BinaryCanonicalExponentDigitCode(e,ell,b,c)

Hygienic expanded first-order definition

(((((e) = 0 /\ (ell) = 1) \/ exists ff_exponent_bl_bd_milestone_length ff_lower_bl_bd_milestone_length ff_upper_bl_bd_milestone_length. (((ell) = S ff_exponent_bl_bd_milestone_length) /\ ((exists ff_positive_bl_bd_milestone_length. ff_positive_bl_bd_milestone_length + 1 = (e)) /\ ((exists pa_b_bl_bd_milestone_length_lower pa_c_bl_bd_milestone_length_lower. ((forall pa_i_bl_bd_milestone_length_lower_repeat. (exists pa_lt_bl_bd_milestone_length_lower_repeat_bound. pa_lt_bl_bd_milestone_length_lower_repeat_bound + S pa_i_bl_bd_milestone_length_lower_repeat = ff_exponent_bl_bd_milestone_length) -> (((exists pa_h_bl_bd_milestone_length_lower_repeat_decoded. pa_h_bl_bd_milestone_length_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_bd_milestone_length_lower_repeat)) * pa_c_bl_bd_milestone_length_lower)) /\ exists pa_q_bl_bd_milestone_length_lower_repeat_decoded. pa_b_bl_bd_milestone_length_lower = pa_q_bl_bd_milestone_length_lower_repeat_decoded * S ((S (pa_i_bl_bd_milestone_length_lower_repeat)) * pa_c_bl_bd_milestone_length_lower) + (2)))) /\ (exists pa_u_bl_bd_milestone_length_lower_product pa_v_bl_bd_milestone_length_lower_product. ((((exists pa_h_bl_bd_milestone_length_lower_product_start. pa_h_bl_bd_milestone_length_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_bd_milestone_length_lower_product)) /\ exists pa_q_bl_bd_milestone_length_lower_product_start. pa_u_bl_bd_milestone_length_lower_product = pa_q_bl_bd_milestone_length_lower_product_start * S ((S (0)) * pa_v_bl_bd_milestone_length_lower_product) + (1))) /\ ((((exists pa_h_bl_bd_milestone_length_lower_product_terminal. pa_h_bl_bd_milestone_length_lower_product_terminal + S (ff_lower_bl_bd_milestone_length) = S ((S (ff_exponent_bl_bd_milestone_length)) * pa_v_bl_bd_milestone_length_lower_product)) /\ exists pa_q_bl_bd_milestone_length_lower_product_terminal. pa_u_bl_bd_milestone_length_lower_product = pa_q_bl_bd_milestone_length_lower_product_terminal * S ((S (ff_exponent_bl_bd_milestone_length)) * pa_v_bl_bd_milestone_length_lower_product) + (ff_lower_bl_bd_milestone_length))) /\ forall pa_i_bl_bd_milestone_length_lower_product. (exists pa_lt_bl_bd_milestone_length_lower_product_bound. pa_lt_bl_bd_milestone_length_lower_product_bound + S pa_i_bl_bd_milestone_length_lower_product = ff_exponent_bl_bd_milestone_length) -> exists pa_p_bl_bd_milestone_length_lower_product pa_r_bl_bd_milestone_length_lower_product pa_s_bl_bd_milestone_length_lower_product. ((((exists pa_h_bl_bd_milestone_length_lower_product_factor. pa_h_bl_bd_milestone_length_lower_product_factor + S (pa_p_bl_bd_milestone_length_lower_product) = S ((S (pa_i_bl_bd_milestone_length_lower_product)) * pa_c_bl_bd_milestone_length_lower)) /\ exists pa_q_bl_bd_milestone_length_lower_product_factor. pa_b_bl_bd_milestone_length_lower = pa_q_bl_bd_milestone_length_lower_product_factor * S ((S (pa_i_bl_bd_milestone_length_lower_product)) * pa_c_bl_bd_milestone_length_lower) + (pa_p_bl_bd_milestone_length_lower_product))) /\ ((((exists pa_h_bl_bd_milestone_length_lower_product_partial. pa_h_bl_bd_milestone_length_lower_product_partial + S (pa_r_bl_bd_milestone_length_lower_product) = S ((S (pa_i_bl_bd_milestone_length_lower_product)) * pa_v_bl_bd_milestone_length_lower_product)) /\ exists pa_q_bl_bd_milestone_length_lower_product_partial. pa_u_bl_bd_milestone_length_lower_product = pa_q_bl_bd_milestone_length_lower_product_partial * S ((S (pa_i_bl_bd_milestone_length_lower_product)) * pa_v_bl_bd_milestone_length_lower_product) + (pa_r_bl_bd_milestone_length_lower_product))) /\ ((((exists pa_h_bl_bd_milestone_length_lower_product_successor. pa_h_bl_bd_milestone_length_lower_product_successor + S (pa_s_bl_bd_milestone_length_lower_product) = S ((S (S pa_i_bl_bd_milestone_length_lower_product)) * pa_v_bl_bd_milestone_length_lower_product)) /\ exists pa_q_bl_bd_milestone_length_lower_product_successor. pa_u_bl_bd_milestone_length_lower_product = pa_q_bl_bd_milestone_length_lower_product_successor * S ((S (S pa_i_bl_bd_milestone_length_lower_product)) * pa_v_bl_bd_milestone_length_lower_product) + (pa_s_bl_bd_milestone_length_lower_product))) /\ pa_s_bl_bd_milestone_length_lower_product = pa_r_bl_bd_milestone_length_lower_product * pa_p_bl_bd_milestone_length_lower_product)))))))) /\ ((exists pa_b_bl_bd_milestone_length_upper pa_c_bl_bd_milestone_length_upper. ((forall pa_i_bl_bd_milestone_length_upper_repeat. (exists pa_lt_bl_bd_milestone_length_upper_repeat_bound. pa_lt_bl_bd_milestone_length_upper_repeat_bound + S pa_i_bl_bd_milestone_length_upper_repeat = ell) -> (((exists pa_h_bl_bd_milestone_length_upper_repeat_decoded. pa_h_bl_bd_milestone_length_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_bd_milestone_length_upper_repeat)) * pa_c_bl_bd_milestone_length_upper)) /\ exists pa_q_bl_bd_milestone_length_upper_repeat_decoded. pa_b_bl_bd_milestone_length_upper = pa_q_bl_bd_milestone_length_upper_repeat_decoded * S ((S (pa_i_bl_bd_milestone_length_upper_repeat)) * pa_c_bl_bd_milestone_length_upper) + (2)))) /\ (exists pa_u_bl_bd_milestone_length_upper_product pa_v_bl_bd_milestone_length_upper_product. ((((exists pa_h_bl_bd_milestone_length_upper_product_start. pa_h_bl_bd_milestone_length_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_bd_milestone_length_upper_product)) /\ exists pa_q_bl_bd_milestone_length_upper_product_start. pa_u_bl_bd_milestone_length_upper_product = pa_q_bl_bd_milestone_length_upper_product_start * S ((S (0)) * pa_v_bl_bd_milestone_length_upper_product) + (1))) /\ ((((exists pa_h_bl_bd_milestone_length_upper_product_terminal. pa_h_bl_bd_milestone_length_upper_product_terminal + S (ff_upper_bl_bd_milestone_length) = S ((S (ell)) * pa_v_bl_bd_milestone_length_upper_product)) /\ exists pa_q_bl_bd_milestone_length_upper_product_terminal. pa_u_bl_bd_milestone_length_upper_product = pa_q_bl_bd_milestone_length_upper_product_terminal * S ((S (ell)) * pa_v_bl_bd_milestone_length_upper_product) + (ff_upper_bl_bd_milestone_length))) /\ forall pa_i_bl_bd_milestone_length_upper_product. (exists