Section trajectory.
Unset Standard Proposition Elimination Names.
Variable intr : Trajectory.
Variable evad : EvaderTrajectory.
Variable T : TimeT.
unfold thetat in |- *; apply (cond_diff (theta intr)).
Qed.
Definition xe : R := x (tr evad) T.
Definition ye : R := y (tr evad) T.
Definition ve : R := v (h (tr evad)).
Definition r_vi : R := r (vi intr).
Definition rho_vi : R := rho (vi intr).
Definition Die (ti te : R) : R :=
sqrt
(Rsqr (xi intr ti - x (tr evad) te) + Rsqr (yi intr ti - y (tr evad) te)).
Definition conflict (t : R) : bool :=
match Rle_dec (Die t t) ConflictRange with
| left _ => true
| right _ => false
end.
intro; unfold xi, vi, thetat in |- *.
assert (H := cond_x intr t).
rewrite <- H.
apply pr_nu.
Qed.
intro; unfold yi, vi, thetat in |- *.
assert (H := cond_y intr t).
rewrite <- H.
apply pr_nu.
Qed.

(* Easy to prove with constructive reals *)
Axiom r_sin_rho : (ConflictRange < r_vi * sin rho_vi)%R.