_candidate_power_product_factor + S (ff_p_bpcce_result_valuation_candidate_power_product) = S ((S (ff_i_bpcce_result_valuation_candidate_power_product)) * bpr_power_scale_bpcce_result_valuation_candidate_power)) /\ exists ff_q_bpcce_result_valuation_candidate_power_product_factor. bpr_power_code_bpcce_result_valuation_candidate_power = ff_q_bpcce_result_valuation_candidate_power_product_factor * S ((S (ff_i_bpcce_result_valuation_candidate_power_product)) * bpr_power_scale_bpcce_result_valuation_candidate_power) + (ff_p_bpcce_result_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcce_result_valuation_candidate_power_product_partial. ff_h_bpcce_result_valuation_candidate_power_product_partial + S (ff_r_bpcce_result_valuation_candidate_power_product) = S ((S (ff_i_bpcce_result_valuation_candidate_power_product)) * ff_v_bpcce_result_valuation_candidate_power_product)) /\ exists ff_q_bpcce_result_valuation_candidate_power_product_partial. ff_u_bpcce_result_valuation_candidate_power_product = ff_q_bpcce_result_valuation_candidate_power_product_partial * S ((S (ff_i_bpcce_result_valuation_candidate_power_product)) * ff_v_bpcce_result_valuation_candidate_power_product) + (ff_r_bpcce_result_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcce_result_valuation_candidate_power_product_successor. ff_h_bpcce_result_valuation_candidate_power_product_successor + S (ff_s_bpcce_result_valuation_candidate_power_product) = S ((S (S ff_i_bpcce_result_valuation_candidate_power_product)) * ff_v_bpcce_result_valuation_candidate_power_product)) /\ exists ff_q_bpcce_result_valuation_candidate_power_product_successor. ff_u_bpcce_result_valuation_candidate_power_product = ff_q_bpcce_result_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcce_result_valuation_candidate_power_product)) * ff_v_bpcce_result_valuation_candidate_power_product) + (ff_s_bpcce_result_valuation_candidate_power_product))) /\ ff_s_bpcce_result_valuation_candidate_power_product = ff_r_bpcce_result_valuation_candidate_power_product * ff_p_bpcce_result_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcce_result_valuation_candidate_divides. n = (bpr_power_value_bpcce_result_valuation_candidate) * bpr_divides_quotient_bpcce_result_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcce_result_valuation_candidate_below. bpr_le_gap_bpcce_result_valuation_candidate_below + (bpr_valuation_candidate_bpcce_result_valuation) = (bpr_choice_exponent_bpcce_result))) /\ (exists bpr_power_code_bpcce_result_power bpr_power_scale_bpcce_result_power. ((forall bpr_power_index_bpcce_result_power. (exists bpr_gap_bpcce_result_power_repeat_bound. bpr_gap_bpcce_result_power_repeat_bound + S (bpr_power_index_bpcce_result_power) = bpr_choice_exponent_bpcce_result) -> (((exists bpr_height_bpcce_result_power_repeat_entry. bpr_height_bpcce_result_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_bpcce_result_power)) * bpr_power_scale_bpcce_result_power)) /\ exists bpr_quotient_bpcce_result_power_repeat_entry. bpr_power_code_bpcce_result_power = bpr_quotient_bpcce_result_power_repeat_entry * S ((S (bpr_power_index_bpcce_result_power)) * bpr_power_scale_bpcce_result_power) + (S (i))))) /\ (exists ff_u_bpcce_result_power_product ff_v_bpcce_result_power_product. ((((exists ff_h_bpcce_result_power_product_start. ff_h_bpcce_result_power_product_start + S (1) = S ((S (0)) * ff_v_bpcce_result_power_product)) /\ exists ff_q_bpcce_result_power_product_start. ff_u_bpcce_result_power_product = ff_q_bpcce_result_power_product_start * S ((S (0)) * ff_v_bpcce_result_power_product) + (1))) /\ ((((exists ff_h_bpcce_result_power_product_terminal. ff_h_bpcce_result_power_product_terminal + S (a) = S ((S (bpr_choice_exponent_bpcce_result)) * ff_v_bpcce_result_power_product)) /\ exists ff_q_bpcce_result_power_product_terminal. ff_u_bpcce_result_power_product = ff_q_bpcce_result_power_product_terminal * S ((S (bpr_choice_exponent_bpcce_result)) * ff_v_bpcce_result_power_product) + (a))) /\ forall ff_i_bpcce_result_power_product. (exists ff_lt_bpcce_result_power_product_bound. ff_lt_bpcce_result_power_product_bound + S ff_i_bpcce_result_power_product = bpr_choice_exponent_bpcce_result) -> exists ff_p_bpcce_result_power_product ff_r_bpcce_result_power_product ff_s_bpcce_result_power_product. ((((exists ff_h_bpcce_result_power_product_factor. ff_h_bpcce_result_power_product_factor + S (ff_p_bpcce_result_power_product) = S ((S (ff_i_bpcce_result_power_product)) * bpr_power_scale_bpcce_result_power)) /\ exists ff_q_bpcce_result_power_product_factor. bpr_power_code_bpcce_result_power = ff_q_bpcce_result_power_product_factor * S ((S (ff_i_bpcce_result_power_product)) * bpr_power_scale_bpcce_result_power) + (ff_p_bpcce_result_power_product))) /\ ((((exists ff_h_bpcce_result_power_product_partial. ff_h_bpcce_result_power_product_partial + S (ff_r_bpcce_result_power_product) = S ((S (ff_i_bpcce_result_power_product)) * ff_v_bpcce_result_power_product)) /\ exists ff_q_bpcce_result_power_product_partial. ff_u_bpcce_result_power_product = ff_q_bpcce_result_power_product_partial * S ((S (ff_i_bpcce_result_power_product)) * ff_v_bpcce_result_power_product) + (ff_r_bpcce_result_power_product))) /\ ((((exists ff_h_bpcce_result_power_product_successor. ff_h_bpcce_result_power_product_successor + S (ff_s_bpcce_result_power_product) = S ((S (S ff_i_bpcce_result_power_product)) * ff_v_bpcce_result_power_product)) /\ exists ff_q_bpcce_result_power_product_successor. ff_u_bpcce_result_power_product = ff_q_bpcce_result_power_product_successor * S ((S (S ff_i_bpcce_result_power_product)) * ff_v_bpcce_result_power_product) + (ff_s_bpcce_result_power_product))) /\ ff_s_bpcce_result_power_product = ff_r_bpcce_result_power_product * ff_p_bpcce_result_power_product)))))))))) \/ (~((~(S (i) = 1) /\ forall bpr_left_bpcce_result_prime bpr_right_bpcce_result_prime. S (i) = bpr_left_bpcce_result_prime * bpr_right_bpcce_result_prime -> bpr_left_bpcce_result_prime = 1 \/ bpr_right_bpcce_result_prime = 1)) /\ a = 1)))

