BT00YW

prime_contribution_prefix_pairwise_coprime

Alpha body-checked · checked-use disabled

Distinct contribution positions decode pairwise-coprime values.

Exact expanded PA statement

forall n b c m. (forall bpr_prefix_index_bpcppc_source. (exists bpr_gap_bpcppc_source_bound. bpr_gap_bpcppc_source_bound + S (bpr_prefix_index_bpcppc_source) = m) -> exists bpr_prefix_value_bpcppc_source. ((((exists bpr_height_bpcppc_source_decoded. bpr_height_bpcppc_source_decoded + S (bpr_prefix_value_bpcppc_source) = S ((S (bpr_prefix_index_bpcppc_source)) * c)) /\ exists bpr_quotient_bpcppc_source_decoded. b = bpr_quotient_bpcppc_source_decoded * S ((S (bpr_prefix_index_bpcppc_source)) * c) + (bpr_prefix_value_bpcppc_source))) /\ (((((~(S (bpr_prefix_index_bpcppc_source) = 1) /\ forall bpr_left_bpcppc_source_choice_prime bpr_right_bpcppc_source_choice_prime. S (bpr_prefix_index_bpcppc_source) = bpr_left_bpcppc_source_choice_prime * bpr_right_bpcppc_source_choice_prime -> bpr_left_bpcppc_source_choice_prime = 1 \/ bpr_right_bpcppc_source_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcppc_source_choice. ((((exists bpr_le_gap_bpcppc_source_choice_valuation_selected_bound. bpr_le_gap_bpcppc_source_choice_valuation_selected_bound + (bpr_choice_exponent_bpcppc_source_choice) = (n)) /\ (exists bpr_power_value_bpcppc_source_choice_valuation_selected. ((exists bpr_power_code_bpcppc_source_choice_valuation_selected_power bpr_power_scale_bpcppc_source_choice_valuation_selected_power. ((forall bpr_power_index_bpcppc_source_choice_valuation_selected_power. (exists bpr_gap_bpcppc_source_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcppc_source_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcppc_source_choice_valuation_selected_power) = bpr_choice_exponent_bpcppc_source_choice) -> (((exists bpr_height_bpcppc_source_choice_valuation_selected_power_repeat_entry. bpr_height_bpcppc_source_choice_valuation_selected_power_repeat_entry + S (S (bpr_prefix_index_bpcppc_source)) = S ((S (bpr_power_index_bpcppc_source_choice_valuation_selected_power)) * bpr_power_scale_bpcppc_source_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcppc_source_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcppc_source_choice_valuation_selected_power = bpr_quotient_bpcppc_source_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcppc_source_choice_valuation_selected_power)) * bpr_power_scale_bpcppc_source_choice_valuation_selected_power) + (S (bpr_prefix_index_bpcppc_source))))) /\ (exists ff_u_bpcppc_source_choice_valuation_selected_power_product ff_v_bpcppc_source_choice_valuation_selected_power_product. ((((exists ff_h_bpcppc_source_choice_valuation_selected_power_product_start. ff_h_bpcppc_source_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_selected_power_product_start. ff_u_bpcppc_source_choice_valuation_selected_power_product = ff_q_bpcppc_source_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcppc_source_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcppc_source_choice_valuation_selected_power_product_terminal. ff_h_bpcppc_source_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcppc_source_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcppc_source_choice)) * ff_v_bpcppc_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_selected_power_product_terminal. ff_u_bpcppc_source_choice_valuation_selected_power_product = ff_q_bpcppc_source_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcppc_source_choice)) * ff_v_bpcppc_source_choice_valuation_selected_power_product) + (bpr_power_value_bpcppc_source_choice_valuation_selected))) /\ forall ff_i_bpcppc_source_choice_valuation_selected_power_product. (exists ff_lt_bpcppc_source_choice_valuation_selected_power_product_bound. ff_lt_bpcppc_source_choice_valuation_selected_power_product_bound + S ff_i_bpcppc_source_choice_valuation_selected_power_product = bpr_choice_exponent_bpcppc_source_choice) -> exists ff_p_bpcppc_source_choice_valuation_selected_power_product ff_r_bpcppc_source_choice_valuation_selected_power_product ff_s_bpcppc_source_choice_valuation_selected_power_product. ((((exists ff_h_bpcppc_source_choice_valuation_selected_power_product_factor. ff_h_bpcppc_source_choice_valuation_selected_power_product_factor + S (ff_p_bpcppc_source_choice_valuation_selected_power_product) = S ((S (ff_i_bpcppc_source_choice_valuation_selected_power_product)) * bpr_power_scale_bpcppc_source_choice_valuation_selected_power)) /\ exists ff_q_bpcppc_source_choice_valuation_selected_power_product_factor. bpr_power_code_bpcppc_source_choice_valuation_selected_power = ff_q_bpcppc_source_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcppc_source_choice_valuation_selected_power_product)) * bpr_power_scale_bpcppc_source_choice_valuation_selected_power) + (ff_p_bpcppc_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcppc_source_choice_valuation_selected_power_product_partial. ff_h_bpcppc_source_choice_valuation_selected_power_product_partial + S (ff_r_bpcppc_source_choice_valuation_selected_power_product) = S ((S (ff_i_bpcppc_source_choice_valuation_selected_power_product)) * ff_v_bpcppc_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_selected_power_product_partial. ff_u_bpcppc_source_choice_valuation_selected_power_product = ff_q_bpcppc_source_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcppc_source_choice_valuation_selected_power_product)) * ff_v_bpcppc_source_choice_valuation_selected_power_product) + (ff_r_bpcppc_source_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcppc_source_choice_valuation_selected_power_product_successor. ff_h_bpcppc_source_choice_valuation_selected_power_product_successor + S (ff_s_bpcppc_source_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcppc_source_choice_valuation_selected_power_product)) * ff_v_bpcppc_source_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_selected_power_product_successor. ff_u_bpcppc_source_choice_valuation_selected_power_product = ff_q_bpcppc_source_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcppc_source_choice_valuation_selected_power_product)) * ff_v_bpcppc_source_choice_valuation_selected_power_product) + (ff_s_bpcppc_source_choice_valuation_selected_power_product))) /\ ff_s_bpcppc_source_choice_valuation_selected_power_product = ff_r_bpcppc_source_choice_valuation_selected_power_product * ff_p_bpcppc_source_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcppc_source_choice_valuation_selected_divides. n = (bpr_power_value_bpcppc_source_choice_valuation_selected) * bpr_divides_quotient_bpcppc_source_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcppc_source_choice_valuation. (exists