BT00YQ

prime_contribution_prefix_exists

Alpha body-checked · checked-use disabled

Every number and finite length has a contribution prefix.

Exact expanded PA statement

forall n m. exists b c. (forall bpr_prefix_index_bpcpx_result. (exists bpr_gap_bpcpx_result_bound. bpr_gap_bpcpx_result_bound + S (bpr_prefix_index_bpcpx_result) = m) -> exists bpr_prefix_value_bpcpx_result. ((((exists bpr_height_bpcpx_result_decoded. bpr_height_bpcpx_result_decoded + S (bpr_prefix_value_bpcpx_result) = S ((S (bpr_prefix_index_bpcpx_result)) * c)) /\ exists bpr_quotient_bpcpx_result_decoded. b = bpr_quotient_bpcpx_result_decoded * S ((S (bpr_prefix_index_bpcpx_result)) * c) + (bpr_prefix_value_bpcpx_result))) /\ (((((~(S (bpr_prefix_index_bpcpx_result) = 1) /\ forall bpr_left_bpcpx_result_choice_prime bpr_right_bpcpx_result_choice_prime. S (bpr_prefix_index_bpcpx_result) = bpr_left_bpcpx_result_choice_prime * bpr_right_bpcpx_result_choice_prime -> bpr_left_bpcpx_result_choice_prime = 1 \/ bpr_right_bpcpx_result_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpx_result_choice. ((((exists bpr_le_gap_bpcpx_result_choice_valuation_selected_bound. bpr_le_gap_bpcpx_result_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpx_result_choice) = (n)) /\ (exists bpr_power_value_bpcpx_result_choice_valuation_selected. ((exists bpr_power_code_bpcpx_result_choice_valuation_selected_power bpr_power_scale_bpcpx_result_choice_valuation_selected_power. ((forall bpr_power_index_bpcpx_result_choice_valuation_selected_power. (exists bpr_gap_bpcpx_result_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpx_result_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpx_result_choice_valuation_selected_power) = bpr_choice_exponent_bpcpx_result_choice) -> (((exists bpr_height_bpcpx_result_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpx_result_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_result)) = S ((S (bpr_power_index_bpcpx_result_choice_valuation_selected_power)) * bpr_power_scale_bpcpx_result_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpx_result_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpx_result_choice_valuation_selected_power = bpr_quotient_bpcpx_result_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpx_result_choice_valuation_selected_power)) * bpr_power_scale_bpcpx_result_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpx_result))))) /\ (exists ff_u_bpcpx_result_choice_valuation_selected_power_product ff_v_bpcpx_result_choice_valuation_selected_power_product. ((((exists ff_h_bpcpx_result_choice_valuation_selected_power_product_start. ff_h_bpcpx_result_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_result_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_selected_power_product_start. ff_u_bpcpx_result_choice_valuation_selected_power_product = ff_q_bpcpx_result_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpx_result_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpx_result_choice_valuation_selected_power_product_terminal. ff_h_bpcpx_result_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpx_result_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpx_result_choice)) * ff_v_bpcpx_result_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_selected_power_product_terminal. ff_u_bpcpx_result_choice_valuation_selected_power_product = ff_q_bpcpx_result_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpx_result_choice)) * ff_v_bpcpx_result_choice_valuation_selected_power_product) + (bpr_power_value_bpcpx_result_choice_valuation_selected))) /\ forall ff_i_bpcpx_result_choice_valuation_selected_power_product. (exists ff_lt_bpcpx_result_choice_valuation_selected_power_product_bound. ff_lt_bpcpx_result_choice_valuation_selected_power_product_bound + S ff_i_bpcpx_result_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpx_result_choice) -> exists ff_p_bpcpx_result_choice_valuation_selected_power_product ff_r_bpcpx_result_choice_valuation_selected_power_product ff_s_bpcpx_result_choice_valuation_selected_power_product. ((((exists ff_h_bpcpx_result_choice_valuation_selected_power_product_factor. ff_h_bpcpx_result_choice_valuation_selected_power_product_factor + S (ff_p_bpcpx_result_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpx_result_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpx_result_choice_valuation_selected_power)) /\ exists ff_q_bpcpx_result_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpx_result_choice_valuation_selected_power = ff_q_bpcpx_result_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpx_result_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpx_result_choice_valuation_selected_power) + (ff_p_bpcpx_result_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpx_result_choice_valuation_selected_power_product_partial. ff_h_bpcpx_result_choice_valuation_selected_power_product_partial + S (ff_r_bpcpx_result_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpx_result_choice_valuation_selected_power_product)) * ff_v_bpcpx_result_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_selected_power_product_partial. ff_u_bpcpx_result_choice_valuation_selected_power_product = ff_q_bpcpx_result_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpx_result_choice_valuation_selected_power_product)) * ff_v_bpcpx_result_choice_valuation_selected_power_product) + (ff_r_bpcpx_result_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpx_result_choice_valuation_selected_power_product_successor. ff_h_bpcpx_result_choice_valuation_selected_power_product_successor + S (ff_s_bpcpx_result_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpx_result_choice_valuation_selected_power_product)) * ff_v_bpcpx_result_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_selected_power_product_successor. ff_u_bpcpx_result_choice_valuation_selected_power_product = ff_q_bpcpx_result_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpx_result_choice_valuation_selected_power_product)) * ff_v_bpcpx_result_choice_valuation_selected_power_product) + (ff_s_bpcpx_result_choice_valuation_selected_power_product))) /\ ff_s_bpcpx_result_choice_valuation_selected_power_product = ff_r_bpcpx_result_choice_valuation_selected_power_product * ff_p_bpcpx_result_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpx_result_choice_valuation_selected_divides. n = (bpr_power_value_bpcpx_result_choice_valuation_selected) * bpr_divides_quotient_bpcpx_result_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpx_result_choice_valuation. (exists bpr_le_gap_bpcpx_result_choice_valuation_candidate_bound. bpr_le_gap_bpcpx_result_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpx_result_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpx_result_choice_valuation_candidate. ((exists bpr_power_code_bpcpx_result_choice_valuation_candidate_power bpr_power_scale_bpcpx_result_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpx_result_choice_valuation_candidate_power. (exists bpr_gap_bpcpx_result_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpx_result_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpx_result_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpx_result_choice_valuation) -> (((exists bpr_height_bpcpx_result_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpx_result_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_result)) = S ((S (bpr_power_index_bpcpx_result_choice_valuation_candidate_power)) * bpr_power_scale_bpcpx_result_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpx_result_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpx_result_choice_valuation_candidate_power = bpr_quotient_bpcpx_result_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpx_result_choice_valuation_candidate_power)) * bpr_power_scale_bpcpx_result_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpx_result))))) /\ (exists ff_u_bpcpx_result_choice_valuation_candidate_power_product ff_v_bpcpx_result_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpx_result_choice_valuation_candidate_power_product_start. ff_h_bpcpx_result_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_candidate_power_product_start. ff_u_bpcpx_result_choice_valuation_candidate_power_product = ff_q_bpcpx_result_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpx_result_choice_valuation_candidate_power_product_terminal. ff_h_bpcpx_result_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpx_result_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpx_result_choice_valuation)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_candidate_power_product_terminal. ff_u_bpcpx_result_choice_valuation_candidate_power_product = ff_q_bpcpx_result_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpx_result_choice_valuation)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpx_result_choice_valuation_candidate))) /\ forall ff_i_bpcpx_result_choice_valuation_candidate_power_product. (exists ff_lt_bpcpx_result_choice_valuation_candidate_power_product_bound. ff_lt_bpcpx_result_choice_valuation_candidate_power_product_bound + S ff_i_bpcpx_result_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpx_result_choice_valuation) -> exists ff_p_bpcpx_result_choice_valuation_candidate_power_product ff_r_bpcpx_result_choice_valuation_candidate_power_product ff_s_bpcpx_result_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpx_result_choice_valuation_candidate_power_product_factor. ff_h_bpcpx_result_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpx_result_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpx_result_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpx_result_choice_valuation_candidate_power)) /\ exists ff_q_bpcpx_result_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpx_result_choice_valuation_candidate_power = ff_q_bpcpx_result_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpx_result_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpx_result_choice_valuation_candidate_power) + (ff_p_bpcpx_result_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpx_result_choice_valuation_candidate_power_product_partial. ff_h_bpcpx_result_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpx_result_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpx_result_choice_valuation_candidate_power_product)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_candidate_power_product_partial. ff_u_bpcpx_result_choice_valuation_candidate_power_product = ff_q_bpcpx_result_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpx_result_choice_valuation_candidate_power_product)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product) + (ff_r_bpcpx_result_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpx_result_choice_valuation_candidate_power_product_successor. ff_h_bpcpx_result_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpx_result_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpx_result_choice_valuation_candidate_power_product)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_result_choice_valuation_candidate_power_product_successor. ff_u_bpcpx_result_choice_valuation_candidate_power_product = ff_q_bpcpx_result_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpx_result_choice_valuation_candidate_power_product)) * ff_v_bpcpx_result_choice_valuation_candidate_power_product) + (ff_s_bpcpx_result_choice_valuation_candidate_power_product))) /\ ff_s_bpcpx_result_choice_valuation_candidate_power_product = ff_r_bpcpx_result_choice_valuation_candidate_power_product * ff_p_bpcpx_result_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpx_result_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpx_result_choice_valuation_candidate) * bpr_divides_quotient_bpcpx_result_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpx_result_choice_valuation_candidate_below. bpr_le_gap_bpcpx_result_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpx_result_choice_valuation) = (bpr_choice_exponent_bpcpx_result_choice))) /\ (exists bpr_power_code_bpcpx_result_choice_power bpr_power_scale_bpcpx_result_choice_power. ((forall bpr_power_index_bpcpx_result_choice_power. (exists bpr_gap_bpcpx_result_choice_power_repeat_bound. bpr_gap_bpcpx_result_choice_power_repeat_bound + S (bpr_power_index_bpcpx_result_choice_power) = bpr_choice_exponent_bpcpx_result_choice) -> (((exists bpr_height_bpcpx_result_choice_power_repeat_entry. bpr_height_bpcpx_result_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_result)) = S ((S (bpr_power_index_bpcpx_result_choice_power)) * bpr_power_scale_bpcpx_result_choice_power)) /\ exists bpr_quotient_bpcpx_result_choice_power_repeat_entry. bpr_power_code_bpcpx_result_choice_power = bpr_quotient_bpcpx_result_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpx_result_choice_power)) * bpr_power_scale_bpcpx_result_choice_power) + (S (bpr_prefix_index_bpcpx_result))))) /\ (exists ff_u_bpcpx_result_choice_power_product ff_v_bpcpx_result_choice_power_product. ((((exists ff_h_bpcpx_result_choice_power_product_start. ff_h_bpcpx_result_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_result_choice_power_product)) /\ exists ff_q_bpcpx_result_choice_power_product_start. ff_u_bpcpx_result_choice_power_product = ff_q_bpcpx_result_choice_power_product_start * S ((S (0)) * ff_v_bpcpx_result_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpx_result_choice_power_product_terminal. ff_h_bpcpx_result_choice_power_product_terminal + S (bpr_prefix_value_bpcpx_result) = S ((S (bpr_choice_exponent_bpcpx_result_choice)) * ff_v_bpcpx_result_choice_power_product)) /\ exists ff_q_bpcpx_result_choice_power_product_terminal. ff_u_bpcpx_result_choice_power_product = ff_q_bpcpx_result_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpx_result_choice)) * ff_v_bpcpx_result_choice_power_product) + (bpr_prefix_value_bpcpx_result))) /\ forall ff_i_bpcpx_result_choice_power_product. (exists ff_lt_bpcpx_result_choice_power_product_bound. ff_lt_bpcpx_result_choice_power_product_bound + S ff_i_bpcpx_result_choice_power_product = bpr_choice_exponent_bpcpx_result_choice) -> exists ff_p_bpcpx_result_choice_power_product ff_r_bpcpx_result_choice_power_product ff_s_bpcpx_result_choice_power_product. ((((exists ff_h_bpcpx_result_choice_power_product_factor. ff_h_bpcpx_result_choice_power_product_factor + S (ff_p_bpcpx_result_choice_power_product) = S ((S (ff_i_bpcpx_result_choice_power_product)) * bpr_power_scale_bpcpx_result_choice_power)) /\ exists ff_q_bpcpx_result_choice_power_product_factor. bpr_power_code_bpcpx_result_choice_power = ff_q_bpcpx_result_choice_power_product_factor * S ((S (ff_i_bpcpx_result_choice_power_product)) * bpr_power_scale_bpcpx_result_choice_power) + (ff_p_bpcpx_result_choice_power_product))) /\ ((((exists ff_h_bpcpx_result_choice_power_product_partial. ff_h_bpcpx_result_choice_power_product_partial + S (ff_r_bpcpx_result_choice_power_product) = S ((S (ff_i_bpcpx_result_choice_power_product)) * ff_v_bpcpx_result_choice_power_product)) /\ exists ff_q_bpcpx_result_choice_power_product_partial. ff_u_bpcpx_result_choice_power_product = ff_q_bpcpx_result_choice_power_product_partial * S ((S (ff_i_bpcpx_result_choice_power_product)) * ff_v_bpcpx_result_choice_power_product) + (ff_r_bpcpx_result_choice_power_product))) /\ ((((exists ff_h_bpcpx_result_choice_power_product_successor. ff_h_bpcpx_result_choice_power_product_successor + S (ff_s_bpcpx_result_choice_power_product) = S ((S (S ff_i_bpcpx_result_choice_power_product)) * ff_v_bpcpx_result_choice_power_product)) /\ exists ff_q_bpcpx_result_choice_power_product_successor. ff_u_bpcpx_result_choice_power_product = ff_q_bpcpx_result_choice_power_product_successor * S ((S (S ff_i_bpcpx_result_choice_power_product)) * ff_v_bpcpx_result_choice_power_product) + (ff_s_bpcpx_result_choice_power_product))) /\ ff_s_bpcpx_result_choice_power_product = ff_r_bpcpx_result_choice_power_product * ff_p_bpcpx_result_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpx_result) = 1) /\ forall bpr_left_bpcpx_result_choice_prime bpr_right_bpcpx_result_choice_prime. S (bpr_prefix_index_bpcpx_result) = bpr_left_bpcpx_result_choice_prime * bpr_right_bpcpx_result_choice_prime -> bpr_left_bpcpx_result_choice_prime = 1 \/ bpr_right_bpcpx_result_choice_prime = 1)) /\ bpr_prefix_value_bpcpx_result = 1)))))

