beta_horner_eval_functional

Exact expanded first-order arithmetic statement

forall b c t l n m. (exists ff_u_ph_root_value ff_v_ph_root_value. ((((exists fs_h_ph_root_value_body_start. fs_h_ph_root_value_body_start + S (0) = S ((S (0)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_start. ff_u_ph_root_value = fs_q_ph_root_value_body_start * S ((S (0)) * ff_v_ph_root_value) + (0))) /\ ((((exists fs_h_ph_root_value_body_terminal. fs_h_ph_root_value_body_terminal + S (n) = S ((S (l)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_terminal. ff_u_ph_root_value = fs_q_ph_root_value_body_terminal * S ((S (l)) * ff_v_ph_root_value) + (n))) /\ forall ff_i_ph_root_value_body_steps. (exists ph_bound_root_value_body_steps. ph_bound_root_value_body_steps + S ff_i_ph_root_value_body_steps = l) -> exists ff_coefficient_ph_root_value_body_steps ff_previous_ph_root_value_body_steps ff_current_ph_root_value_body_steps. ((((exists fs_h_ph_root_value_body_steps_coefficient. fs_h_ph_root_value_body_steps_coefficient + S (ff_coefficient_ph_root_value_body_steps) = S ((S (ff_i_ph_root_value_body_steps)) * c)) /\ exists fs_q_ph_root_value_body_steps_coefficient. b = fs_q_ph_root_value_body_steps_coefficient * S ((S (ff_i_ph_root_value_body_steps)) * c) + (ff_coefficient_ph_root_value_body_steps))) /\ ((((exists fs_h_ph_root_value_body_steps_before. fs_h_ph_root_value_body_steps_before + S (ff_previous_ph_root_value_body_steps) = S ((S (ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_steps_before. ff_u_ph_root_value = fs_q_ph_root_value_body_steps_before * S ((S (ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value) + (ff_previous_ph_root_value_body_steps))) /\ ((((exists fs_h_ph_root_value_body_steps_after. fs_h_ph_root_value_body_steps_after + S (ff_current_ph_root_value_body_steps) = S ((S (S ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value)) /\ exists fs_q_ph_root_value_body_steps_after. ff_u_ph_root_value = fs_q_ph_root_value_body_steps_after * S ((S (S ff_i_ph_root_value_body_steps)) * ff_v_ph_root_value) + (ff_current_ph_root_value_body_steps))) /\ ff_current_ph_root_value_body_steps = ff_previous_ph_root_value_body_steps * t + ff_coefficient_ph_root_value_body_steps)))))) -> (exists ff_u_ph_root_other ff_v_ph_root_other. ((((exists fs_h_ph_root_other_body_start. fs_h_ph_root_other_body_start + S (0) = S ((S (0)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_start. ff_u_ph_root_other = fs_q_ph_root_other_body_start * S ((S (0)) * ff_v_ph_root_other) + (0))) /\ ((((exists fs_h_ph_root_other_body_terminal. fs_h_ph_root_other_body_terminal + S (m) = S ((S (l)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_terminal. ff_u_ph_root_other = fs_q_ph_root_other_body_terminal * S ((S (l)) * ff_v_ph_root_other) + (m))) /\ forall ff_i_ph_root_other_body_steps. (exists ph_bound_root_other_body_steps. ph_bound_root_other_body_steps + S ff_i_ph_root_other_body_steps = l) -> exists ff_coefficient_ph_root_other_body_steps ff_previous_ph_root_other_body_steps ff_current_ph_root_other_body_steps. ((((exists fs_h_ph_root_other_body_steps_coefficient. fs_h_ph_root_other_body_steps_coefficient + S (ff_coefficient_ph_root_other_body_steps) = S ((S (ff_i_ph_root_other_body_steps)) * c)) /\ exists fs_q_ph_root_other_body_steps_coefficient. b = fs_q_ph_root_other_body_steps_coefficient * S ((S (ff_i_ph_root_other_body_steps)) * c) + (ff_coefficient_ph_root_other_body_steps))) /\ ((((exists fs_h_ph_root_other_body_steps_before. fs_h_ph_root_other_body_steps_before + S (ff_previous_ph_root_other_body_steps) = S ((S (ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_steps_before. ff_u_ph_root_other = fs_q_ph_root_other_body_steps_before * S ((S (ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other) + (ff_previous_ph_root_other_body_steps))) /\ ((((exists fs_h_ph_root_other_body_steps_after. fs_h_ph_root_other_body_steps_after + S (ff_current_ph_root_other_body_steps) = S ((S (S ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other)) /\ exists fs_q_ph_root_other_body_steps_after. ff_u_ph_root_other = fs_q_ph_root_other_body_steps_after * S ((S (S ff_i_ph_root_other_body_steps)) * ff_v_ph_root_other) + (ff_current_ph_root_other_body_steps))) /\ ff_current_ph_root_other_body_steps = ff_previous_ph_root_other_body_steps * t + ff_coefficient_ph_root_other_body_steps)))))) -> n = m