Definition alerting_distance (s1 s2 : State) : Prop :=
  (distance s1 s2 <= AlertRange)%R.Definition conflict_distance (s1 s2 : State) : Prop :=
  (distance s1 s2 <= ConflictRange)%R.Definition chkrange (range tpred : R) : Prop :=
  (range <= AlertRange)%R /\ (tpred <= AlertTime)%R.Definition chktrack (intruder evader : State) (tpred : R) : Prop :=
  let tau := tau intruder evader 0 in
  match Rle_dec tau 0 with
  | left _ => chkrange (RR intruder evader 0) tpred
  | right _ =>
      match Rlt_dec AlertTime (tpred + tau) with
      | left _ => (RR intruder evader AlertTime <= AlertRange)%R
      | right _ => (RR intruder evader tau <= AlertRange)%R
      end
  end.Definition arc_loop (intruder evader : State) (arcrad trkrate : R)
  (idtrk iarc : nat) : Prop :=
  match iarc with
  | S p => False
  | O =>
      let tpred := (INR iarc * tstep)%R in
      let xloc := (xt evader + v evaderSpeed * tpred)%R in
      let yloc := yt evader in
      let xtrk :=
        match Rlt_le_dec 0 trkrate with
        | left _ =>
            (xt intruder +
             arcrad *
             (sind (heading intruder + trkrate * tpred) -
              sind (heading intruder)))%R
        | right _ =>
            (xt intruder +
             arcrad *
             (sind (heading intruder) - sind (heading intruder) +
              trkrate * tpred))%R
        end in
      let ytrk :=
        match Rlt_le_dec 0 trkrate with
        | left _ =>
            (yt intruder +
             arcrad *
             (cosd (heading intruder) -
              cosd (heading intruder + trkrate * tpred)))%R
        | right _ =>
            (yt intruder +
             arcrad *
             (cosd (heading intruder + trkrate * tpred) -
              cosd (heading intruder)))%R
        end in
      match mod_ iarc idtrk with
      | O =>
          let tantrk := (heading intruder + tpred * trkrate)%R in
          let int := mkState xtrk ytrk tantrk (bank intruder) in
          let eva := mkState xloc yloc (heading evader) (bank evader) in
          chktrack int eva tpred
      | _ =>
          let range := sqrt (Rsqr (xtrk - xloc) + Rsqr (ytrk - yloc)) in
          chkrange range tpred
      end
  end.Definition ails_alert (intruder evader : State) : Prop :=
  let phi := bank intruder in
  let trkrate := trkrate phi in
  match Req_EM_var trkrate 0 with
  | left _ => chktrack intruder evader 0
  | right _ =>
      let arcrad := (Rsqr (v intruderSpeed) / (g * tand (r phi)))%R in
      let idtrk :=
        match Rle_dec 3 trkrate with
        | left _ => 1
        | right _ =>
            match Rle_dec (3 / 2) trkrate with
            | left _ => 2
            | right _ =>
                match Rle_dec (3 / 4) trkrate with
                | left _ => 4
                | right _ => 8
                end
            end
        end in
      arc_loop intruder evader arcrad trkrate idtrk 0
  end.

Lemma step2 : forall (evader intruder : State) (x : nat), (distance evader intruder <= AlertRange)%R -> trkrate (bank intruder) <> 0%R -> arc_loop intruder evader (Rsqr intruderSpeed / (g * tand (bank intruder))) (trkrate (bank intruder)) x 0.