bpr_le_gap_bpcppc_source_choice_valuation_candidate_bound. bpr_le_gap_bpcppc_source_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcppc_source_choice_valuation) = (n)) -> (exists bpr_power_value_bpcppc_source_choice_valuation_candidate. ((exists bpr_power_code_bpcppc_source_choice_valuation_candidate_power bpr_power_scale_bpcppc_source_choice_valuation_candidate_power. ((forall bpr_power_index_bpcppc_source_choice_valuation_candidate_power. (exists bpr_gap_bpcppc_source_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcppc_source_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcppc_source_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcppc_source_choice_valuation) -> (((exists bpr_height_bpcppc_source_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcppc_source_choice_valuation_candidate_power_repeat_entry + S (S (bpr_prefix_index_bpcppc_source)) = S ((S (bpr_power_index_bpcppc_source_choice_valuation_candidate_power)) * bpr_power_scale_bpcppc_source_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcppc_source_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcppc_source_choice_valuation_candidate_power = bpr_quotient_bpcppc_source_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcppc_source_choice_valuation_candidate_power)) * bpr_power_scale_bpcppc_source_choice_valuation_candidate_power) + (S (bpr_prefix_index_bpcppc_source))))) /\ (exists ff_u_bpcppc_source_choice_valuation_candidate_power_product ff_v_bpcppc_source_choice_valuation_candidate_power_product. ((((exists ff_h_bpcppc_source_choice_valuation_candidate_power_product_start. ff_h_bpcppc_source_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_candidate_power_product_start. ff_u_bpcppc_source_choice_valuation_candidate_power_product = ff_q_bpcppc_source_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcppc_source_choice_valuation_candidate_power_product_terminal. ff_h_bpcppc_source_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcppc_source_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcppc_source_choice_valuation)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_candidate_power_product_terminal. ff_u_bpcppc_source_choice_valuation_candidate_power_product = ff_q_bpcppc_source_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcppc_source_choice_valuation)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product) + (bpr_power_value_bpcppc_source_choice_valuation_candidate))) /\ forall ff_i_bpcppc_source_choice_valuation_candidate_power_product. (exists ff_lt_bpcppc_source_choice_valuation_candidate_power_product_bound. ff_lt_bpcppc_source_choice_valuation_candidate_power_product_bound + S ff_i_bpcppc_source_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcppc_source_choice_valuation) -> exists ff_p_bpcppc_source_choice_valuation_candidate_power_product ff_r_bpcppc_source_choice_valuation_candidate_power_product ff_s_bpcppc_source_choice_valuation_candidate_power_product. ((((exists ff_h_bpcppc_source_choice_valuation_candidate_power_product_factor. ff_h_bpcppc_source_choice_valuation_candidate_power_product_factor + S (ff_p_bpcppc_source_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcppc_source_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcppc_source_choice_valuation_candidate_power)) /\ exists ff_q_bpcppc_source_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcppc_source_choice_valuation_candidate_power = ff_q_bpcppc_source_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcppc_source_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcppc_source_choice_valuation_candidate_power) + (ff_p_bpcppc_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcppc_source_choice_valuation_candidate_power_product_partial. ff_h_bpcppc_source_choice_valuation_candidate_power_product_partial + S (ff_r_bpcppc_source_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcppc_source_choice_valuation_candidate_power_product)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_candidate_power_product_partial. ff_u_bpcppc_source_choice_valuation_candidate_power_product = ff_q_bpcppc_source_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcppc_source_choice_valuation_candidate_power_product)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product) + (ff_r_bpcppc_source_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcppc_source_choice_valuation_candidate_power_product_successor. ff_h_bpcppc_source_choice_valuation_candidate_power_product_successor + S (ff_s_bpcppc_source_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcppc_source_choice_valuation_candidate_power_product)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_source_choice_valuation_candidate_power_product_successor. ff_u_bpcppc_source_choice_valuation_candidate_power_product = ff_q_bpcppc_source_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcppc_source_choice_valuation_candidate_power_product)) * ff_v_bpcppc_source_choice_valuation_candidate_power_product) + (ff_s_bpcppc_source_choice_valuation_candidate_power_product))) /\ ff_s_bpcppc_source_choice_valuation_candidate_power_product = ff_r_bpcppc_source_choice_valuation_candidate_power_product * ff_p_bpcppc_source_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcppc_source_choice_valuation_candidate_divides. n = (bpr_power_value_bpcppc_source_choice_valuation_candidate) * bpr_divides_quotient_bpcppc_source_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcppc_source_choice_valuation_candidate_below. bpr_le_gap_bpcppc_source_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcppc_source_choice_valuation) = (bpr_choice_exponent_bpcppc_source_choice))) /\ (exists bpr_power_code_bpcppc_source_choice_power bpr_power_scale_bpcppc_source_choice_power. ((forall bpr_power_index_bpcppc_source_choice_power. (exists bpr_gap_bpcppc_source_choice_power_repeat_bound. bpr_gap_bpcppc_source_choice_power_repeat_bound + S (bpr_power_index_bpcppc_source_choice_power) = bpr_choice_exponent_bpcppc_source_choice) -> (((exists bpr_height_bpcppc_source_choice_power_repeat_entry. bpr_height_bpcppc_source_choice_power_repeat_entry + S (S (bpr_prefix_index_bpcppc_source)) = S ((S (bpr_power_index_bpcppc_source_choice_power)) * bpr_power_scale_bpcppc_source_choice_power)) /\ exists bpr_quotient_bpcppc_source_choice_power_repeat_entry. bpr_power_code_bpcppc_source_choice_power = bpr_quotient_bpcppc_source_choice_power_repeat_entry * S ((S (bpr_power_index_bpcppc_source_choice_power)) * bpr_power_scale_bpcppc_source_choice_power) + (S (bpr_prefix_index_bpcppc_source))))) /\ (exists ff_u_bpcppc_source_choice_power_product ff_v_bpcppc_source_choice_power_product. ((((exists