Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Definition in prerequisite notation
∃ hpl_value_secondwave. ∃ hpl_derivative_secondwave. HornerDerivative(b,c,a,l,hpl_value_secondwave,hpl_derivative_secondwave) ∧ (ModEq(m,hpl_value_secondwave,0) ∧ Coprime(hpl_derivative_secondwave,p))
Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists hpl_value_secondwave hpl_derivative_secondwave. ((exists ff_u_hd_hpl_secondwave ff_v_hd_hpl_secondwave ff_d_hd_hpl_secondwave ff_e_hd_hpl_secondwave. ((((((exists fs_h_ph_hd_hpl_secondwave_body_value_start. fs_h_ph_hd_hpl_secondwave_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_value_start. ff_u_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_value_start * S ((S (0)) * ff_v_hd_hpl_secondwave) + (0))) /\ ((((exists fs_h_ph_hd_hpl_secondwave_body_value_terminal. fs_h_ph_hd_hpl_secondwave_body_value_terminal + S (hpl_value_secondwave) = S ((S (l)) * ff_v_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_value_terminal. ff_u_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_secondwave) + (hpl_value_secondwave))) /\ forall ff_i_ph_hd_hpl_secondwave_body_value_steps. (exists ph_bound_hd_hpl_secondwave_body_value_steps. ph_bound_hd_hpl_secondwave_body_value_steps + S ff_i_ph_hd_hpl_secondwave_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_secondwave_body_value_steps ff_previous_ph_hd_hpl_secondwave_body_value_steps ff_current_ph_hd_hpl_secondwave_body_value_steps. ((((exists fs_h_ph_hd_hpl_secondwave_body_value_steps_coefficient. fs_h_ph_hd_hpl_secondwave_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_secondwave_body_value_steps) = S ((S (ff_i_ph_hd_hpl_secondwave_body_value_steps)) * c)) /\ exists fs_q_ph_hd_hpl_secondwave_body_value_steps_coefficient. b = fs_q_ph_hd_hpl_secondwave_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_secondwave_body_value_steps)) * c) + (ff_coefficient_ph_hd_hpl_secondwave_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_secondwave_body_value_steps_before. fs_h_ph_hd_hpl_secondwave_body_value_steps_before + S (ff_previous_ph_hd_hpl_secondwave_body_value_steps) = S ((S (ff_i_ph_hd_hpl_secondwave_body_value_steps)) * ff_v_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_value_steps_before. ff_u_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_secondwave_body_value_steps)) * ff_v_hd_hpl_secondwave) + (ff_previous_ph_hd_hpl_secondwave_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_secondwave_body_value_steps_after. fs_h_ph_hd_hpl_secondwave_body_value_steps_after + S (ff_current_ph_hd_hpl_secondwave_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_secondwave_body_value_steps)) * ff_v_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_value_steps_after. ff_u_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_secondwave_body_value_steps)) * ff_v_hd_hpl_secondwave) + (ff_current_ph_hd_hpl_secondwave_body_value_steps))) /\ ff_current_ph_hd_hpl_secondwave_body_value_steps = ff_previous_ph_hd_hpl_secondwave_body_value_steps * a + ff_coefficient_ph_hd_hpl_secondwave_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_secondwave_body_derivative_start. fs_h_ph_hd_hpl_secondwave_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_derivative_start. ff_d_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_secondwave) + (0))) /\ ((((exists fs_h_ph_hd_hpl_secondwave_body_derivative_terminal. fs_h_ph_hd_hpl_secondwave_body_derivative_terminal + S (hpl_derivative_secondwave) = S ((S (l)) * ff_e_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_derivative_terminal. ff_d_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_secondwave) + (hpl_derivative_secondwave))) /\ forall ff_i_ph_hd_hpl_secondwave_body_derivative_steps. (exists ph_bound_hd_hpl_secondwave_body_derivative_steps. ph_bound_hd_hpl_secondwave_body_derivative_steps + S ff_i_ph_hd_hpl_secondwave_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_secondwave_body_derivative_steps ff_previous_ph_hd_hpl_secondwave_body_derivative_steps ff_current_ph_hd_hpl_secondwave_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_secondwave_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_secondwave_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_secondwave_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_secondwave_body_derivative_steps)) * ff_v_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_derivative_steps_coefficient. ff_u_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_secondwave_body_derivative_steps)) * ff_v_hd_hpl_secondwave) + (ff_coefficient_ph_hd_hpl_secondwave_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_secondwave_body_derivative_steps_before. fs_h_ph_hd_hpl_secondwave_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_secondwave_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_secondwave_body_derivative_steps)) * ff_e_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_derivative_steps_before. ff_d_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_secondwave_body_derivative_steps)) * ff_e_hd_hpl_secondwave) + (ff_previous_ph_hd_hpl_secondwave_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_secondwave_body_derivative_steps_after. fs_h_ph_hd_hpl_secondwave_body_derivative_steps_after + S (ff_current_ph_hd_hpl_secondwave_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_secondwave_body_derivative_steps)) * ff_e_hd_hpl_secondwave)) /\ exists fs_q_ph_hd_hpl_secondwave_body_derivative_steps_after. ff_d_hd_hpl_secondwave = fs_q_ph_hd_hpl_secondwave_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_secondwave_body_derivative_steps)) * ff_e_hd_hpl_secondwave) + (ff_current_ph_hd_hpl_secondwave_body_derivative_steps))) /\ ff_current_ph_hd_hpl_secondwave_body_derivative_steps = ff_previous_ph_hd_hpl_secondwave_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_secondwave_body_derivative_steps)))))))) /\ ((exists hgcrt_mod_left_hpl_secondwave hgcrt_mod_right_hpl_secondwave. hpl_value_secondwave + m * hgcrt_mod_left_hpl_secondwave = 0 + m * hgcrt_mod_right_hpl_secondwave) /\ (forall hmi_divisor_hpl_secondwave. (exists hmi_left_factor_hpl_secondwave. hpl_derivative_secondwave = hmi_divisor_hpl_secondwave * hmi_left_factor_hpl_secondwave) -> (exists hmi_right_factor_hpl_secondwave. p = hmi_divisor_hpl_secondwave * hmi_right_factor_hpl_secondwave) -> hmi_divisor_hpl_secondwave = 1)))
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
none