binary_modular_exponent_coded_execution_power_correct

Exact expanded first-order arithmetic statement

forall n a m l b c r. (exists ff_modulus_gap_binary_bd_modulus. ff_modulus_gap_binary_bd_modulus + S 1 = m) -> ((((((n) = 0 /\ (l) = 1) \/ exists ff_exponent_bl_bd_modular_canonical_length ff_lower_bl_bd_modular_canonical_length ff_upper_bl_bd_modular_canonical_length. (((l) = S ff_exponent_bl_bd_modular_canonical_length) /\ ((exists ff_positive_bl_bd_modular_canonical_length. ff_positive_bl_bd_modular_canonical_length + 1 = (n)) /\ ((exists pa_b_bl_bd_modular_canonical_length_lower pa_c_bl_bd_modular_canonical_length_lower. ((forall pa_i_bl_bd_modular_canonical_length_lower_repeat. (exists pa_lt_bl_bd_modular_canonical_length_lower_repeat_bound. pa_lt_bl_bd_modular_canonical_length_lower_repeat_bound + S pa_i_bl_bd_modular_canonical_length_lower_repeat = ff_exponent_bl_bd_modular_canonical_length) -> (((exists pa_h_bl_bd_modular_canonical_length_lower_repeat_decoded. pa_h_bl_bd_modular_canonical_length_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_bd_modular_canonical_length_lower_repeat)) * pa_c_bl_bd_modular_canonical_length_lower)) /\ exists pa_q_bl_bd_modular_canonical_length_lower_repeat_decoded. pa_b_bl_bd_modular_canonical_length_lower = pa_q_bl_bd_modular_canonical_length_lower_repeat_decoded * S ((S (pa_i_bl_bd_modular_canonical_length_lower_repeat)) * pa_c_bl_bd_modular_canonical_length_lower) + (2)))) /\ (exists pa_u_bl_bd_modular_canonical_length_lower_product pa_v_bl_bd_modular_canonical_length_lower_product. ((((exists pa_h_bl_bd_modular_canonical_length_lower_product_start. pa_h_bl_bd_modular_canonical_length_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_bd_modular_canonical_length_lower_product)) /\ exists pa_q_bl_bd_modular_canonical_length_lower_product_start. pa_u_bl_bd_modular_canonical_length_lower_product = pa_q_bl_bd_modular_canonical_length_lower_product_start * S ((S (0)) * pa_v_bl_bd_modular_canonical_length_lower_product) + (1))) /\ ((((exists pa_h_bl_bd_modular_canonical_length_lower_product_terminal. pa_h_bl_bd_modular_canonical_length_lower_product_terminal + S (ff_lower_bl_bd_modular_canonical_length) = S ((S (ff_exponent_bl_bd_modular_canonical_length)) * pa_v_bl_bd_modular_canonical_length_lower_product)) /\ exists pa_q_bl_bd_modular_canonical_length_lower_product_terminal. pa_u_bl_bd_modular_canonical_length_lower_product = pa_q_bl_bd_modular_canonical_length_lower_product_terminal * S ((S (ff_exponent_bl_bd_modular_canonical_length)) * pa_v_bl_bd_modular_canonical_length_lower_product) + (ff_lower_bl_bd_modular_canonical_length))) /\ forall pa_i_bl_bd_modular_canonical_length_lower_product. (exists pa_lt_bl_bd_modular_canonical_length_lower_product_bound. pa_lt_bl_bd_modular_canonical_length_lower_product_bound + S pa_i_bl_bd_modular_canonical_length_lower_product = ff_exponent_bl_bd_modular_canonical_length) -> exists pa_p_bl_bd_modular_canonical_length_lower_product pa_r_bl_bd_modular_canonical_length_lower_product pa_s_bl_bd_modular_canonical_length_lower_product. ((((exists pa_h_bl_bd_modular_canonical_length_lower_product_factor. pa_h_bl_bd_modular_canonical_length_lower_product_factor + S (pa_p_bl_bd_modular_canonical_length_lower_product) = S ((S (pa_i_bl_bd_modular_canonical_length_lower_product)) * pa_c_bl_bd_modular_canonical_length_lower)) /\ exists pa_q_bl_bd_modular_canonical_length_lower_product_factor. pa_b_bl_bd_modular_canonical_length_lower = pa_q_bl_bd_modular_canonical_length_lower_product_factor * S ((S (pa_i_bl_bd_modular_canonical_length_lower_product)) * pa_c_bl_bd_modular_canonical_length_lower) + (pa_p_bl_bd_modular_canonical_length_lower_product))) /\ ((((exists pa_h_bl_bd_modular_canonical_length_lower_product_partial. pa_h_bl_bd_modular_canonical_length_lower_product_partial + S (pa_r_bl_bd_modular_canonical_length_lower_product) = S ((S (pa_i_bl_bd_modular_canonical_length_lower_product)) * pa_v_bl_bd_modular_canonical_length_lower_product)) /\ exists pa_q_bl_bd_modular_canonical_length_lower_product_partial. pa_u_bl_bd_modular_canonical_length_lower_product = pa_q_bl_bd_modular_canonical_length_lower_product_partial * S ((S (pa_i_bl_bd_modular_canonical_length_lower_product)) * pa_v_bl_bd_modular_canonical_length_lower_product) + (pa_r_bl_bd_modular_canonical_length_lower_product))) /\ ((((exists pa_h_bl_bd_modular_canonical_length_lower_product_successor. pa_h_bl_bd_modular_canonical_length_lower_product_successor + S (pa_s_bl_bd_modular_canonical_length_lower_product) = S ((S (S pa_i_bl_bd_modular_canonical_length_lower_product)) * pa_v_bl_bd_modular_canonical_length_lower_product)) /\ exists pa_q_bl_bd_modular_canonical_length_lower_product_successor. pa_u_bl_bd_modular_canonical_length_lower_product = pa_q_bl_bd_modular_canonical_length_lower_product_successor * S ((S (S pa_i_bl_bd_modular_canonical_length_lower_product)) * pa_v_bl_bd_modular_canonical_length_lower_product) + (pa_s_bl_bd_modular_canonical_length_lower_product))) /\ pa_s_bl_bd_modular_canonical_length_lower_product = pa_r_bl_bd_modular_canonical_length_lower_product * pa_p_bl_bd_modular_canonical_length_lower_product)))))))) /\ ((exists pa_b_bl_bd_modular_canonical_length_upper pa_c_bl_bd_modular_canonical_length_upper. ((forall pa_i_bl_bd_modular_canonical_length_upper_repeat. (exists pa_lt_bl_bd_modular_canonical_length_upper_repeat_bound. pa_lt_bl_bd_modular_canonical_length_upper_repeat_bound + S pa_i_bl_bd_modular_canonical_length_upper_repeat = l) -> (((exists pa_h_bl_bd_modular_canonical_length_upper_repeat_decoded. pa_h_bl_bd_modular_canonical_length_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_bd_modular_canonical_length_upper_repeat)) * pa_c_bl_bd_modular_canonical_length_upper)) /\ exists pa_q_bl_bd_modular_canonical_length_upper_repeat_decoded. pa_b_bl_bd_modular_canonical_length_upper = pa_q_bl_bd_modular_canonical_length_upper_repeat_decoded * S ((S (pa_i_bl_bd_modular_canonical_length_upper_repeat)) * pa_c_bl_bd_modular_canonical_length_upper) + (2)))) /\ (exists pa_u_bl_bd_modular_canonical_length_upper_product pa_v_bl_bd_modular_canonical_length_upper_product. ((((exists pa_h_bl_bd_modular_canonical_length_upper_product_start. pa_h_bl_bd_modular_canonical_length_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_bd_modular_canonical_length_upper_product)) /\ exists pa_q_bl_bd_modular_canonical_length_upper_product_start. pa_u_bl_bd_modular_canonical_length_upper_product = pa_q_bl_bd_modular_canonical_length_upper_product_start * S ((S (0)) * pa_v_bl_bd_modular_canonical_length_upper_product) + (1))) /\ ((((exists pa_h_bl_bd_modular_canonical_length_upper_product_terminal. pa_h_bl_bd_modular_canonical_length_upper_product_terminal + S (ff_upper_bl_bd_modular_canonical_length) = S ((S (l)) * pa_v_bl_bd_modular_canonical_length_upper_product)) /\ exists pa_q_bl_bd_modular_canonical_length_upper_product_terminal. pa_u_bl_bd_modular_canonical_length_upper_product = pa_q_bl_bd_modular_canonical_length_upper_product_terminal * S ((S (l)) * pa_v_bl_bd_modular_canonical_length_upper_product) + (ff_upper_bl_bd_modular_canonical_length))) /\ forall pa_i_bl_bd_modular_canonical_length_upper_product. (exists pa_lt_bl_bd_modular_canonical_length_upper_product_bound. pa_lt_bl_bd_modular_canonical_length_upper_product_bound + S pa_i_bl_bd_modular_canonical_length_upper_product = l) -> exists pa_p_bl_bd_modular_canonical_length_upper_product pa_r_bl_bd_modular_canonical_length_upper_product pa_s_bl_bd_modular_canonical_length_upper_product. ((((exists pa_h_bl_bd_modular_canonical_length_upper_product_factor. pa_h_bl_bd_modular_canonical_length_upper_product_factor + S (pa_p_bl_bd_modular_canonical_length_upper_product) = S ((S (pa_i_bl_bd_modular_canonical_length_upper_product)) * pa_c_bl_bd_modular_canonical_length_upper)) /\ exists pa_q_bl_bd_modular_canonical_length_upper_product_factor. pa_b_bl_bd_modular_canonical_length_upper = pa_q_bl_bd_modular_canonical_length_upper_product_factor * S ((S (pa_i_bl_bd_modular_canonical_length_upper_product)) * pa_c_bl_bd_modular_canonical_length_upper) + (pa_p_bl_bd_modular_canonical_length_upper_product))) /\ ((((exists pa_h_bl_bd_modular_canonical_length_upper_product_partial. pa_h_bl_bd_modular_canonical_length_upper_product_partial + S (pa_r_bl_bd_modular_canonical_length_upper_product) = S ((S (pa_i_bl_bd_modular_canonical_length_upper_product)) * pa_v_bl_bd_modular_canonical_length_upper_product)) /\ exists pa_q_bl_bd_modular_canonical_length_upper_product_partial. pa_u_bl_bd_modular_canonical_length_upper_product = pa_q_bl_bd_modular_canonical_length_upper_product_partial * S ((S (pa_i_bl_bd_modular_canonical_length_upper_product)) * pa_v_bl_bd_modular_canonical_length_upper_product) + (pa_r_bl_bd_modular_canonical_length_upper_product))) /\ ((((exists pa_h_bl_bd_modular_canonical_length_upper_product_successor. pa_h_bl_bd_modular_canonical_length_upper_product_successor + S (pa_s_bl_bd_modular_canonical_length_upper_product) = S ((S (S pa_i_bl_bd_modular_canonical_length_upper_product)) * pa_v_bl_bd_modular_canonical_length_upper_product)) /\ exists pa_q_bl_bd_modular_canonical_length_upper_product_successor. pa_u_bl_bd_modular_canonical_length_upper_product = pa_q_bl_bd_modular_canonical_length_upper_product_successor * S ((S (S pa_i_bl_bd_modular_canonical_length_upper_product)) * pa_v_bl_bd_modular_canonical_length_upper_product) + (pa_s_bl_bd_modular_canonical_length_upper_product))) /\ pa_s_bl_bd_modular_canonical_length_upper_product = pa_r_bl_bd_modular_canonical_length_upper_product * pa_p_bl_bd_modular_canonical_length_upper_product)))))))) /\ ((exists ff_lower_gap_bl_bd_modular_canonical_length. ff_lower_gap_bl_bd_modular_canonical_length + (ff_lower_bl_bd_modular_canonical_length) = (n)) /\ (exists ff_upper_gap_bl_bd_modular_canonical_length. ff_upper_gap_bl_bd_modular_canonical_length + S (n) = (ff_upper_bl_bd_modular_canonical_length))))))))) /\ (((forall ff_index_be_bd_modular_canonical_code_digits ff_digit_be_bd_modular_canonical_code_digits. (exists ff_lt_be_bd_modular_canonical_code_digits_bound. ff_lt_be_bd_modular_canonical_code_digits_bound + S ff_index_be_bd_modular_canonical_code_digits = l) -> (((exists ff_h_be_bd_modular_canonical_code_digits_digit. ff_h_be_bd_modular_canonical_code_digits_digit + S (ff_digit_be_bd_modular_canonical_code_digits) = S ((S (ff_index_be_bd_modular_canonical_code_digits)) * c)) /\ exists ff_q_be_bd_modular_canonical_code_digits_digit. b = ff_q_be_bd_modular_canonical_code_digits_digit * S ((S (ff_index_be_bd_modular_canonical_code_digits)) * c) + (ff_digit_be_bd_modular_canonical_code_digits))) -> (ff_digit_be_bd_modular_canonical_code_digits = 0 \/ ff_digit_be_bd_modular_canonical_code_digits = 1)) /\ (exists ff_u_ph_bd_modular_canonical_code_horner ff_v_ph_bd_modular_canonical_code_horner. ((((exists fs_h_ph_bd_modular_canonical_code_horner_body_start. fs_h_ph_bd_modular_canonical_code_horner_body_start + S (0) = S ((S (0)) * ff_v_ph_bd_modular_canonical_code_horner)) /\ exists fs_q_ph_bd_modular_canonical_code_horner_body_start. ff_u_ph_bd_modular_canonical_code_horner = fs_q_ph_bd_modular_canonical_code_horner_body_start * S ((S (0)) * ff_v_ph_bd_modular_canonical_code_horner) + (0))) /\ ((((exists fs_h_ph_bd_modular_canonical_code_horner_body_terminal. fs_h_ph_bd_modular_canonical_code_horner_body_terminal + S (n) = S ((S (l)) * ff_v_ph_bd_modular_canonical_code_horner)) /\ exists fs_q_ph_bd_modular_canonical_code_horner_body_terminal. ff_u_ph_bd_modular_canonical_code_horner = fs_q_ph_bd_modular_canonical_code_horner_body_terminal * S ((S (l)) * ff_v_ph_bd_modular_canonical_code_horner) + (n))) /\ forall ff_i_ph_bd_modular_canonical_code_horner_body_steps. (exists ph_bound_bd_modular_canonical_code_horner_body_steps. ph_bound_bd_modular_canonical_code_horner_body_steps + S ff_i_ph_bd_modular_canonical_code_horner_body_steps = l) -> exists ff_coefficient_ph_bd_modular_canonical_code_horner_body_steps ff_previous_ph_bd_modular_canonical_code_horner_body_steps ff_current_ph_bd_modular_canonical_code_horner_body_steps. ((((exists fs_h_ph_bd_modular_canonical_code_horner_body_steps_coefficient. fs_h_ph_bd_modular_canonical_code_horner_body_steps_coefficient + S (ff_coefficient_ph_bd_modular_canonical_code_horner_body_steps) = S ((S (ff_i_ph_bd_modular_canonical_code_horner_body_steps)) * c)) /\ exists fs_q_ph_bd_modular_canonical_code_horner_body_steps_coefficient. b = fs_q_ph_bd_modular_canonical_code_horner_body_steps_coefficient * S ((S (ff_i_ph_bd_modular_canonical_code_horner_body_steps)) * c) + (ff_coefficient_ph_bd_modular_canonical_code_horner_body_steps))) /\ ((((exists fs_h_ph_bd_modular_canonical_code_horner_body_steps_before. fs_h_ph_bd_modular_canonical_code_horner_body_steps_before + S (ff_previous_ph_bd_modular_canonical_code_horner_body_steps) = S ((S (ff_i_ph_bd_modular_canonical_code_horner_body_steps)) * ff_v_ph_bd_modular_canonical_code_horner)) /\ exists fs_q_ph_bd_modular_canonical_code_horner_body_steps_before. ff_u_ph_bd_modular_canonical_code_horner = fs_q_ph_bd_modular_canonical_code_horner_body_steps_before * S ((S (ff_i_ph_bd_modular_canonical_code_horner_body_steps)) * ff_v_ph_bd_modular_canonical_code_horner) + (ff_previous_ph_bd_modular_canonical_code_horner_body_steps))) /\ ((((exists fs_h_ph_bd_modular_canonical_code_horner_body_steps_after. fs_h_ph_bd_modular_canonical_code_horner_body_steps_after + S (ff_current_ph_bd_modular_canonical_code_horner_body_steps) = S ((S (S ff_i_ph_bd_modular_canonical_code_horner_body_steps)) * ff_v_ph_bd_modular_canonical_code_horner)) /\ exists fs_q_ph_bd_modular_canonical_code_horner_body_steps_after. ff_u_ph_bd_modular_canonical_code_horner = fs_q_ph_bd_modular_canonical_code_horner_body_steps_after * S ((S (S ff_i_ph_bd_modular_canonical_code_horner_body_steps)) * ff_v_ph_bd_modular_canonical_code_horner) + (ff_current_ph_bd_modular_canonical_code_horner_body_steps))) /\ ff_current_ph_bd_modular_canonical_code_horner_body_steps = ff_previous_ph_bd_modular_canonical_code_horner_body_steps * 2 + ff_coefficient_ph_bd_modular_canonical_code_horner_body_steps)))))))))) -> (exists ff_trace_code_be_bd_modular_run ff_trace_scale_be_bd_modular_run. ((((((exists ff_h_be_bd_modular_run_trace_start. ff_h_be_bd_modular_run_trace_start + S (1) = S ((S (0)) * ff_trace_scale_be_bd_modular_run)) /\ exists ff_q_be_bd_modular_run_trace_start. ff_trace_code_be_bd_modular_run = ff_q_be_bd_modular_run_trace_start * S ((S (0)) * ff_trace_scale_be_bd_modular_run) + (1))) /\ forall ff_index_be_bd_modular_run_trace. (exists ff_lt_be_bd_modular_run_trace_bound. ff_lt_be_bd_modular_run_trace_bound + S ff_index_be_bd_modular_run_trace = l) -> exists ff_digit_be_bd_modular_run_trace ff_previous_be_bd_modular_run_trace ff_current_be_bd_modular_run_trace. ((((exists ff_h_be_bd_modular_run_trace_source. ff_h_be_bd_modular_run_trace_source + S (ff_digit_be_bd_modular_run_trace) = S ((S (ff_index_be_bd_modular_run_trace)) * c)) /\ exists ff_q_be_bd_modular_run_trace_source. b = ff_q_be_bd_modular_run_trace_source * S ((S (ff_index_be_bd_modular_run_trace)) * c) + (ff_digit_be_bd_modular_run_trace))) /\ ((((exists ff_h_be_bd_modular_run_trace_before. ff_h_be_bd_modular_run_trace_before + S (ff_previous_be_bd_modular_run_trace) = S ((S (ff_index_be_bd_modular_run_trace)) * ff_trace_scale_be_bd_modular_run)) /\ exists ff_q_be_bd_modular_run_trace_before. ff_trace_code_be_bd_modular_run = ff_q_be_bd_modular_run_trace_before * S ((S (ff_index_be_bd_modular_run_trace)) * ff_trace_scale_be_bd_modular_run) + (ff_previous_be_bd_modular_run_trace))) /\ ((((exists ff_h_be_bd_modular_run_trace_after. ff_h_be_bd_modular_run_trace_after + S (ff_current_be_bd_modular_run_trace) = S ((S (S ff_index_be_bd_modular_run_trace)) * ff_trace_scale_be_bd_modular_run)) /\ exists ff_q_be_bd_modular_run_trace_after. ff_trace_code_be_bd_modular_run = ff_q_be_bd_modular_run_trace_after * S ((S (S ff_index_be_bd_modular_run_trace)) * ff_trace_scale_be_bd_modular_run) + (ff_current_be_bd_modular_run_trace))) /\ ((((ff_digit_be_bd_modular_run_trace = 0) /\ (((exists ff_gap_binary_be_bd_modular_run_trace_transition_square. ff_gap_binary_be_bd_modular_run_trace_transition_square + S (ff_current_be_bd_modular_run_trace) = m) /\ (exists ff_left_binary_be_bd_modular_run_trace_transition_square_congruence ff_right_binary_be_bd_modular_run_trace_transition_square_congruence. (ff_previous_be_bd_modular_run_trace * ff_previous_be_bd_modular_run_trace) + m * ff_left_binary_be_bd_modular_run_trace_transition_square_congruence = (ff_current_be_bd_modular_run_trace) + m * ff_right_binary_be_bd_modular_run_trace_transition_square_congruence)))) \/ ((ff_digit_be_bd_modular_run_trace = 1) /\ (((exists ff_gap_binary_be_bd_modular_run_trace_transition_multiply. ff_gap_binary_be_bd_modular_run_trace_transition_multiply + S (ff_current_be_bd_modular_run_trace) = m) /\ (exists ff_left_binary_be_bd_modular_run_trace_transition_multiply_congruence ff_right_binary_be_bd_modular_run_trace_transition_multiply_congruence. ((ff_previous_be_bd_modular_run_trace * ff_previous_be_bd_modular_run_trace) * a) + m * ff_left_binary_be_bd_modular_run_trace_transition_multiply_congruence = (ff_current_be_bd_modular_run_trace) + m * ff_right_binary_be_bd_modular_run_trace_transition_multiply_congruence))))))))))) /\ (((exists ff_h_be_bd_modular_run_terminal. ff_h_be_bd_modular_run_terminal + S (r) = S ((S (l)) * ff_trace_scale_be_bd_modular_run)) /\ exists ff_q_be_bd_modular_run_terminal. ff_trace_code_be_bd_modular_run = ff_q_be_bd_modular_run_terminal * S ((S (l)) * ff_trace_scale_be_bd_modular_run) + (r))))) -> (exists ff_power_binary_bd_modular_power. ((exists ff_b_binary_bd_modular_power_value ff_c_binary_bd_modular_power_value. ((forall ff_i_binary_bd_modular_power_value_repeat. (exists ff_lt_binary_bd_modular_power_value_repeat_bound. ff_lt_binary_bd_modular_power_value_repeat_bound + S ff_i_binary_bd_modular_power_value_repeat = n) -> (((exists ff_h_binary_bd_modular_power_value_repeat_decoded. ff_h_binary_bd_modular_power_value_repeat_decoded + S (a) = S ((S (ff_i_binary_bd_modular_power_value_repeat)) * ff_c_binary_bd_modular_power_value)) /\ exists ff_q_binary_bd_modular_power_value_repeat_decoded. ff_b_binary_bd_modular_power_value = ff_q_binary_bd_modular_power_value_repeat_decoded * S ((S (ff_i_binary_bd_modular_power_value_repeat)) * ff_c_binary_bd_modular_power_value) + (a)))) /\ (exists ff_u_binary_bd_modular_power_value_product ff_v_binary_bd_modular_power_value_product. ((((exists ff_h_binary_bd_modular_power_value_product_start. ff_h_binary_bd_modular_power_value_product_start + S (1) = S ((S (0)) * ff_v_binary_bd_modular_power_value_product)) /\ exists ff_q_binary_bd_modular_power_value_product_start. ff_u_binary_bd_modular_power_value_product = ff_q_binary_bd_modular_power_value_product_start * S ((S (0)) * ff_v_binary_bd_modular_power_value_product) + (1))) /\ ((((exists ff_h_binary_bd_modular_power_value_product_terminal. ff_h_binary_bd_modular_power_value_product_terminal + S (ff_power_binary_bd_modular_power) = S ((S (n)) * ff_v_binary_bd_modular_power_value_product)) /\ exists ff_q_binary_bd_modular_power_value_product_terminal. ff_u_binary_bd_modular_power_value_product = ff_q_binary_bd_modular_power_value_product_terminal * S ((S (n)) * ff_v_binary_bd_modular_power_value_product) + (ff_power_binary_bd_modular_power))) /\ forall ff_i_binary_bd_modular_power_value_product. (exists ff_lt_binary_bd_modular_power_value_product_bound. ff_lt_binary_bd_modular_power_value_product_bound + S ff_i_binary_bd_modular_power_value_product = n) -> exists ff_p_binary_bd_modular_power_value_product ff_r_binary_bd_modular_power_value_product ff_s_binary_bd_modular_power_value_product. ((((exists ff_h_binary_bd_modular_power_value_product_factor. ff_h_binary_bd_modular_power_value_product_factor + S (ff_p_binary_bd_modular_power_value_product) = S ((S (ff_i_binary_bd_modular_power_value_product)) * ff_c_binary_bd_modular_power_value)) /\ exists ff_q_binary_bd_modular_power_value_product_factor. ff_b_binary_bd_modular_power_value = ff_q_binary_bd_modular_power_value_product_factor * S ((S (ff_i_binary_bd_modular_power_value_product)) * ff_c_binary_bd_modular_power_value) + (ff_p_binary_bd_modular_power_value_product))) /\ ((((exists ff_h_binary_bd_modular_power_value_product_partial. ff_h_binary_bd_modular_power_value_product_partial + S (ff_r_binary_bd_modular_power_value_product) = S ((S (ff_i_binary_bd_modular_power_value_product)) * ff_v_binary_bd_modular_power_value_product)) /\ exists ff_q_binary_bd_modular_power_value_product_partial. ff_u_binary_bd_modular_power_value_product = ff_q_binary_bd_modular_power_value_product_partial * S ((S (ff_i_binary_bd_modular_power_value_product)) * ff_v_binary_bd_modular_power_value_product) + (ff_r_binary_bd_modular_power_value_product))) /\ ((((exists ff_h_binary_bd_modular_power_value_product_successor. ff_h_binary_bd_modular_power_value_product_successor + S (ff_s_binary_bd_modular_power_value_product) = S ((S (S ff_i_binary_bd_modular_power_value_product)) * ff_v_binary_bd_modular_power_value_product)) /\ exists ff_q_binary_bd_modular_power_value_product_successor. ff_u_binary_bd_modular_power_value_product = ff_q_binary_bd_modular_power_value_product_successor * S ((S (S ff_i_binary_bd_modular_power_value_product)) * ff_v_binary_bd_modular_power_value_product) + (ff_s_binary_bd_modular_power_value_product))) /\ ff_s_binary_bd_modular_power_value_product = ff_r_binary_bd_modular_power_value_product * ff_p_binary_bd_modular_power_value_product)))))))) /\ (((exists ff_gap_binary_bd_modular_power_residue. ff_gap_binary_bd_modular_power_residue + S (r) = m) /\ (exists ff_left_binary_bd_modular_power_residue_congruence ff_right_binary_bd_modular_power_residue_congruence. (ff_power_binary_bd_modular_power) + m * ff_left_binary_bd_modular_power_residue_congruence = (r) + m * ff_right_binary_bd_modular_power_residue_congruence)))))