pa_lt_bl_bd_milestone_length_upper_product_bound. pa_lt_bl_bd_milestone_length_upper_product_bound + S pa_i_bl_bd_milestone_length_upper_product = ell) -> exists pa_p_bl_bd_milestone_length_upper_product pa_r_bl_bd_milestone_length_upper_product pa_s_bl_bd_milestone_length_upper_product. ((((exists pa_h_bl_bd_milestone_length_upper_product_factor. pa_h_bl_bd_milestone_length_upper_product_factor + S (pa_p_bl_bd_milestone_length_upper_product) = S ((S (pa_i_bl_bd_milestone_length_upper_product)) * pa_c_bl_bd_milestone_length_upper)) /\ exists pa_q_bl_bd_milestone_length_upper_product_factor. pa_b_bl_bd_milestone_length_upper = pa_q_bl_bd_milestone_length_upper_product_factor * S ((S (pa_i_bl_bd_milestone_length_upper_product)) * pa_c_bl_bd_milestone_length_upper) + (pa_p_bl_bd_milestone_length_upper_product))) /\ ((((exists pa_h_bl_bd_milestone_length_upper_product_partial. pa_h_bl_bd_milestone_length_upper_product_partial + S (pa_r_bl_bd_milestone_length_upper_product) = S ((S (pa_i_bl_bd_milestone_length_upper_product)) * pa_v_bl_bd_milestone_length_upper_product)) /\ exists pa_q_bl_bd_milestone_length_upper_product_partial. pa_u_bl_bd_milestone_length_upper_product = pa_q_bl_bd_milestone_length_upper_product_partial * S ((S (pa_i_bl_bd_milestone_length_upper_product)) * pa_v_bl_bd_milestone_length_upper_product) + (pa_r_bl_bd_milestone_length_upper_product))) /\ ((((exists pa_h_bl_bd_milestone_length_upper_product_successor. pa_h_bl_bd_milestone_length_upper_product_successor + S (pa_s_bl_bd_milestone_length_upper_product) = S ((S (S pa_i_bl_bd_milestone_length_upper_product)) * pa_v_bl_bd_milestone_length_upper_product)) /\ exists pa_q_bl_bd_milestone_length_upper_product_successor. pa_u_bl_bd_milestone_length_upper_product = pa_q_bl_bd_milestone_length_upper_product_successor * S ((S (S pa_i_bl_bd_milestone_length_upper_product)) * pa_v_bl_bd_milestone_length_upper_product) + (pa_s_bl_bd_milestone_length_upper_product))) /\ pa_s_bl_bd_milestone_length_upper_product = pa_r_bl_bd_milestone_length_upper_product * pa_p_bl_bd_milestone_length_upper_product)))))))) /\ ((exists ff_lower_gap_bl_bd_milestone_length. ff_lower_gap_bl_bd_milestone_length + (ff_lower_bl_bd_milestone_length) = (e)) /\ (exists ff_upper_gap_bl_bd_milestone_length. ff_upper_gap_bl_bd_milestone_length + S (e) = (ff_upper_bl_bd_milestone_length))))))))) /\ (((forall ff_index_be_bd_milestone_code_digits ff_digit_be_bd_milestone_code_digits. (exists ff_lt_be_bd_milestone_code_digits_bound. ff_lt_be_bd_milestone_code_digits_bound + S ff_index_be_bd_milestone_code_digits = ell) -> (((exists ff_h_be_bd_milestone_code_digits_digit. ff_h_be_bd_milestone_code_digits_digit + S (ff_digit_be_bd_milestone_code_digits) = S ((S (ff_index_be_bd_milestone_code_digits)) * c)) /\ exists ff_q_be_bd_milestone_code_digits_digit. b = ff_q_be_bd_milestone_code_digits_digit * S ((S (ff_index_be_bd_milestone_code_digits)) * c) + (ff_digit_be_bd_milestone_code_digits))) -> (ff_digit_be_bd_milestone_code_digits = 0 \/ ff_digit_be_bd_milestone_code_digits = 1)) /\ (exists