Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record Trajectory : Type := mkTrajectory
  {x : Differential;
   y : Differential;
   theta : Differential;
   phi : Differential;
   h : TypeSpeed;
   cond_x :
    forall t : R, derive_pt x t (cond_diff x t) = (h * cos (theta t))%R;
   cond_y :
    forall t : R, derive_pt y t (cond_diff y t) = (h * sin (theta t))%R;
   cond_phi : forall t : R, (- MaxBank <= phi t <= MaxBank)%R;
   cond_theta :
    forall t : R,
    derive_pt theta t (cond_diff theta t) = (g * (tan (phi t) / h))%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.Record EvaderTrajectory : Type := mkEvaderTrajectory
  {tr : Trajectory;
   tr_cond1 : forall t : R, x tr t = (x tr 0 + h tr * t)%R;
   tr_cond2 : forall t : R, y tr t = y tr 0%R;
   tr_cond3 : forall t : R, theta tr t = 0%R;
   tr_cond4 : forall t : R, phi tr t = 0%R}.

Lemma init_tr_cond_y : forall (h : TypeSpeed) (t : R), derive_pt (fun t : R => 0%R) t (init_tr_derivable_y h t) = (h * sin ((fun t : R => 0) t))%R.