Structural proof guide

Every index has its complete prime-power contribution or one.

Direct prerequisites: prime_decidable, power_valuation_exists, pow_exists. The authored body proceeds by case analysis (3), intermediate claims (2).

Proof neighborhood

Direct dependencies

BT003H prime_decidable BT00Q7 power_valuation_exists BT0080 pow_exists

Direct dependents

BT00YP prime_contribution_prefix_extend BT010K prime_contribution_interval_prefix_extend

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

0001intro n
0002intro i
0003specialize prime_decidable (S i)
0004cases prime_decidable
0005have hvaluation : exists e. (((exists bpr_le_gap_bpcce_valuation_selected_bound. bpr_le_gap_bpcce_valuation_selected_bound + (e) = (n)) /\ (exists bpr_power_value_bpcce_valuation_selected. ((exists bpr_power_code_bpcce_valuation_selected_power bpr_power_scale_bpcce_valuation_selected_power. ((forall bpr_power_index_bpcce_valuation_selected_power. (exists bpr_gap_bpcce_valuation_selected_power_repeat_bound. bpr_gap_bpcce_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcce_valuation_selected_power) = e) -> (((exists bpr_height_bpcce_valuation_selected_power_repeat_entry. bpr_height_bpcce_valuation_selected_power_repeat_entry + S (S i) = S ((S (bpr_power_index_bpcce_valuation_selected_power)) * bpr_power_scale_bpcce_valuation_selected_power)) /\ exists bpr_quotient_bpcce_valuation_selected_power_repeat_entry. bpr_power_code_bpcce_valuation_selected_power = bpr_quotient_bpcce_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcce_valuation_selected_power)) * bpr_power_scale_bpcce_valuation_selected_power) + (S i)))) /\ (exists ff_u_bpcce_valuation_selected_power_product ff_v_bpcce_valuation_selected_power_product. ((((exists ff_h_bpcce_valuation_selected_power_product_start. ff_h_bpcce_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcce_valuation_selected_power_product)) /\ exists ff_q_bpcce_valuation_selected_power_product_start. ff_u_bpcce_valuation_selected_power_product = ff_q_bpcce_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcce_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcce_valuation_selected_power_product_terminal. ff_h_bpcce_valuation_selected_power_product_terminal + S (bpr_power_value_bpcce_valuation_selected) = S ((S (e)) * ff_v_bpcce_valuation_selected_power_product)) /\ exists ff_q_bpcce_valuation_selected_power_product_terminal. ff_u_bpcce_valuation_selected_power_product = ff_q_bpcce_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bpcce_valuation_selected_power_product) + (bpr_power_value_bpcce_valuation_selected))) /\ forall ff_i_bpcce_valuation_selected_power_product. (exists ff_lt_bpcce_valuation_selected_power_product_bound. ff_lt_bpcce_valuation_selected_power_product_bound + S ff_i_bpcce_valuation_selected_power_product = e) -> exists ff_p_bpcce_valuation_selected_power_product ff_r_bpcce_valuation_selected_power_product ff_s_bpcce_valuation_selected_power_product. ((((exists ff_h_bpcce_valuation_selected_power_product_factor. ff_h_bpcce_valuation_selected_power_product_factor + S (ff_p_bpcce_valuation_selected_power_product) = S ((S (ff_i_bpcce_valuation_selected_power_product)) * bpr_power_scale_bpcce_valuation_selected_power)) /\ exists ff_q_bpcce_valuation_selected_power_product_factor. bpr_power_code_bpcce_valuation_selected_power = ff_q_bpcce_valuation_selected_power_product_factor * S ((S (ff_i_bpcce_valuation_selected_power_product)) * bpr_power_scale_bpcce_valuation_selected_power) + (ff_p_bpcce_valuation_selected_power_product))) /\ ((((exists ff_h_bpcce_valuation_selected_power_product_partial. ff_h_bpcce_valuation_selected_power_product_partial + S (ff_r_bpcce_valuation_selected_power_product) = S ((S (ff_i_bpcce_valuation_selected_power_product)) * ff_v_bpcce_valuation_selected_power_product)) /\ exists ff_q_bpcce_valuation_selected_power_product_partial. ff_u_bpcce_valuation_selected_power_product = ff_q_bpcce_valuation_selected_power_product_partial * S ((S (ff_i_bpcce_valuation_selected_power_product)) * ff_v_bpcce_valuation_selected_power_product) + (ff_r_bpcce_valuation_selected_power_product))) /\ ((((exists ff_h_bpcce_valuation_selected_power_product_successor. ff_h_bpcce_valuation_selected_power_product_successor + S (ff_s_bpcce_valuation_selected_power_product) = S ((S (S