ff_u_ph_bd_milestone_code_horner ff_v_ph_bd_milestone_code_horner. ((((exists fs_h_ph_bd_milestone_code_horner_body_start. fs_h_ph_bd_milestone_code_horner_body_start + S (0) = S ((S (0)) * ff_v_ph_bd_milestone_code_horner)) /\ exists fs_q_ph_bd_milestone_code_horner_body_start. ff_u_ph_bd_milestone_code_horner = fs_q_ph_bd_milestone_code_horner_body_start * S ((S (0)) * ff_v_ph_bd_milestone_code_horner) + (0))) /\ ((((exists fs_h_ph_bd_milestone_code_horner_body_terminal. fs_h_ph_bd_milestone_code_horner_body_terminal + S (e) = S ((S (ell)) * ff_v_ph_bd_milestone_code_horner)) /\ exists fs_q_ph_bd_milestone_code_horner_body_terminal. ff_u_ph_bd_milestone_code_horner = fs_q_ph_bd_milestone_code_horner_body_terminal * S ((S (ell)) * ff_v_ph_bd_milestone_code_horner) + (e))) /\ forall ff_i_ph_bd_milestone_code_horner_body_steps. (exists ph_bound_bd_milestone_code_horner_body_steps. ph_bound_bd_milestone_code_horner_body_steps + S ff_i_ph_bd_milestone_code_horner_body_steps = ell) -> exists ff_coefficient_ph_bd_milestone_code_horner_body_steps ff_previous_ph_bd_milestone_code_horner_body_steps ff_current_ph_bd_milestone_code_horner_body_steps. ((((exists fs_h_ph_bd_milestone_code_horner_body_steps_coefficient. fs_h_ph_bd_milestone_code_horner_body_steps_coefficient + S (ff_coefficient_ph_bd_milestone_code_horner_body_steps) = S ((S (ff_i_ph_bd_milestone_code_horner_body_steps)) * c)) /\ exists fs_q_ph_bd_milestone_code_horner_body_steps_coefficient. b = fs_q_ph_bd_milestone_code_horner_body_steps_coefficient * S ((S (ff_i_ph_bd_milestone_code_horner_body_steps)) * c) + (ff_coefficient_ph_bd_milestone_code_horner_body_steps))) /\ ((((exists fs_h_ph_bd_milestone_code_horner_body_steps_before. fs_h_ph_bd_milestone_code_horner_body_steps_before + S (ff_previous_ph_bd_milestone_code_horner_body_steps) = S ((S (ff_i_ph_bd_milestone_code_horner_body_steps)) * ff_v_ph_bd_milestone_code_horner)) /\ exists fs_q_ph_bd_milestone_code_horner_body_steps_before. ff_u_ph_bd_milestone_code_horner = fs_q_ph_bd_milestone_code_horner_body_steps_before * S ((S (ff_i_ph_bd_milestone_code_horner_body_steps)) * ff_v_ph_bd_milestone_code_horner) + (ff_previous_ph_bd_milestone_code_horner_body_steps))) /\ ((((exists fs_h_ph_bd_milestone_code_horner_body_steps_after. fs_h_ph_bd_milestone_code_horner_body_steps_after + S (ff_current_ph_bd_milestone_code_horner_body_steps) = S ((S (S ff_i_ph_bd_milestone_code_horner_body_steps)) * ff_v_ph_bd_milestone_code_horner)) /\ exists fs_q_ph_bd_milestone_code_horner_body_steps_after. ff_u_ph_bd_milestone_code_horner = fs_q_ph_bd_milestone_code_horner_body_steps_after * S ((S (S ff_i_ph_bd_milestone_code_horner_body_steps)) * ff_v_ph_bd_milestone_code_horner) + (ff_current_ph_bd_milestone_code_horner_body_steps))) /\ ff_current_ph_bd_milestone_code_horner_body_steps = ff_previous_ph_bd_milestone_code_horner_body_steps * 2 + ff_coefficient_ph_bd_milestone_code_horner_body_steps)))))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Definitions depending on this notation

Checked theorems using this definition