ff_h_bpcppc_source_choice_power_product_start. ff_h_bpcppc_source_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_source_choice_power_product)) /\ exists ff_q_bpcppc_source_choice_power_product_start. ff_u_bpcppc_source_choice_power_product = ff_q_bpcppc_source_choice_power_product_start * S ((S (0)) * ff_v_bpcppc_source_choice_power_product) + (1))) /\ ((((exists ff_h_bpcppc_source_choice_power_product_terminal. ff_h_bpcppc_source_choice_power_product_terminal + S (bpr_prefix_value_bpcppc_source) = S ((S (bpr_choice_exponent_bpcppc_source_choice)) * ff_v_bpcppc_source_choice_power_product)) /\ exists ff_q_bpcppc_source_choice_power_product_terminal. ff_u_bpcppc_source_choice_power_product = ff_q_bpcppc_source_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcppc_source_choice)) * ff_v_bpcppc_source_choice_power_product) + (bpr_prefix_value_bpcppc_source))) /\ forall ff_i_bpcppc_source_choice_power_product. (exists ff_lt_bpcppc_source_choice_power_product_bound. ff_lt_bpcppc_source_choice_power_product_bound + S ff_i_bpcppc_source_choice_power_product = bpr_choice_exponent_bpcppc_source_choice) -> exists ff_p_bpcppc_source_choice_power_product ff_r_bpcppc_source_choice_power_product ff_s_bpcppc_source_choice_power_product. ((((exists ff_h_bpcppc_source_choice_power_product_factor. ff_h_bpcppc_source_choice_power_product_factor + S (ff_p_bpcppc_source_choice_power_product) = S ((S (ff_i_bpcppc_source_choice_power_product)) * bpr_power_scale_bpcppc_source_choice_power)) /\ exists ff_q_bpcppc_source_choice_power_product_factor. bpr_power_code_bpcppc_source_choice_power = ff_q_bpcppc_source_choice_power_product_factor * S ((S (ff_i_bpcppc_source_choice_power_product)) * bpr_power_scale_bpcppc_source_choice_power) + (ff_p_bpcppc_source_choice_power_product))) /\ ((((exists ff_h_bpcppc_source_choice_power_product_partial. ff_h_bpcppc_source_choice_power_product_partial + S (ff_r_bpcppc_source_choice_power_product) = S ((S (ff_i_bpcppc_source_choice_power_product)) * ff_v_bpcppc_source_choice_power_product)) /\ exists ff_q_bpcppc_source_choice_power_product_partial. ff_u_bpcppc_source_choice_power_product = ff_q_bpcppc_source_choice_power_product_partial * S ((S (ff_i_bpcppc_source_choice_power_product)) * ff_v_bpcppc_source_choice_power_product) + (ff_r_bpcppc_source_choice_power_product))) /\ ((((exists ff_h_bpcppc_source_choice_power_product_successor. ff_h_bpcppc_source_choice_power_product_successor + S (ff_s_bpcppc_source_choice_power_product) = S ((S (S ff_i_bpcppc_source_choice_power_product)) * ff_v_bpcppc_source_choice_power_product)) /\ exists ff_q_bpcppc_source_choice_power_product_successor. ff_u_bpcppc_source_choice_power_product = ff_q_bpcppc_source_choice_power_product_successor * S ((S (S ff_i_bpcppc_source_choice_power_product)) * ff_v_bpcppc_source_choice_power_product) + (ff_s_bpcppc_source_choice_power_product))) /\ ff_s_bpcppc_source_choice_power_product = ff_r_bpcppc_source_choice_power_product * ff_p_bpcppc_source_choice_power_product)))))))))) \/ (~((~(S (bpr_prefix_index_bpcppc_source) = 1) /\ forall bpr_left_bpcppc_source_choice_prime bpr_right_bpcppc_source_choice_prime. S (bpr_prefix_index_bpcppc_source) = bpr_left_bpcppc_source_choice_prime * bpr_right_bpcppc_source_choice_prime -> bpr_left_bpcppc_source_choice_prime = 1 \/ bpr_right_bpcppc_source_choice_prime = 1)) /\ bpr_prefix_value_bpcppc_source = 1))))) -> (forall bpr_pair_left_index_bpcppc_result bpr_pair_right_index_bpcppc_result bpr_pair_left_bpcppc_result bpr_pair_right_bpcppc_result. (exists bpr_gap_bpcppc_result_left_bound. bpr_gap_bpcppc_result_left_bound + S (bpr_pair_left_index_bpcppc_result) = m) -> (exists bpr_gap_bpcppc_result_right_bound. bpr_gap_bpcppc_result_right_bound + S (bpr_pair_right_index_bpcppc_result) = m) -> (((exists bpr_height_bpcppc_result_left_entry. bpr_height_bpcppc_result_left_entry + S (bpr_pair_left_bpcppc_result) = S ((S (bpr_pair_left_index_bpcppc_result)) * c)) /\ exists bpr_quotient_bpcppc_result_left_entry. b = bpr_quotient_bpcppc_result_left_entry * S ((S (bpr_pair_left_index_bpcppc_result)) * c) + (bpr_pair_left_bpcppc_result))) -> (((exists bpr_height_bpcppc_result_right_entry. bpr_height_bpcppc_result_right_entry + S (bpr_pair_right_bpcppc_result) = S ((S (bpr_pair_right_index_bpcppc_result)) * c)) /\ exists bpr_quotient_bpcppc_result_right_entry. b = bpr_quotient_bpcppc_result_right_entry * S ((S (bpr_pair_right_index_bpcppc_result)) * c) + (bpr_pair_right_bpcppc_result))) -> ~(bpr_pair_left_index_bpcppc_result = bpr_pair_right_index_bpcppc_result) -> (forall bpr_coprime_divisor_bpcppc_result_coprime. (exists bpr_coprime_left_bpcppc_result_coprime. bpr_pair_left_bpcppc_result = bpr_coprime_divisor_bpcppc_result_coprime * bpr_coprime_left_bpcppc_result_coprime) -> (exists bpr_coprime_right_bpcppc_result_coprime. bpr_pair_right_bpcppc_result = bpr_coprime_divisor_bpcppc_result_coprime * bpr_coprime_right_bpcppc_result_coprime) -> bpr_coprime_divisor_bpcppc_result_coprime = 1))

Structural proof guide

Distinct contribution positions decode pairwise-coprime values.

Direct prerequisites: beta_at_unique, distinct_primes_coprime, coprime_one_left, coprime_one_right, coprime_powers. The authored body proceeds by case analysis (16), intermediate claims (7), equality transport (5).

Proof neighborhood

Direct dependencies

BT0042 beta_at_unique BT008S distinct_primes_coprime BT002Y coprime_one_left BT002X coprime_one_right BT00YV coprime_powers

Direct dependents

BT00YY prime_contribution_product_divides

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. 0002intro b
  3. 0003intro c
  4. 0004intro m
  5. 0005intro hprefix
  6. 0006intro i
  7. 0007intro j
  8. 0008intro a
  9. 0009intro z
  10. 0010intro hi
  11. 0011intro hj
  12. 0012intro ha
  13. 0013intro hz
  14. 0014intro hij
