Section ycngftys.
Unset Standard Proposition Elimination Names.
Variable intr : Trajectory.
(**********)
Definition xi : R -> R := x intr.
Definition yi : R -> R := y intr.
Definition vi : TypeSpeed := h intr.
Definition xP (t : R) : R := (xi t - xi 0)%R.
Definition yP (t : R) : R := (yi t - yi 0)%R.
Definition u (eps : posreal) (t : R) : R :=
sqrt (Rsqr (xP t) + Rsqr (yP t) + Rsqr eps).
Definition u0 (t : R) : R := sqrt (Rsqr (xP t) + Rsqr (yP t)).
unfold xi in |- *; apply (cond_diff (x intr)).
Qed.
unfold yi in |- *; apply (cond_diff (y intr)).
Qed.
unfold xP in |- *.
reg; apply xi_derivable.
Qed.
unfold yP in |- *.
reg; apply yi_derivable.
Qed.
intro.
unfold u in |- *; unfold derivable in |- *; intro.
reg.
apply yP_derivable.
apply xP_derivable.
apply Rle_lt_trans with (Rsqr (xP x) + Rsqr (yP x))%R.
apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite Rplus_assoc.
apply Rplus_lt_compat_l.
pattern (Rsqr (yP x)) at 1 in |- *; rewrite <- Rplus_0_r.
apply Rplus_lt_compat_l.
apply Rsqr_pos_lt.
red in |- *; intro; assert (H0 := cond_pos eps); rewrite H in H0;
elim (Rlt_irrefl _ H0).
Qed.
intros; set (f := u eps); reg.
apply (d_u eps).
Qed.
intros; set (f := u eps).
cut (derivable_pt f t); [ intro H | apply (d_u eps t) ].
cut (derive_pt f t H = derive_pt f t (d_u eps t)); [ intro | apply pr_nu ].
reg.
Qed.
intro; unfold xP in |- *; reg.
apply pr_nu.
apply xi_derivable.
Qed.
intro; unfold yP in |- *; reg.
apply pr_nu.
apply yi_derivable.
Qed.
intros; reg.
apply yP_derivable.
apply xP_derivable.
Qed.

(**********)
Lemma D_Rsqr_u_epsilon_2 :
forall (eps : posreal) (t : R),
(derive_pt (fun y : R => Rsqr (u eps y)) t (fct_der2 eps t) <= 2 * u0 t * vi)%R.