Structural proof guide

Every number and finite length has a contribution prefix.

Direct prerequisites: add_eq_zero_right, succ_ne_zero, prime_contribution_prefix_extend. The authored body proceeds by structural induction (1), case analysis (3), intermediate claims (3).

Proof neighborhood

Direct dependencies

BT000L add_eq_zero_right BT000C succ_ne_zero BT00YP prime_contribution_prefix_extend

Direct dependents

BT00YS prime_contribution_product_exists

Formal native tactic body

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

  1. 0001intro n
  2. 0002induction m
  3. 0003exists 0
  4. 0004exists 0
  5. 0005intro i
  6. 0006intro hi
  7. 0007exfalso
  8. 0008cases hi
  9. 0009have hsi : S i = 0
  10. 0010apply add_eq_zero_right
  11. 0011exact hi_witness
  12. 0012apply succ_ne_zero
  13. 0013exact hsi
  14. 0014have hprevious : exists b c. (forall bpr_prefix_index_bpcpx_previous. (exists bpr_gap_bpcpx_previous_bound. bpr_gap_bpcpx_previous_bound + S (bpr_prefix_index_bpcpx_previous) = m) -> exists bpr_prefix_value_bpcpx_previous. ((((exists bpr_height_bpcpx_previous_decoded. bpr_height_bpcpx_previous_decoded + S (bpr_prefix_value_bpcpx_previous) = S ((S (bpr_prefix_index_bpcpx_previous)) * c)) /\ exists bpr_quotient_bpcpx_previous_decoded. b = bpr_quotient_bpcpx_previous_decoded * S ((S (bpr_prefix_index_bpcpx_previous)) * c) + (bpr_prefix_value_bpcpx_previous))) /\ (((((~(S (bpr_prefix_index_bpcpx_previous) = 1) /\ forall bpr_left_bpcpx_previous_choice_prime bpr_right_bpcpx_previous_choice_prime. S (bpr_prefix_index_bpcpx_previous) = bpr_left_bpcpx_previous_choice_prime * bpr_right_bpcpx_previous_choice_prime -> bpr_left_bpcpx_previous_choice_prime = 1 \/ bpr_right_bpcpx_previous_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpx_previous_choice. ((((exists bpr_le_gap_bpcpx_previous_choice_valuation_selected_bound. bpr_le_gap_bpcpx_previous_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpx_previous_choice) = (n)) /\ (exists bpr_power_value_bpcpx_previous_choice_valuation_selected. ((exists bpr_power_code_bpcpx_previous_choice_valuation_selected_power bpr_power_scale_bpcpx_previous_choice_valuation_selected_power. ((forall bpr_power_index_bpcpx_previous_choice_valuation_selected_power. (exists bpr_gap_bpcpx_previous_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpx_previous_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpx_previous_choice_valuation_selected_power) = bpr_choice_exponent_bpcpx_previous_choice) -> (((exists bpr_height_bpcpx_previous_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpx_previous_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_previous)) = S ((S (bpr_power_index_bpcpx_previous_choice_valuation_selected_power)) * bpr_power_scale_bpcpx_previous_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpx_previous_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpx_previous_choice_valuation_selected_power = bpr_quotient_bpcpx_previous_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpx_previous_choice_valuation_selected_power)) * bpr_power_scale_bpcpx_previous_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpx_previous))))) /\ (exists ff_u_bpcpx_previous_choice_valuation_selected_power_product ff_v_bpcpx_previous_choice_valuation_selected_power_product. ((((exists ff_h_bpcpx_previous_choice_valuation_selected_power_product_start. ff_h_bpcpx_previous_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_selected_power_product_start. ff_u_bpcpx_previous_choice_valuation_selected_power_product = ff_q_bpcpx_previous_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpx_previous_choice_valuation_selected_power_product_terminal. ff_h_bpcpx_previous_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpx_previous_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpx_previous_choice)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_selected_power_product_terminal. ff_u_bpcpx_previous_choice_valuation_selected_power_product = ff_q_bpcpx_previous_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpx_previous_choice)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product) + (bpr_power_value_bpcpx_previous_choice_valuation_selected))) /\ forall ff_i_bpcpx_previous_choice_valuation_selected_power_product. (exists ff_lt_bpcpx_previous_choice_valuation_selected_power_product_bound. ff_lt_bpcpx_previous_choice_valuation_selected_power_product_bound + S ff_i_bpcpx_previous_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpx_previous_choice) -> exists ff_p_bpcpx_previous_choice_valuation_selected_power_product ff_r_bpcpx_previous_choice_valuation_selected_power_product ff_s_bpcpx_previous_choice_valuation_selected_power_product. ((((exists ff_h_bpcpx_previous_choice_valuation_selected_power_product_factor. ff_h_bpcpx_previous_choice_valuation_selected_power_product_factor + S (ff_p_bpcpx_previous_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpx_previous_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpx_previous_choice_valuation_selected_power)) /\ exists ff_q_bpcpx_previous_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpx_previous_choice_valuation_selected_power = ff_q_bpcpx_previous_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpx_previous_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpx_previous_choice_valuation_selected_power) + (ff_p_bpcpx_previous_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpx_previous_choice_valuation_selected_power_product_partial. ff_h_bpcpx_previous_choice_valuation_selected_power_product_partial + S (ff_r_bpcpx_previous_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpx_previous_choice_valuation_selected_power_product)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_selected_power_product_partial. ff_u_bpcpx_previous_choice_valuation_selected_power_product = ff_q_bpcpx_previous_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpx_previous_choice_valuation_selected_power_product)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product) + (ff_r_bpcpx_previous_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpx_previous_choice_valuation_selected_power_product_successor. ff_h_bpcpx_previous_choice_valuation_selected_power_product_successor + S (ff_s_bpcpx_previous_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpx_previous_choice_valuation_selected_power_product)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_selected_power_product_successor. ff_u_bpcpx_previous_choice_valuation_selected_power_product = ff_q_bpcpx_previous_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpx_previous_choice_valuation_selected_power_product)) * ff_v_bpcpx_previous_choice_valuation_selected_power_product) + (ff_s_bpcpx_previous_choice_valuation_selected_power_product))) /\ ff_s_bpcpx_previous_choice_valuation_selected_power_product = ff_r_bpcpx_previous_choice_valuation_selected_power_product * ff_p_bpcpx_previous_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpx_previous_choice_valuation_selected_divides. n = (bpr_power_value_bpcpx_previous_choice_valuation_selected) * bpr_divides_quotient_bpcpx_previous_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpx_previous_choice_valuation. (exists bpr_le_gap_bpcpx_previous_choice_valuation_candidate_bound. bpr_le_gap_bpcpx_previous_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpx_previous_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpx_previous_choice_valuation_candidate. ((exists bpr_power_code_bpcpx_previous_choice_valuation_candidate_power bpr_power_scale_bpcpx_previous_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpx_previous_choice_valuation_candidate_power. (exists bpr_gap_bpcpx_previous_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpx_previous_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpx_previous_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpx_previous_choice_valuation) -> (((exists bpr_height_bpcpx_previous_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpx_previous_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_previous)) = S ((S (bpr_power_index_bpcpx_previous_choice_valuation_candidate_power)) * bpr_power_scale_bpcpx_previous_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpx_previous_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpx_previous_choice_valuation_candidate_power = bpr_quotient_bpcpx_previous_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpx_previous_choice_valuation_candidate_power)) * bpr_power_scale_bpcpx_previous_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpx_previous))))) /\ (exists ff_u_bpcpx_previous_choice_valuation_candidate_power_product ff_v_bpcpx_previous_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpx_previous_choice_valuation_candidate_power_product_start. ff_h_bpcpx_previous_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_candidate_power_product_start. ff_u_bpcpx_previous_choice_valuation_candidate_power_product = ff_q_bpcpx_previous_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpx_previous_choice_valuation_candidate_power_product_terminal. ff_h_bpcpx_previous_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpx_previous_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpx_previous_choice_valuation)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_candidate_power_product_terminal. ff_u_bpcpx_previous_choice_valuation_candidate_power_product = ff_q_bpcpx_previous_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpx_previous_choice_valuation)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpx_previous_choice_valuation_candidate))) /\ forall ff_i_bpcpx_previous_choice_valuation_candidate_power_product. (exists ff_lt_bpcpx_previous_choice_valuation_candidate_power_product_bound. ff_lt_bpcpx_previous_choice_valuation_candidate_power_product_bound + S ff_i_bpcpx_previous_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpx_previous_choice_valuation) -> exists ff_p_bpcpx_previous_choice_valuation_candidate_power_product ff_r_bpcpx_previous_choice_valuation_candidate_power_product ff_s_bpcpx_previous_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpx_previous_choice_valuation_candidate_power_product_factor. ff_h_bpcpx_previous_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpx_previous_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpx_previous_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpx_previous_choice_valuation_candidate_power)) /\ exists ff_q_bpcpx_previous_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpx_previous_choice_valuation_candidate_power = ff_q_bpcpx_previous_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpx_previous_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpx_previous_choice_valuation_candidate_power) + (ff_p_bpcpx_previous_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpx_previous_choice_valuation_candidate_power_product_partial. ff_h_bpcpx_previous_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpx_previous_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpx_previous_choice_valuation_candidate_power_product)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_candidate_power_product_partial. ff_u_bpcpx_previous_choice_valuation_candidate_power_product = ff_q_bpcpx_previous_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpx_previous_choice_valuation_candidate_power_product)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product) + (ff_r_bpcpx_previous_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpx_previous_choice_valuation_candidate_power_product_successor. ff_h_bpcpx_previous_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpx_previous_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpx_previous_choice_valuation_candidate_power_product)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_previous_choice_valuation_candidate_power_product_successor. ff_u_bpcpx_previous_choice_valuation_candidate_power_product = ff_q_bpcpx_previous_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpx_previous_choice_valuation_candidate_power_product)) * ff_v_bpcpx_previous_choice_valuation_candidate_power_product) + (ff_s_bpcpx_previous_choice_valuation_candidate_power_product))) /\ ff_s_bpcpx_previous_choice_valuation_candidate_power_product = ff_r_bpcpx_previous_choice_valuation_candidate_power_product * ff_p_bpcpx_previous_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpx_previous_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpx_previous_choice_valuation_candidate) * bpr_divides_quotient_bpcpx_previous_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpx_previous_choice_valuation_candidate_below. bpr_le_gap_bpcpx_previous_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpx_previous_choice_valuation) = (bpr_choice_exponent_bpcpx_previous_choice))) /\ (exists bpr_power_code_bpcpx_previous_choice_power bpr_power_scale_bpcpx_previous_choice_power. ((forall bpr_power_index_bpcpx_previous_choice_power. (exists bpr_gap_bpcpx_previous_choice_power_repeat_bound. bpr_gap_bpcpx_previous_choice_power_repeat_bound + S (bpr_power_index_bpcpx_previous_choice_power) = bpr_choice_exponent_bpcpx_previous_choice) -> (((exists bpr_height_bpcpx_previous_choice_power_repeat_entry. bpr_height_bpcpx_previous_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_previous)) = S ((S (bpr_power_index_bpcpx_previous_choice_power)) * bpr_power_scale_bpcpx_previous_choice_power)) /\ exists bpr_quotient_bpcpx_previous_choice_power_repeat_entry. bpr_power_code_bpcpx_previous_choice_power = bpr_quotient_bpcpx_previous_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpx_previous_choice_power)) * bpr_power_scale_bpcpx_previous_choice_power) + (S (bpr_prefix_index_bpcpx_previous))))) /\ (exists ff_u_bpcpx_previous_choice_power_product ff_v_bpcpx_previous_choice_power_product. ((((exists ff_h_bpcpx_previous_choice_power_product_start. ff_h_bpcpx_previous_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_previous_choice_power_product)) /\ exists ff_q_bpcpx_previous_choice_power_product_start. ff_u_bpcpx_previous_choice_power_product = ff_q_bpcpx_previous_choice_power_product_start * S ((S (0)) * ff_v_bpcpx_previous_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpx_previous_choice_power_product_terminal. ff_h_bpcpx_previous_choice_power_product_terminal + S (bpr_prefix_value_bpcpx_previous) = S ((S (bpr_choice_exponent_bpcpx_previous_choice)) * ff_v_bpcpx_previous_choice_power_product)) /\ exists ff_q_bpcpx_previous_choice_power_product_terminal. ff_u_bpcpx_previous_choice_power_product = ff_q_bpcpx_previous_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpx_previous_choice)) * ff_v_bpcpx_previous_choice_power_product) + (bpr_prefix_value_bpcpx_previous))) /\ forall ff_i_bpcpx_previous_choice_power_product. (exists ff_lt_bpcpx_previous_choice_power_product_bound. ff_lt_bpcpx_previous_choice_power_product_bound + S ff_i_bpcpx_previous_choice_power_product = bpr_choice_exponent_bpcpx_previous_choice) -> exists ff_p_bpcpx_previous_choice_power_product ff_r_bpcpx_previous_choice_power_product ff_s_bpcpx_previous_choice_power_product. ((((exists ff_h_bpcpx_previous_choice_power_product_factor. ff_h_bpcpx_previous_choice_power_product_factor + S (ff_p_bpcpx_previous_choice_power_product) = S ((S (ff_i_bpcpx_previous_choice_power_product)) * bpr_power_scale_bpcpx_previous_choice_power)) /\ exists ff_q_bpcpx_previous_choice_power_product_factor. bpr_power_code_bpcpx_previous_choice_power = ff_q_bpcpx_previous_choice_power_product_factor * S ((S (ff_i_bpcpx_previous_choice_power_product)) * bpr_power_scale_bpcpx_previous_choice_power) + (ff_p_bpcpx_previous_choice_power_product))) /\ ((((exists ff_h_bpcpx_previous_choice_power_product_partial. ff_h_bpcpx_previous_choice_power_product_partial + S (ff_r_bpcpx_previous_choice_power_product) = S ((S (ff_i_bpcpx_previous_choice_power_product)) * ff_v_bpcpx_previous_choice_power_product)) /\ exists ff_q_bpcpx_previous_choice_power_product_partial. ff_u_bpcpx_previous_choice_power_product = ff_q_bpcpx_previous_choice_power_product_partial * S ((S (ff_i_bpcpx_previous_choice_power_product)) * ff_v_bpcpx_previous_choice_power_product) + (ff_r_bpcpx_previous_choice_power_product))) /\ ((((exists ff_h_bpcpx_previous_choice_power_product_successor. ff_h_bpcpx_previous_choice_power_product_successor + S (ff_s_bpcpx_previous_choice_power_product) = S ((S (S ff_i_bpcpx_previous_choice_power_product)) * ff_v_bpcpx_previous_choice_power_product)) /\ exists ff_q_bpcpx_previous_choice_power_product_successor. ff_u_bpcpx_previous_choice_power_product = ff_q_bpcpx_previous_choice_power_product_successor * S ((S (S ff_i_bpcpx_previous_choice_power_product)) * ff_v_bpcpx_previous_choice_power_product) + (ff_s_bpcpx_previous_choice_power_product))) /\ ff_s_bpcpx_previous_choice_power_product = ff_r_bpcpx_previous_choice_power_product * ff_p_bpcpx_previous_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpx_previous) = 1) /\ forall bpr_left_bpcpx_previous_choice_prime bpr_right_bpcpx_previous_choice_prime. S (bpr_prefix_index_bpcpx_previous) = bpr_left_bpcpx_previous_choice_prime * bpr_right_bpcpx_previous_choice_prime -> bpr_left_bpcpx_previous_choice_prime = 1 \/ bpr_right_bpcpx_previous_choice_prime = 1)) /\ bpr_prefix_value_bpcpx_previous = 1)))))