  15. 0015have hleft : exists x. (((exists bpr_height_bpcppc_left_entry. bpr_height_bpcppc_left_entry + S (x) = S ((S (i)) * c)) /\ exists bpr_quotient_bpcppc_left_entry. b = bpr_quotient_bpcppc_left_entry * S ((S (i)) * c) + (x))) /\ (((((~(S (i) = 1) /\ forall bpr_left_bpcppc_left_choice_prime bpr_right_bpcppc_left_choice_prime. S (i) = bpr_left_bpcppc_left_choice_prime * bpr_right_bpcppc_left_choice_prime -> bpr_left_bpcppc_left_choice_prime = 1 \/ bpr_right_bpcppc_left_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcppc_left_choice. ((((exists bpr_le_gap_bpcppc_left_choice_valuation_selected_bound. bpr_le_gap_bpcppc_left_choice_valuation_selected_bound + (bpr_choice_exponent_bpcppc_left_choice) = (n)) /\ (exists bpr_power_value_bpcppc_left_choice_valuation_selected. ((exists bpr_power_code_bpcppc_left_choice_valuation_selected_power bpr_power_scale_bpcppc_left_choice_valuation_selected_power. ((forall bpr_power_index_bpcppc_left_choice_valuation_selected_power. (exists bpr_gap_bpcppc_left_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcppc_left_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcppc_left_choice_valuation_selected_power) = bpr_choice_exponent_bpcppc_left_choice) -> (((exists bpr_height_bpcppc_left_choice_valuation_selected_power_repeat_entry. bpr_height_bpcppc_left_choice_valuation_selected_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_bpcppc_left_choice_valuation_selected_power)) * bpr_power_scale_bpcppc_left_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcppc_left_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcppc_left_choice_valuation_selected_power = bpr_quotient_bpcppc_left_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcppc_left_choice_valuation_selected_power)) * bpr_power_scale_bpcppc_left_choice_valuation_selected_power) + (S (i))))) /\ (exists ff_u_bpcppc_left_choice_valuation_selected_power_product ff_v_bpcppc_left_choice_valuation_selected_power_product. ((((exists ff_h_bpcppc_left_choice_valuation_selected_power_product_start. ff_h_bpcppc_left_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_left_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_selected_power_product_start. ff_u_bpcppc_left_choice_valuation_selected_power_product = ff_q_bpcppc_left_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcppc_left_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcppc_left_choice_valuation_selected_power_product_terminal. ff_h_bpcppc_left_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcppc_left_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcppc_left_choice)) * ff_v_bpcppc_left_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_selected_power_product_terminal. ff_u_bpcppc_left_choice_valuation_selected_power_product = ff_q_bpcppc_left_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcppc_left_choice)) * ff_v_bpcppc_left_choice_valuation_selected_power_product) + (bpr_power_value_bpcppc_left_choice_valuation_selected))) /\ forall ff_i_bpcppc_left_choice_valuation_selected_power_product. (exists ff_lt_bpcppc_left_choice_valuation_selected_power_product_bound. ff_lt_bpcppc_left_choice_valuation_selected_power_product_bound + S ff_i_bpcppc_left_choice_valuation_selected_power_product = bpr_choice_exponent_bpcppc_left_choice) -> exists ff_p_bpcppc_left_choice_valuation_selected_power_product ff_r_bpcppc_left_choice_valuation_selected_power_product ff_s_bpcppc_left_choice_valuation_selected_power_product. ((((exists ff_h_bpcppc_left_choice_valuation_selected_power_product_factor. ff_h_bpcppc_left_choice_valuation_selected_power_product_factor + S (ff_p_bpcppc_left_choice_valuation_selected_power_product) = S ((S (ff_i_bpcppc_left_choice_valuation_selected_power_product)) * bpr_power_scale_bpcppc_left_choice_valuation_selected_power)) /\ exists ff_q_bpcppc_left_choice_valuation_selected_power_product_factor. bpr_power_code_bpcppc_left_choice_valuation_selected_power = ff_q_bpcppc_left_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcppc_left_choice_valuation_selected_power_product)) * bpr_power_scale_bpcppc_left_choice_valuation_selected_power) + (ff_p_bpcppc_left_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcppc_left_choice_valuation_selected_power_product_partial. ff_h_bpcppc_left_choice_valuation_selected_power_product_partial + S (ff_r_bpcppc_left_choice_valuation_selected_power_product) = S ((S (ff_i_bpcppc_left_choice_valuation_selected_power_product)) * ff_v_bpcppc_left_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_selected_power_product_partial. ff_u_bpcppc_left_choice_valuation_selected_power_product = ff_q_bpcppc_left_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcppc_left_choice_valuation_selected_power_product)) * ff_v_bpcppc_left_choice_valuation_selected_power_product) + (ff_r_bpcppc_left_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcppc_left_choice_valuation_selected_power_product_successor. ff_h_bpcppc_left_choice_valuation_selected_power_product_successor + S (ff_s_bpcppc_left_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcppc_left_choice_valuation_selected_power_product)) * ff_v_bpcppc_left_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_selected_power_product_successor. ff_u_bpcppc_left_choice_valuation_selected_power_product = ff_q_bpcppc_left_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcppc_left_choice_valuation_selected_power_product)) * ff_v_bpcppc_left_choice_valuation_selected_power_product) + (ff_s_bpcppc_left_choice_valuation_selected_power_product))) /\ ff_s_bpcppc_left_choice_valuation_selected_power_product = ff_r_bpcppc_left_choice_valuation_selected_power_product * ff_p_bpcppc_left_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcppc_left_choice_valuation_selected_divides. n = (bpr_power_value_bpcppc_left_choice_valuation_selected) * bpr_divides_quotient_bpcppc_left_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcppc_left_choice_valuation. (exists bpr_le_gap_bpcppc_left_choice_valuation_candidate_bound. bpr_le_gap_bpcppc_left_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcppc_left_choice_valuation) = (n)) -> (exists bpr_power_value_bpcppc_left_choice_valuation_candidate. ((exists bpr_power_code_bpcppc_left_choice_valuation_candidate_power bpr_power_scale_bpcppc_left_choice_valuation_candidate_power. ((forall bpr_power_index_bpcppc_left_choice_valuation_candidate_power. (exists bpr_gap_bpcppc_left_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcppc_left_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcppc_left_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcppc_left_choice_valuation) -> (((exists bpr_height_bpcppc_left_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcppc_left_choice_valuation_candidate_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_bpcppc_left_choice_valuation_candidate_power)) * bpr_power_scale_bpcppc_left_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcppc_left_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcppc_left_choice_valuation_candidate_power = bpr_quotient_bpcppc_left_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcppc_left_choice_valuation_candidate_power)) * bpr_power_scale_bpcppc_left_choice_valuation_candidate_power) + (S (i))))) /\ (exists ff_u_bpcppc_left_choice_valuation_candidate_power_product ff_v_bpcppc_left_choice_valuation_candidate_power_product. ((((exists