ff_i_bpcce_valuation_selected_power_product)) * ff_v_bpcce_valuation_selected_power_product)) /\ exists ff_q_bpcce_valuation_selected_power_product_successor. ff_u_bpcce_valuation_selected_power_product = ff_q_bpcce_valuation_selected_power_product_successor * S ((S (S ff_i_bpcce_valuation_selected_power_product)) * ff_v_bpcce_valuation_selected_power_product) + (ff_s_bpcce_valuation_selected_power_product))) /\ ff_s_bpcce_valuation_selected_power_product = ff_r_bpcce_valuation_selected_power_product * ff_p_bpcce_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcce_valuation_selected_divides. n = (bpr_power_value_bpcce_valuation_selected) * bpr_divides_quotient_bpcce_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcce_valuation. (exists bpr_le_gap_bpcce_valuation_candidate_bound. bpr_le_gap_bpcce_valuation_candidate_bound + (bpr_valuation_candidate_bpcce_valuation) = (n)) -> (exists bpr_power_value_bpcce_valuation_candidate. ((exists bpr_power_code_bpcce_valuation_candidate_power bpr_power_scale_bpcce_valuation_candidate_power. ((forall bpr_power_index_bpcce_valuation_candidate_power. (exists bpr_gap_bpcce_valuation_candidate_power_repeat_bound. bpr_gap_bpcce_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcce_valuation_candidate_power) = bpr_valuation_candidate_bpcce_valuation) -> (((exists bpr_height_bpcce_valuation_candidate_power_repeat_entry. bpr_height_bpcce_valuation_candidate_power_repeat_entry + S (S i) = S ((S (bpr_power_index_bpcce_valuation_candidate_power)) * bpr_power_scale_bpcce_valuation_candidate_power)) /\ exists bpr_quotient_bpcce_valuation_candidate_power_repeat_entry. bpr_power_code_bpcce_valuation_candidate_power = bpr_quotient_bpcce_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcce_valuation_candidate_power)) * bpr_power_scale_bpcce_valuation_candidate_power) + (S i)))) /\ (exists ff_u_bpcce_valuation_candidate_power_product ff_v_bpcce_valuation_candidate_power_product. ((((exists ff_h_bpcce_valuation_candidate_power_product_start. ff_h_bpcce_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcce_valuation_candidate_power_product)) /\ exists ff_q_bpcce_valuation_candidate_power_product_start. ff_u_bpcce_valuation_candidate_power_product = ff_q_bpcce_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcce_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcce_valuation_candidate_power_product_terminal. ff_h_bpcce_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcce_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcce_valuation)) * ff_v_bpcce_valuation_candidate_power_product)) /\ exists ff_q_bpcce_valuation_candidate_power_product_terminal. ff_u_bpcce_valuation_candidate_power_product = ff_q_bpcce_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcce_valuation)) * ff_v_bpcce_valuation_candidate_power_product) + (bpr_power_value_bpcce_valuation_candidate))) /\ forall ff_i_bpcce_valuation_candidate_power_product. (exists ff_lt_bpcce_valuation_candidate_power_product_bound. ff_lt_bpcce_valuation_candidate_power_product_bound + S ff_i_bpcce_valuation_candidate_power_product = bpr_valuation_candidate_bpcce_valuation) -> exists ff_p_bpcce_valuation_candidate_power_product ff_r_bpcce_valuation_candidate_power_product ff_s_bpcce_valuation_candidate_power_product. ((((exists ff_h_bpcce_valuation_candidate_power_product_factor. ff_h_bpcce_valuation_candidate_power_product_factor + S (ff_p_bpcce_valuation_candidate_power_product) = S ((S (ff_i_bpcce_valuation_candidate_power_product)) * bpr_power_scale_bpcce_valuation_candidate_power)) /\ exists ff_q_bpcce_valuation_candidate_power_product_factor. bpr_power_code_bpcce_valuation_candidate_power = ff_q_bpcce_valuation_candidate_power_product_factor * S ((S (ff_i_bpcce_valuation_candidate_power_product)) * bpr_power_scale_bpcce_valuation_candidate_power) + (ff_p_bpcce_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcce_valuation_candidate_power_product_partial. ff_h_bpcce_valuation_candidate_power_product_partial + S (ff_r_bpcce_valuation_candidate_power_product) = S ((S (ff_i_bpcce_valuation_candidate_power_product)) * ff_v_bpcce_valuation_candidate_power_product)) /\ exists ff_q_bpcce_valuation_candidate_power_product_partial. ff_u_bpcce_valuation_candidate_power_product = ff_q_bpcce_valuation_candidate_power_product_partial * S ((S (ff_i_bpcce_valuation_candidate_power_product)) * ff_v_bpcce_valuation_candidate_power_product) + (ff_r_bpcce_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcce_valuation_candidate_power_product_successor. ff_h_bpcce_valuation_candidate_power_product_successor + S (ff_s_bpcce_valuation_candidate_power_product) = S ((S (S ff_i_bpcce_valuation_candidate_power_product)) * ff_v_bpcce_valuation_candidate_power_product)) /\ exists ff_q_bpcce_valuation_candidate_power_product_successor. ff_u_bpcce_valuation_candidate_power_product = ff_q_bpcce_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcce_valuation_candidate_power_product)) * ff_v_bpcce_valuation_candidate_power_product) + (ff_s_bpcce_valuation_candidate_power_product))) /\ ff_s_bpcce_valuation_candidate_power_product = ff_r_bpcce_valuation_candidate_power_product * ff_p_bpcce_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcce_valuation_candidate_divides. n = (bpr_power_value_bpcce_valuation_candidate) * bpr_divides_quotient_bpcce_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcce_valuation_candidate_below. bpr_le_gap_bpcce_valuation_candidate_below + (bpr_valuation_candidate_bpcce_valuation) = (e)))
0006specialize power_valuation_exists (S i)
0007specialize power_valuation_exists n
0008exact power_valuation_exists
0009cases hvaluation
0010have hpower : exists a. (exists bpr_power_code_bpcce_power bpr_power_scale_bpcce_power. ((forall bpr_power_index_bpcce_power. (exists bpr_gap_bpcce_power_repeat_bound. bpr_gap_bpcce_power_repeat_bound + S (bpr_power_index_bpcce_power) = x) -> (((exists bpr_height_bpcce_power_repeat_entry. bpr_height_bpcce_power_repeat_entry + S (S i) = S ((S (bpr_power_index_bpcce_power)) * bpr_power_scale_bpcce_power)) /\ exists bpr_quotient_bpcce_power_repeat_entry. bpr_power_code_bpcce_power = bpr_quotient_bpcce_power_repeat_entry * S ((S (bpr_power_index_bpcce_power)) * bpr_power_scale_bpcce_power) + (S i)))) /\ (exists ff_u_bpcce_power_product ff_v_bpcce_power_product. ((((exists ff_h_bpcce_power_product_start. ff_h_bpcce_power_product_start + S (1) = S ((S (0)) * ff_v_bpcce_power_product)) /\ exists ff_q_bpcce_power_product_start. ff_u_bpcce_power_product = ff_q_bpcce_power_product_start * S ((S (0)) * ff_v_bpcce_power_product) + (1))) /\ ((((exists ff_h_bpcce_power_product_terminal. ff_h_bpcce_power_product_terminal + S (a) = S ((S (x)) * ff_v_bpcce_power_product)) /\ exists ff_q_bpcce_power_product_terminal. ff_u_bpcce_power_product = ff_q_bpcce_power_product_terminal * S ((S (x)) * ff_v_bpcce_power_product) + (a))) /\ forall ff_i_bpcce_power_product. (exists ff_lt_bpcce_power_product_bound. ff_lt_bpcce_power_product_bound + S ff_i_bpcce_power_product = x) -> exists ff_p_bpcce_power_product ff_r_bpcce_power_product ff_s_bpcce_power_product. ((((exists ff_h_bpcce_power_product_factor. ff_h_bpcce_power_product_factor + S (ff_p_bpcce_power_product) = S ((S (ff_i_bpcce_power_product)) * bpr_power_scale_bpcce_power)) /\ exists ff_q_bpcce_power_product_factor. bpr_power_code_bpcce_power = ff_q_bpcce_power_product_factor * S ((S (ff_i_bpcce_power_product)) * bpr_power_scale_bpcce_power) + (ff_p_bpcce_power_product))) /\ ((((exists ff_h_bpcce_power_product_partial. ff_h_bpcce_power_product_partial + S (ff_r_bpcce_power_product) = S ((S (ff_i_bpcce_power_product)) * ff_v_bpcce_power_product)) /\ exists ff_q_bpcce_power_product_partial. ff_u_bpcce_power_product = ff_q_bpcce_power_product_partial * S ((S (ff_i_bpcce_power_product)) * ff_v_bpcce_power_product) + (ff_r_bpcce_power_product))) /\ ((((exists ff_h_bpcce_power_product_successor. ff_h_bpcce_power_product_successor + S (ff_s_bpcce_power_product) = S ((S (S ff_i_bpcce_power_product)) * ff_v_bpcce_power_product)) /\ exists ff_q_bpcce_power_product_successor. ff_u_bpcce_power_product = ff_q_bpcce_power_product_successor * S ((S (S ff_i_bpcce_power_product)) * ff_v_bpcce_power_product) + (ff_s_bpcce_power_product))) /\ ff_s_bpcce_power_product = ff_r_bpcce_power_product * ff_p_bpcce_power_product))))))))
0011specialize pow_exists (S i)
0012specialize pow_exists x
0013exact pow_exists
0014cases hpower
0015exists x1
0016left
0017split
0018exact prime_decidable_left
0019exists x
0020split
0021exact hvaluation_witness
0022exact hpower_witness
0023exists 1
0024right
0025split
0026exact prime_decidable_right
0027refl