  15. 0015exact IH
  16. 0016cases hprevious
  17. 0017cases hprevious_witness
  18. 0018have hnext : exists b c. (forall bpr_prefix_index_bpcpx_successor. (exists bpr_gap_bpcpx_successor_bound. bpr_gap_bpcpx_successor_bound + S (bpr_prefix_index_bpcpx_successor) = S m) -> exists bpr_prefix_value_bpcpx_successor. ((((exists bpr_height_bpcpx_successor_decoded. bpr_height_bpcpx_successor_decoded + S (bpr_prefix_value_bpcpx_successor) = S ((S (bpr_prefix_index_bpcpx_successor)) * c)) /\ exists bpr_quotient_bpcpx_successor_decoded. b = bpr_quotient_bpcpx_successor_decoded * S ((S (bpr_prefix_index_bpcpx_successor)) * c) + (bpr_prefix_value_bpcpx_successor))) /\ (((((~(S (bpr_prefix_index_bpcpx_successor) = 1) /\ forall bpr_left_bpcpx_successor_choice_prime bpr_right_bpcpx_successor_choice_prime. S (bpr_prefix_index_bpcpx_successor) = bpr_left_bpcpx_successor_choice_prime * bpr_right_bpcpx_successor_choice_prime -> bpr_left_bpcpx_successor_choice_prime = 1 \/ bpr_right_bpcpx_successor_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcpx_successor_choice. ((((exists bpr_le_gap_bpcpx_successor_choice_valuation_selected_bound. bpr_le_gap_bpcpx_successor_choice_valuation_selected_bound + (bpr_choice_exponent_bpcpx_successor_choice) = (n)) /\ (exists bpr_power_value_bpcpx_successor_choice_valuation_selected. ((exists bpr_power_code_bpcpx_successor_choice_valuation_selected_power bpr_power_scale_bpcpx_successor_choice_valuation_selected_power. ((forall bpr_power_index_bpcpx_successor_choice_valuation_selected_power. (exists bpr_gap_bpcpx_successor_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcpx_successor_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcpx_successor_choice_valuation_selected_power) = bpr_choice_exponent_bpcpx_successor_choice) -> (((exists bpr_height_bpcpx_successor_choice_valuation_selected_power_repeat_entry. bpr_height_bpcpx_successor_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_successor)) = S ((S (bpr_power_index_bpcpx_successor_choice_valuation_selected_power)) * bpr_power_scale_bpcpx_successor_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcpx_successor_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcpx_successor_choice_valuation_selected_power = bpr_quotient_bpcpx_successor_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcpx_successor_choice_valuation_selected_power)) * bpr_power_scale_bpcpx_successor_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcpx_successor))))) /\ (exists ff_u_bpcpx_successor_choice_valuation_selected_power_product ff_v_bpcpx_successor_choice_valuation_selected_power_product. ((((exists ff_h_bpcpx_successor_choice_valuation_selected_power_product_start. ff_h_bpcpx_successor_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_selected_power_product_start. ff_u_bpcpx_successor_choice_valuation_selected_power_product = ff_q_bpcpx_successor_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcpx_successor_choice_valuation_selected_power_product_terminal. ff_h_bpcpx_successor_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcpx_successor_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcpx_successor_choice)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_selected_power_product_terminal. ff_u_bpcpx_successor_choice_valuation_selected_power_product = ff_q_bpcpx_successor_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcpx_successor_choice)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product) + (bpr_power_value_bpcpx_successor_choice_valuation_selected))) /\ forall ff_i_bpcpx_successor_choice_valuation_selected_power_product. (exists ff_lt_bpcpx_successor_choice_valuation_selected_power_product_bound. ff_lt_bpcpx_successor_choice_valuation_selected_power_product_bound + S ff_i_bpcpx_successor_choice_valuation_selected_power_product = bpr_choice_exponent_bpcpx_successor_choice) -> exists ff_p_bpcpx_successor_choice_valuation_selected_power_product ff_r_bpcpx_successor_choice_valuation_selected_power_product ff_s_bpcpx_successor_choice_valuation_selected_power_product. ((((exists ff_h_bpcpx_successor_choice_valuation_selected_power_product_factor. ff_h_bpcpx_successor_choice_valuation_selected_power_product_factor + S (ff_p_bpcpx_successor_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpx_successor_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpx_successor_choice_valuation_selected_power)) /\ exists ff_q_bpcpx_successor_choice_valuation_selected_power_product_factor. bpr_power_code_bpcpx_successor_choice_valuation_selected_power = ff_q_bpcpx_successor_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcpx_successor_choice_valuation_selected_power_product)) * bpr_power_scale_bpcpx_successor_choice_valuation_selected_power) + (ff_p_bpcpx_successor_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpx_successor_choice_valuation_selected_power_product_partial. ff_h_bpcpx_successor_choice_valuation_selected_power_product_partial + S (ff_r_bpcpx_successor_choice_valuation_selected_power_product) = S ((S (ff_i_bpcpx_successor_choice_valuation_selected_power_product)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_selected_power_product_partial. ff_u_bpcpx_successor_choice_valuation_selected_power_product = ff_q_bpcpx_successor_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcpx_successor_choice_valuation_selected_power_product)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product) + (ff_r_bpcpx_successor_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcpx_successor_choice_valuation_selected_power_product_successor. ff_h_bpcpx_successor_choice_valuation_selected_power_product_successor + S (ff_s_bpcpx_successor_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcpx_successor_choice_valuation_selected_power_product)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_selected_power_product_successor. ff_u_bpcpx_successor_choice_valuation_selected_power_product = ff_q_bpcpx_successor_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcpx_successor_choice_valuation_selected_power_product)) * ff_v_bpcpx_successor_choice_valuation_selected_power_product) + (ff_s_bpcpx_successor_choice_valuation_selected_power_product))) /\ ff_s_bpcpx_successor_choice_valuation_selected_power_product = ff_r_bpcpx_successor_choice_valuation_selected_power_product * ff_p_bpcpx_successor_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpx_successor_choice_valuation_selected_divides. n = (bpr_power_value_bpcpx_successor_choice_valuation_selected) * bpr_divides_quotient_bpcpx_successor_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcpx_successor_choice_valuation. (exists bpr_le_gap_bpcpx_successor_choice_valuation_candidate_bound. bpr_le_gap_bpcpx_successor_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcpx_successor_choice_valuation) = (n)) -> (exists bpr_power_value_bpcpx_successor_choice_valuation_candidate. ((exists bpr_power_code_bpcpx_successor_choice_valuation_candidate_power bpr_power_scale_bpcpx_successor_choice_valuation_candidate_power. ((forall bpr_power_index_bpcpx_successor_choice_valuation_candidate_power. (exists bpr_gap_bpcpx_successor_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcpx_successor_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcpx_successor_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcpx_successor_choice_valuation) -> (((exists bpr_height_bpcpx_successor_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcpx_successor_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_successor)) = S ((S (bpr_power_index_bpcpx_successor_choice_valuation_candidate_power)) * bpr_power_scale_bpcpx_successor_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcpx_successor_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcpx_successor_choice_valuation_candidate_power = bpr_quotient_bpcpx_successor_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcpx_successor_choice_valuation_candidate_power)) * bpr_power_scale_bpcpx_successor_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcpx_successor))))) /\ (exists ff_u_bpcpx_successor_choice_valuation_candidate_power_product ff_v_bpcpx_successor_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpx_successor_choice_valuation_candidate_power_product_start. ff_h_bpcpx_successor_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_candidate_power_product_start. ff_u_bpcpx_successor_choice_valuation_candidate_power_product = ff_q_bpcpx_successor_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcpx_successor_choice_valuation_candidate_power_product_terminal. ff_h_bpcpx_successor_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcpx_successor_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcpx_successor_choice_valuation)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_candidate_power_product_terminal. ff_u_bpcpx_successor_choice_valuation_candidate_power_product = ff_q_bpcpx_successor_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcpx_successor_choice_valuation)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product) + (bpr_power_value_bpcpx_successor_choice_valuation_candidate))) /\ forall ff_i_bpcpx_successor_choice_valuation_candidate_power_product. (exists ff_lt_bpcpx_successor_choice_valuation_candidate_power_product_bound. ff_lt_bpcpx_successor_choice_valuation_candidate_power_product_bound + S ff_i_bpcpx_successor_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcpx_successor_choice_valuation) -> exists ff_p_bpcpx_successor_choice_valuation_candidate_power_product ff_r_bpcpx_successor_choice_valuation_candidate_power_product ff_s_bpcpx_successor_choice_valuation_candidate_power_product. ((((exists ff_h_bpcpx_successor_choice_valuation_candidate_power_product_factor. ff_h_bpcpx_successor_choice_valuation_candidate_power_product_factor + S (ff_p_bpcpx_successor_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpx_successor_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpx_successor_choice_valuation_candidate_power)) /\ exists ff_q_bpcpx_successor_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcpx_successor_choice_valuation_candidate_power = ff_q_bpcpx_successor_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcpx_successor_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcpx_successor_choice_valuation_candidate_power) + (ff_p_bpcpx_successor_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpx_successor_choice_valuation_candidate_power_product_partial. ff_h_bpcpx_successor_choice_valuation_candidate_power_product_partial + S (ff_r_bpcpx_successor_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcpx_successor_choice_valuation_candidate_power_product)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_candidate_power_product_partial. ff_u_bpcpx_successor_choice_valuation_candidate_power_product = ff_q_bpcpx_successor_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcpx_successor_choice_valuation_candidate_power_product)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product) + (ff_r_bpcpx_successor_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcpx_successor_choice_valuation_candidate_power_product_successor. ff_h_bpcpx_successor_choice_valuation_candidate_power_product_successor + S (ff_s_bpcpx_successor_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcpx_successor_choice_valuation_candidate_power_product)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcpx_successor_choice_valuation_candidate_power_product_successor. ff_u_bpcpx_successor_choice_valuation_candidate_power_product = ff_q_bpcpx_successor_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcpx_successor_choice_valuation_candidate_power_product)) * ff_v_bpcpx_successor_choice_valuation_candidate_power_product) + (ff_s_bpcpx_successor_choice_valuation_candidate_power_product))) /\ ff_s_bpcpx_successor_choice_valuation_candidate_power_product = ff_r_bpcpx_successor_choice_valuation_candidate_power_product * ff_p_bpcpx_successor_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcpx_successor_choice_valuation_candidate_divides. n = (bpr_power_value_bpcpx_successor_choice_valuation_candidate) * bpr_divides_quotient_bpcpx_successor_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcpx_successor_choice_valuation_candidate_below. bpr_le_gap_bpcpx_successor_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcpx_successor_choice_valuation) = (bpr_choice_exponent_bpcpx_successor_choice))) /\ (exists bpr_power_code_bpcpx_successor_choice_power bpr_power_scale_bpcpx_successor_choice_power. ((forall bpr_power_index_bpcpx_successor_choice_power. (exists bpr_gap_bpcpx_successor_choice_power_repeat_bound. bpr_gap_bpcpx_successor_choice_power_repeat_bound + S (bpr_power_index_bpcpx_successor_choice_power) = bpr_choice_exponent_bpcpx_successor_choice) -> (((exists bpr_height_bpcpx_successor_choice_power_repeat_entry. bpr_height_bpcpx_successor_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcpx_successor)) = S ((S (bpr_power_index_bpcpx_successor_choice_power)) * bpr_power_scale_bpcpx_successor_choice_power)) /\ exists bpr_quotient_bpcpx_successor_choice_power_repeat_entry. bpr_power_code_bpcpx_successor_choice_power = bpr_quotient_bpcpx_successor_choice_power_repeat_entry * S ((S (bpr_power_index_bpcpx_successor_choice_power)) * bpr_power_scale_bpcpx_successor_choice_power) + (S (bpr_prefix_index_bpcpx_successor))))) /\ (exists ff_u_bpcpx_successor_choice_power_product ff_v_bpcpx_successor_choice_power_product. ((((exists ff_h_bpcpx_successor_choice_power_product_start. ff_h_bpcpx_successor_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcpx_successor_choice_power_product)) /\ exists ff_q_bpcpx_successor_choice_power_product_start. ff_u_bpcpx_successor_choice_power_product = ff_q_bpcpx_successor_choice_power_product_start * S ((S (0)) * ff_v_bpcpx_successor_choice_power_product) + (1))) /\ ((((exists ff_h_bpcpx_successor_choice_power_product_terminal. ff_h_bpcpx_successor_choice_power_product_terminal + S (bpr_prefix_value_bpcpx_successor) = S ((S (bpr_choice_exponent_bpcpx_successor_choice)) * ff_v_bpcpx_successor_choice_power_product)) /\ exists ff_q_bpcpx_successor_choice_power_product_terminal. ff_u_bpcpx_successor_choice_power_product = ff_q_bpcpx_successor_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcpx_successor_choice)) * ff_v_bpcpx_successor_choice_power_product) + (bpr_prefix_value_bpcpx_successor))) /\ forall ff_i_bpcpx_successor_choice_power_product. (exists ff_lt_bpcpx_successor_choice_power_product_bound. ff_lt_bpcpx_successor_choice_power_product_bound + S ff_i_bpcpx_successor_choice_power_product = bpr_choice_exponent_bpcpx_successor_choice) -> exists ff_p_bpcpx_successor_choice_power_product ff_r_bpcpx_successor_choice_power_product ff_s_bpcpx_successor_choice_power_product. ((((exists ff_h_bpcpx_successor_choice_power_product_factor. ff_h_bpcpx_successor_choice_power_product_factor + S (ff_p_bpcpx_successor_choice_power_product) = S ((S (ff_i_bpcpx_successor_choice_power_product)) * bpr_power_scale_bpcpx_successor_choice_power)) /\ exists ff_q_bpcpx_successor_choice_power_product_factor. bpr_power_code_bpcpx_successor_choice_power = ff_q_bpcpx_successor_choice_power_product_factor * S ((S (ff_i_bpcpx_successor_choice_power_product)) * bpr_power_scale_bpcpx_successor_choice_power) + (ff_p_bpcpx_successor_choice_power_product))) /\ ((((exists ff_h_bpcpx_successor_choice_power_product_partial. ff_h_bpcpx_successor_choice_power_product_partial + S (ff_r_bpcpx_successor_choice_power_product) = S ((S (ff_i_bpcpx_successor_choice_power_product)) * ff_v_bpcpx_successor_choice_power_product)) /\ exists ff_q_bpcpx_successor_choice_power_product_partial. ff_u_bpcpx_successor_choice_power_product = ff_q_bpcpx_successor_choice_power_product_partial * S ((S (ff_i_bpcpx_successor_choice_power_product)) * ff_v_bpcpx_successor_choice_power_product) + (ff_r_bpcpx_successor_choice_power_product))) /\ ((((exists ff_h_bpcpx_successor_choice_power_product_successor. ff_h_bpcpx_successor_choice_power_product_successor + S (ff_s_bpcpx_successor_choice_power_product) = S ((S (S ff_i_bpcpx_successor_choice_power_product)) * ff_v_bpcpx_successor_choice_power_product)) /\ exists ff_q_bpcpx_successor_choice_power_product_successor. ff_u_bpcpx_successor_choice_power_product = ff_q_bpcpx_successor_choice_power_product_successor * S ((S (S ff_i_bpcpx_successor_choice_power_product)) * ff_v_bpcpx_successor_choice_power_product) + (ff_s_bpcpx_successor_choice_power_product))) /\ ff_s_bpcpx_successor_choice_power_product = ff_r_bpcpx_successor_choice_power_product * ff_p_bpcpx_successor_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcpx_successor) = 1) /\ forall bpr_left_bpcpx_successor_choice_prime bpr_right_bpcpx_successor_choice_prime. S (bpr_prefix_index_bpcpx_successor) = bpr_left_bpcpx_successor_choice_prime * bpr_right_bpcpx_successor_choice_prime -> bpr_left_bpcpx_successor_choice_prime = 1 \/ bpr_right_bpcpx_successor_choice_prime = 1)) /\ bpr_prefix_value_bpcpx_successor = 1)))))
  19. 0019apply prime_contribution_prefix_extend
  20. 0020exact hprevious_witness_witness
  21. 0021exact hnext