ff_h_bpcppc_left_choice_valuation_candidate_power_product_start. ff_h_bpcppc_left_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_candidate_power_product_start. ff_u_bpcppc_left_choice_valuation_candidate_power_product = ff_q_bpcppc_left_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcppc_left_choice_valuation_candidate_power_product_terminal. ff_h_bpcppc_left_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcppc_left_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcppc_left_choice_valuation)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_candidate_power_product_terminal. ff_u_bpcppc_left_choice_valuation_candidate_power_product = ff_q_bpcppc_left_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcppc_left_choice_valuation)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product) + (bpr_power_value_bpcppc_left_choice_valuation_candidate))) /\ forall ff_i_bpcppc_left_choice_valuation_candidate_power_product. (exists ff_lt_bpcppc_left_choice_valuation_candidate_power_product_bound. ff_lt_bpcppc_left_choice_valuation_candidate_power_product_bound + S ff_i_bpcppc_left_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcppc_left_choice_valuation) -> exists ff_p_bpcppc_left_choice_valuation_candidate_power_product ff_r_bpcppc_left_choice_valuation_candidate_power_product ff_s_bpcppc_left_choice_valuation_candidate_power_product. ((((exists ff_h_bpcppc_left_choice_valuation_candidate_power_product_factor. ff_h_bpcppc_left_choice_valuation_candidate_power_product_factor + S (ff_p_bpcppc_left_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcppc_left_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcppc_left_choice_valuation_candidate_power)) /\ exists ff_q_bpcppc_left_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcppc_left_choice_valuation_candidate_power = ff_q_bpcppc_left_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcppc_left_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcppc_left_choice_valuation_candidate_power) + (ff_p_bpcppc_left_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcppc_left_choice_valuation_candidate_power_product_partial. ff_h_bpcppc_left_choice_valuation_candidate_power_product_partial + S (ff_r_bpcppc_left_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcppc_left_choice_valuation_candidate_power_product)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_candidate_power_product_partial. ff_u_bpcppc_left_choice_valuation_candidate_power_product = ff_q_bpcppc_left_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcppc_left_choice_valuation_candidate_power_product)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product) + (ff_r_bpcppc_left_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcppc_left_choice_valuation_candidate_power_product_successor. ff_h_bpcppc_left_choice_valuation_candidate_power_product_successor + S (ff_s_bpcppc_left_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcppc_left_choice_valuation_candidate_power_product)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_left_choice_valuation_candidate_power_product_successor. ff_u_bpcppc_left_choice_valuation_candidate_power_product = ff_q_bpcppc_left_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcppc_left_choice_valuation_candidate_power_product)) * ff_v_bpcppc_left_choice_valuation_candidate_power_product) + (ff_s_bpcppc_left_choice_valuation_candidate_power_product))) /\ ff_s_bpcppc_left_choice_valuation_candidate_power_product = ff_r_bpcppc_left_choice_valuation_candidate_power_product * ff_p_bpcppc_left_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcppc_left_choice_valuation_candidate_divides. n = (bpr_power_value_bpcppc_left_choice_valuation_candidate) * bpr_divides_quotient_bpcppc_left_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcppc_left_choice_valuation_candidate_below. bpr_le_gap_bpcppc_left_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcppc_left_choice_valuation) = (bpr_choice_exponent_bpcppc_left_choice))) /\ (exists bpr_power_code_bpcppc_left_choice_power bpr_power_scale_bpcppc_left_choice_power. ((forall bpr_power_index_bpcppc_left_choice_power. (exists bpr_gap_bpcppc_left_choice_power_repeat_bound. bpr_gap_bpcppc_left_choice_power_repeat_bound + S (bpr_power_index_bpcppc_left_choice_power) = bpr_choice_exponent_bpcppc_left_choice) -> (((exists bpr_height_bpcppc_left_choice_power_repeat_entry. bpr_height_bpcppc_left_choice_power_repeat_entry + S (S (i)) = S ((S (bpr_power_index_bpcppc_left_choice_power)) * bpr_power_scale_bpcppc_left_choice_power)) /\ exists bpr_quotient_bpcppc_left_choice_power_repeat_entry. bpr_power_code_bpcppc_left_choice_power = bpr_quotient_bpcppc_left_choice_power_repeat_entry * S ((S (bpr_power_index_bpcppc_left_choice_power)) * bpr_power_scale_bpcppc_left_choice_power) + (S (i))))) /\ (exists ff_u_bpcppc_left_choice_power_product ff_v_bpcppc_left_choice_power_product. ((((exists ff_h_bpcppc_left_choice_power_product_start. ff_h_bpcppc_left_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_left_choice_power_product)) /\ exists ff_q_bpcppc_left_choice_power_product_start. ff_u_bpcppc_left_choice_power_product = ff_q_bpcppc_left_choice_power_product_start * S ((S (0)) * ff_v_bpcppc_left_choice_power_product) + (1))) /\ ((((exists ff_h_bpcppc_left_choice_power_product_terminal. ff_h_bpcppc_left_choice_power_product_terminal + S (x) = S ((S (bpr_choice_exponent_bpcppc_left_choice)) * ff_v_bpcppc_left_choice_power_product)) /\ exists ff_q_bpcppc_left_choice_power_product_terminal. ff_u_bpcppc_left_choice_power_product = ff_q_bpcppc_left_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcppc_left_choice)) * ff_v_bpcppc_left_choice_power_product) + (x))) /\ forall ff_i_bpcppc_left_choice_power_product. (exists ff_lt_bpcppc_left_choice_power_product_bound. ff_lt_bpcppc_left_choice_power_product_bound + S ff_i_bpcppc_left_choice_power_product = bpr_choice_exponent_bpcppc_left_choice) -> exists ff_p_bpcppc_left_choice_power_product ff_r_bpcppc_left_choice_power_product ff_s_bpcppc_left_choice_power_product. ((((exists ff_h_bpcppc_left_choice_power_product_factor. ff_h_bpcppc_left_choice_power_product_factor + S (ff_p_bpcppc_left_choice_power_product) = S ((S (ff_i_bpcppc_left_choice_power_product)) * bpr_power_scale_bpcppc_left_choice_power)) /\ exists ff_q_bpcppc_left_choice_power_product_factor. bpr_power_code_bpcppc_left_choice_power = ff_q_bpcppc_left_choice_power_product_factor * S ((S (ff_i_bpcppc_left_choice_power_product)) * bpr_power_scale_bpcppc_left_choice_power) + (ff_p_bpcppc_left_choice_power_product))) /\ ((((exists ff_h_bpcppc_left_choice_power_product_partial. ff_h_bpcppc_left_choice_power_product_partial + S (ff_r_bpcppc_left_choice_power_product) = S ((S (ff_i_bpcppc_left_choice_power_product)) * ff_v_bpcppc_left_choice_power_product)) /\ exists ff_q_bpcppc_left_choice_power_product_partial. ff_u_bpcppc_left_choice_power_product = ff_q_bpcppc_left_choice_power_product_partial * S ((S (ff_i_bpcppc_left_choice_power_product)) * ff_v_bpcppc_left_choice_power_product) + (ff_r_bpcppc_left_choice_power_product))) /\ ((((exists ff_h_bpcppc_left_choice_power_product_successor. ff_h_bpcppc_left_choice_power_product_successor + S (ff_s_bpcppc_left_choice_power_product) = S ((S (S ff_i_bpcppc_left_choice_power_product)) * ff_v_bpcppc_left_choice_power_product)) /\ exists ff_q_bpcppc_left_choice_power_product_successor. ff_u_bpcppc_left_choice_power_product = ff_q_bpcppc_left_choice_power_product_successor * S ((S (S ff_i_bpcppc_left_choice_power_product)) * ff_v_bpcppc_left_choice_power_product) + (ff_s_bpcppc_left_choice_power_product))) /\ ff_s_bpcppc_left_choice_power_product = ff_r_bpcppc_left_choice_power_product * ff_p_bpcppc_left_choice_power_product)))))))))) \/ (~((~(S (i) = 1) /\ forall bpr_left_bpcppc_left_choice_prime bpr_right_bpcppc_left_choice_prime. S (i) = bpr_left_bpcppc_left_choice_prime * bpr_right_bpcppc_left_choice_prime -> bpr_left_bpcppc_left_choice_prime = 1 \/ bpr_right_bpcppc_left_choice_prime = 1)) /\ x = 1)))
  16. 0016apply hprefix
  17. 0017exact hi
  18. 0018cases hleft
  19. 0019cases hleft_witness
  20. 0020have hright : exists x1. (((exists bpr_height_bpcppc_right_entry. bpr_height_bpcppc_right_entry + S (x1) = S ((S (j)) * c)) /\ exists bpr_quotient_bpcppc_right_entry. b = bpr_quotient_bpcppc_right_entry * S ((S (j)) * c) + (x1))) /\ (((((~(S (j) = 1) /\ forall bpr_left_bpcppc_right_choice_prime bpr_right_bpcppc_right_choice_prime. S (j) = bpr_left_bpcppc_right_choice_prime * bpr_right_bpcppc_right_choice_prime -> bpr_left_bpcppc_right_choice_prime = 1 \/ bpr_right_bpcppc_right_choice_prime = 1)) /\ exists bpr_choice_exponent_bpcppc_right_choice. ((((exists bpr_le_gap_bpcppc_right_choice_valuation_selected_bound. bpr_le_gap_bpcppc_right_choice_valuation_selected_bound + (bpr_choice_exponent_bpcppc_right_choice) = (n)) /\ (exists bpr_power_value_bpcppc_right_choice_valuation_selected. ((exists bpr_power_code_bpcppc_right_choice_valuation_selected_power bpr_power_scale_bpcppc_right_choice_valuation_selected_power. ((forall bpr_power_index_bpcppc_right_choice_valuation_selected_power. (exists bpr_gap_bpcppc_right_choice_valuation_selected_power_repeat_bound. bpr_gap_bpcppc_right_choice_valuation_selected_power_repeat_bound + S (bpr_power_index_bpcppc_right_choice_valuation_selected_power) = bpr_choice_exponent_bpcppc_right_choice) -> (((exists bpr_height_bpcppc_right_choice_valuation_selected_power_repeat_entry. bpr_height_bpcppc_right_choice_valuation_selected_power_repeat_entry + S (S (j)) = S ((S (bpr_power_index_bpcppc_right_choice_valuation_selected_power)) * bpr_power_scale_bpcppc_right_choice_valuation_selected_power)) /\ exists bpr_quotient_bpcppc_right_choice_valuation_selected_power_repeat_entry. bpr_power_code_bpcppc_right_choice_valuation_selected_power = bpr_quotient_bpcppc_right_choice_valuation_selected_power_repeat_entry * S ((S (bpr_power_index_bpcppc_right_choice_valuation_selected_power)) * bpr_power_scale_bpcppc_right_choice_valuation_selected_power) + (S (j))))) /\ (exists ff_u_bpcppc_right_choice_valuation_selected_power_product ff_v_bpcppc_right_choice_valuation_selected_power_product. ((((exists ff_h_bpcppc_right_choice_valuation_selected_power_product_start. ff_h_bpcppc_right_choice_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_right_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_selected_power_product_start. ff_u_bpcppc_right_choice_valuation_selected_power_product = ff_q_bpcppc_right_choice_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpcppc_right_choice_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bpcppc_right_choice_valuation_selected_power_product_terminal. ff_h_bpcppc_right_choice_valuation_selected_power_product_terminal + S (bpr_power_value_bpcppc_right_choice_valuation_selected) = S ((S (bpr_choice_exponent_bpcppc_right_choice)) * ff_v_bpcppc_right_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_selected_power_product_terminal. ff_u_bpcppc_right_choice_valuation_selected_power_product = ff_q_bpcppc_right_choice_valuation_selected_power_product_terminal * S ((S (bpr_choice_exponent_bpcppc_right_choice)) * ff_v_bpcppc_right_choice_valuation_selected_power_product) + (bpr_power_value_bpcppc_right_choice_valuation_selected))) /\ forall ff_i_bpcppc_right_choice_valuation_selected_power_product. (exists ff_lt_bpcppc_right_choice_valuation_selected_power_product_bound. ff_lt_bpcppc_right_choice_valuation_selected_power_product_bound + S ff_i_bpcppc_right_choice_valuation_selected_power_product = bpr_choice_exponent_bpcppc_right_choice) -> exists ff_p_bpcppc_right_choice_valuation_selected_power_product ff_r_bpcppc_right_choice_valuation_selected_power_product ff_s_bpcppc_right_choice_valuation_selected_power_product. ((((exists ff_h_bpcppc_right_choice_valuation_selected_power_product_factor. ff_h_bpcppc_right_choice_valuation_selected_power_product_factor + S (ff_p_bpcppc_right_choice_valuation_selected_power_product) = S ((S (ff_i_bpcppc_right_choice_valuation_selected_power_product)) * bpr_power_scale_bpcppc_right_choice_valuation_selected_power)) /\ exists ff_q_bpcppc_right_choice_valuation_selected_power_product_factor. bpr_power_code_bpcppc_right_choice_valuation_selected_power = ff_q_bpcppc_right_choice_valuation_selected_power_product_factor * S ((S (ff_i_bpcppc_right_choice_valuation_selected_power_product)) * bpr_power_scale_bpcppc_right_choice_valuation_selected_power) + (ff_p_bpcppc_right_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcppc_right_choice_valuation_selected_power_product_partial. ff_h_bpcppc_right_choice_valuation_selected_power_product_partial + S (ff_r_bpcppc_right_choice_valuation_selected_power_product) = S ((S (ff_i_bpcppc_right_choice_valuation_selected_power_product)) * ff_v_bpcppc_right_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_selected_power_product_partial. ff_u_bpcppc_right_choice_valuation_selected_power_product = ff_q_bpcppc_right_choice_valuation_selected_power_product_partial * S ((S (ff_i_bpcppc_right_choice_valuation_selected_power_product)) * ff_v_bpcppc_right_choice_valuation_selected_power_product) + (ff_r_bpcppc_right_choice_valuation_selected_power_product))) /\ ((((exists ff_h_bpcppc_right_choice_valuation_selected_power_product_successor. ff_h_bpcppc_right_choice_valuation_selected_power_product_successor + S (ff_s_bpcppc_right_choice_valuation_selected_power_product) = S ((S (S ff_i_bpcppc_right_choice_valuation_selected_power_product)) * ff_v_bpcppc_right_choice_valuation_selected_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_selected_power_product_successor. ff_u_bpcppc_right_choice_valuation_selected_power_product = ff_q_bpcppc_right_choice_valuation_selected_power_product_successor * S ((S (S ff_i_bpcppc_right_choice_valuation_selected_power_product)) * ff_v_bpcppc_right_choice_valuation_selected_power_product) + (ff_s_bpcppc_right_choice_valuation_selected_power_product))) /\ ff_s_bpcppc_right_choice_valuation_selected_power_product = ff_r_bpcppc_right_choice_valuation_selected_power_product * ff_p_bpcppc_right_choice_valuation_selected_power_product)))))))) /\ (exists bpr_divides_quotient_bpcppc_right_choice_valuation_selected_divides. n = (bpr_power_value_bpcppc_right_choice_valuation_selected) * bpr_divides_quotient_bpcppc_right_choice_valuation_selected_divides)))) /\ forall bpr_valuation_candidate_bpcppc_right_choice_valuation. (exists bpr_le_gap_bpcppc_right_choice_valuation_candidate_bound. bpr_le_gap_bpcppc_right_choice_valuation_candidate_bound + (bpr_valuation_candidate_bpcppc_right_choice_valuation) = (n)) -> (exists bpr_power_value_bpcppc_right_choice_valuation_candidate. ((exists bpr_power_code_bpcppc_right_choice_valuation_candidate_power bpr_power_scale_bpcppc_right_choice_valuation_candidate_power. ((forall bpr_power_index_bpcppc_right_choice_valuation_candidate_power. (exists bpr_gap_bpcppc_right_choice_valuation_candidate_power_repeat_bound. bpr_gap_bpcppc_right_choice_valuation_candidate_power_repeat_bound + S (bpr_power_index_bpcppc_right_choice_valuation_candidate_power) = bpr_valuation_candidate_bpcppc_right_choice_valuation) -> (((exists bpr_height_bpcppc_right_choice_valuation_candidate_power_repeat_entry. bpr_height_bpcppc_right_choice_valuation_candidate_power_repeat_entry + S (S (j)) = S ((S (bpr_power_index_bpcppc_right_choice_valuation_candidate_power)) * bpr_power_scale_bpcppc_right_choice_valuation_candidate_power)) /\ exists bpr_quotient_bpcppc_right_choice_valuation_candidate_power_repeat_entry. bpr_power_code_bpcppc_right_choice_valuation_candidate_power = bpr_quotient_bpcppc_right_choice_valuation_candidate_power_repeat_entry * S ((S (bpr_power_index_bpcppc_right_choice_valuation_candidate_power)) * bpr_power_scale_bpcppc_right_choice_valuation_candidate_power) + (S (j))))) /\ (exists ff_u_bpcppc_right_choice_valuation_candidate_power_product ff_v_bpcppc_right_choice_valuation_candidate_power_product. ((((exists ff_h_bpcppc_right_choice_valuation_candidate_power_product_start. ff_h_bpcppc_right_choice_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_candidate_power_product_start. ff_u_bpcppc_right_choice_valuation_candidate_power_product = ff_q_bpcppc_right_choice_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bpcppc_right_choice_valuation_candidate_power_product_terminal. ff_h_bpcppc_right_choice_valuation_candidate_power_product_terminal + S (bpr_power_value_bpcppc_right_choice_valuation_candidate) = S ((S (bpr_valuation_candidate_bpcppc_right_choice_valuation)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_candidate_power_product_terminal. ff_u_bpcppc_right_choice_valuation_candidate_power_product = ff_q_bpcppc_right_choice_valuation_candidate_power_product_terminal * S ((S (bpr_valuation_candidate_bpcppc_right_choice_valuation)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product) + (bpr_power_value_bpcppc_right_choice_valuation_candidate))) /\ forall ff_i_bpcppc_right_choice_valuation_candidate_power_product. (exists ff_lt_bpcppc_right_choice_valuation_candidate_power_product_bound. ff_lt_bpcppc_right_choice_valuation_candidate_power_product_bound + S ff_i_bpcppc_right_choice_valuation_candidate_power_product = bpr_valuation_candidate_bpcppc_right_choice_valuation) -> exists ff_p_bpcppc_right_choice_valuation_candidate_power_product ff_r_bpcppc_right_choice_valuation_candidate_power_product ff_s_bpcppc_right_choice_valuation_candidate_power_product. ((((exists ff_h_bpcppc_right_choice_valuation_candidate_power_product_factor. ff_h_bpcppc_right_choice_valuation_candidate_power_product_factor + S (ff_p_bpcppc_right_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcppc_right_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcppc_right_choice_valuation_candidate_power)) /\ exists ff_q_bpcppc_right_choice_valuation_candidate_power_product_factor. bpr_power_code_bpcppc_right_choice_valuation_candidate_power = ff_q_bpcppc_right_choice_valuation_candidate_power_product_factor * S ((S (ff_i_bpcppc_right_choice_valuation_candidate_power_product)) * bpr_power_scale_bpcppc_right_choice_valuation_candidate_power) + (ff_p_bpcppc_right_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcppc_right_choice_valuation_candidate_power_product_partial. ff_h_bpcppc_right_choice_valuation_candidate_power_product_partial + S (ff_r_bpcppc_right_choice_valuation_candidate_power_product) = S ((S (ff_i_bpcppc_right_choice_valuation_candidate_power_product)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_candidate_power_product_partial. ff_u_bpcppc_right_choice_valuation_candidate_power_product = ff_q_bpcppc_right_choice_valuation_candidate_power_product_partial * S ((S (ff_i_bpcppc_right_choice_valuation_candidate_power_product)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product) + (ff_r_bpcppc_right_choice_valuation_candidate_power_product))) /\ ((((exists ff_h_bpcppc_right_choice_valuation_candidate_power_product_successor. ff_h_bpcppc_right_choice_valuation_candidate_power_product_successor + S (ff_s_bpcppc_right_choice_valuation_candidate_power_product) = S ((S (S ff_i_bpcppc_right_choice_valuation_candidate_power_product)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product)) /\ exists ff_q_bpcppc_right_choice_valuation_candidate_power_product_successor. ff_u_bpcppc_right_choice_valuation_candidate_power_product = ff_q_bpcppc_right_choice_valuation_candidate_power_product_successor * S ((S (S ff_i_bpcppc_right_choice_valuation_candidate_power_product)) * ff_v_bpcppc_right_choice_valuation_candidate_power_product) + (ff_s_bpcppc_right_choice_valuation_candidate_power_product))) /\ ff_s_bpcppc_right_choice_valuation_candidate_power_product = ff_r_bpcppc_right_choice_valuation_candidate_power_product * ff_p_bpcppc_right_choice_valuation_candidate_power_product)))))))) /\ (exists bpr_divides_quotient_bpcppc_right_choice_valuation_candidate_divides. n = (bpr_power_value_bpcppc_right_choice_valuation_candidate) * bpr_divides_quotient_bpcppc_right_choice_valuation_candidate_divides))) -> (exists bpr_le_gap_bpcppc_right_choice_valuation_candidate_below. bpr_le_gap_bpcppc_right_choice_valuation_candidate_below + (bpr_valuation_candidate_bpcppc_right_choice_valuation) = (bpr_choice_exponent_bpcppc_right_choice))) /\ (exists bpr_power_code_bpcppc_right_choice_power bpr_power_scale_bpcppc_right_choice_power. ((forall bpr_power_index_bpcppc_right_choice_power. (exists bpr_gap_bpcppc_right_choice_power_repeat_bound. bpr_gap_bpcppc_right_choice_power_repeat_bound + S (bpr_power_index_bpcppc_right_choice_power) = bpr_choice_exponent_bpcppc_right_choice) -> (((exists bpr_height_bpcppc_right_choice_power_repeat_entry. bpr_height_bpcppc_right_choice_power_repeat_entry + S (S (j)) = S ((S (bpr_power_index_bpcppc_right_choice_power)) * bpr_power_scale_bpcppc_right_choice_power)) /\ exists bpr_quotient_bpcppc_right_choice_power_repeat_entry. bpr_power_code_bpcppc_right_choice_power = bpr_quotient_bpcppc_right_choice_power_repeat_entry * S ((S (bpr_power_index_bpcppc_right_choice_power)) * bpr_power_scale_bpcppc_right_choice_power) + (S (j))))) /\ (exists ff_u_bpcppc_right_choice_power_product ff_v_bpcppc_right_choice_power_product. ((((exists ff_h_bpcppc_right_choice_power_product_start. ff_h_bpcppc_right_choice_power_product_start + S (1) = S ((S (0)) * ff_v_bpcppc_right_choice_power_product)) /\ exists ff_q_bpcppc_right_choice_power_product_start. ff_u_bpcppc_right_choice_power_product = ff_q_bpcppc_right_choice_power_product_start * S ((S (0)) * ff_v_bpcppc_right_choice_power_product) + (1))) /\ ((((exists ff_h_bpcppc_right_choice_power_product_terminal. ff_h_bpcppc_right_choice_power_product_terminal + S (x1) = S ((S (bpr_choice_exponent_bpcppc_right_choice)) * ff_v_bpcppc_right_choice_power_product)) /\ exists ff_q_bpcppc_right_choice_power_product_terminal. ff_u_bpcppc_right_choice_power_product = ff_q_bpcppc_right_choice_power_product_terminal * S ((S (bpr_choice_exponent_bpcppc_right_choice)) * ff_v_bpcppc_right_choice_power_product) + (x1))) /\ forall ff_i_bpcppc_right_choice_power_product. (exists ff_lt_bpcppc_right_choice_power_product_bound. ff_lt_bpcppc_right_choice_power_product_bound + S ff_i_bpcppc_right_choice_power_product = bpr_choice_exponent_bpcppc_right_choice) -> exists ff_p_bpcppc_right_choice_power_product ff_r_bpcppc_right_choice_power_product ff_s_bpcppc_right_choice_power_product. ((((exists ff_h_bpcppc_right_choice_power_product_factor. ff_h_bpcppc_right_choice_power_product_factor + S (ff_p_bpcppc_right_choice_power_product) = S ((S (ff_i_bpcppc_right_choice_power_product)) * bpr_power_scale_bpcppc_right_choice_power)) /\ exists ff_q_bpcppc_right_choice_power_product_factor. bpr_power_code_bpcppc_right_choice_power = ff_q_bpcppc_right_choice_power_product_factor * S ((S (ff_i_bpcppc_right_choice_power_product)) * bpr_power_scale_bpcppc_right_choice_power) + (ff_p_bpcppc_right_choice_power_product))) /\ ((((exists ff_h_bpcppc_right_choice_power_product_partial. ff_h_bpcppc_right_choice_power_product_partial + S (ff_r_bpcppc_right_choice_power_product) = S ((S (ff_i_bpcppc_right_choice_power_product)) * ff_v_bpcppc_right_choice_power_product)) /\ exists ff_q_bpcppc_right_choice_power_product_partial. ff_u_bpcppc_right_choice_power_product = ff_q_bpcppc_right_choice_power_product_partial * S ((S (ff_i_bpcppc_right_choice_power_product)) * ff_v_bpcppc_right_choice_power_product) + (ff_r_bpcppc_right_choice_power_product))) /\ ((((exists ff_h_bpcppc_right_choice_power_product_successor. ff_h_bpcppc_right_choice_power_product_successor + S (ff_s_bpcppc_right_choice_power_product) = S ((S (S ff_i_bpcppc_right_choice_power_product)) * ff_v_bpcppc_right_choice_power_product)) /\ exists ff_q_bpcppc_right_choice_power_product_successor. ff_u_bpcppc_right_choice_power_product = ff_q_bpcppc_right_choice_power_product_successor * S ((S (S ff_i_bpcppc_right_choice_power_product)) * ff_v_bpcppc_right_choice_power_product) + (ff_s_bpcppc_right_choice_power_product))) /\ ff_s_bpcppc_right_choice_power_product = ff_r_bpcppc_right_choice_power_product * ff_p_bpcppc_right_choice_power_product)))))))))) \/ (~((~(S (j) = 1) /\ forall bpr_left_bpcppc_right_choice_prime bpr_right_bpcppc_right_choice_prime. S (j) = bpr_left_bpcppc_right_choice_prime * bpr_right_bpcppc_right_choice_prime -> bpr_left_bpcppc_right_choice_prime = 1 \/ bpr_right_bpcppc_right_choice_prime = 1)) /\ x1 = 1)))
  21. 0021apply hprefix
  22. 0022exact hj
  23. 0023cases hright
  24. 0024cases hright_witness
  25. 0025have hax : x = a
  26. 0026apply beta_at_unique
  27. 0027exact hleft_witness_left
  28. 0028exact ha
  29. 0029have hxz : x1 = z
  30. 0030apply beta_at_unique
  31. 0031exact hright_witness_left
  32. 0032exact hz
  33. 0033have hcoprime : forall d. (exists u. x = d * u) -> (exists v. x1 = d * v) -> d = 1
  34. 0034cases hleft_witness_right
  35. 0035cases hleft_witness_right_left
  36. 0036cases hleft_witness_right_left_right
  37. 0037cases hleft_witness_right_left_right_witness
  38. 0038cases hright_witness_right
  39. 0039cases hright_witness_right_left
  40. 0040cases hright_witness_right_left_right
  41. 0041cases hright_witness_right_left_right_witness
  42. 0042have hbase_ne : ~(S i = S j)
  43. 0043intro hbase
  44. 0044apply hij
  45. 0045apply PA2
  46. 0046exact hbase
  47. 0047have hbase_coprime : forall d. (exists u. S i = d * u) -> (exists v. S j = d * v) -> d = 1
  48. 0048specialize distinct_primes_coprime (S i)
  49. 0049specialize distinct_primes_coprime (S j)
  50. 0050apply distinct_primes_coprime
  51. 0051exact hleft_witness_right_left_left
  52. 0052exact hright_witness_right_left_left
  53. 0053exact hbase_ne
  54. 0054specialize coprime_powers (S i)
  55. 0055specialize coprime_powers (S j)
  56. 0056specialize coprime_powers x2
  57. 0057specialize coprime_powers x3
  58. 0058specialize coprime_powers x
  59. 0059specialize coprime_powers x1
  60. 0060apply coprime_powers
  61. 0061exact hbase_coprime
  62. 0062exact hleft_witness_right_left_right_witness_right
  63. 0063exact hright_witness_right_left_right_witness_right
  64. 0064cases hright_witness_right_right
  65. 0065rewrite hright_witness_right_right_right
  66. 0066specialize coprime_one_right x
  67. 0067apply coprime_one_right
  68. 0068cases hleft_witness_right_right
  69. 0069cases hright_witness_right
  70. 0070rewrite hleft_witness_right_right_right
  71. 0071specialize coprime_one_left x1
  72. 0072apply coprime_one_left
  73. 0073cases hright_witness_right_right
  74. 0074rewrite hleft_witness_right_right_right
  75. 0075specialize coprime_one_left x1
  76. 0076apply coprime_one_left
  77. 0077rewrite <- hax
  78. 0078rewrite <- hxz
  79. 0079exact hcoprime