Section ycngstys.
Unset Standard Proposition Elimination Names.
Variable intr : Trajectory.
Definition thetat : R -> R := theta intr.
Definition Rs (t : R) : R :=
sqrt (Rsqr (xi intr t - xi intr 0) + Rsqr (yi intr t - yi intr 0)).
Axiom
alphas_exists :
{ f : Differential_D2 |
forall t : R,
xi intr t = (Rs t * cos (f t) + xi intr 0)%R /\
yi intr t = (Rs t * sin (f t) + yi intr 0)%R }.
Definition alphas := let (a,_) := alphas_exists in a.
unfold alphas in |- *; case alphas_exists; trivial.
Qed.
intros; generalize (alphas_def t); intro H; elim H; intros;
rewrite Rplus_comm; assumption.
Qed.
intros; generalize (alphas_def t); intro H; elim H; intros;
rewrite Rplus_comm; assumption.
Qed.
intros; rewrite Rsx; unfold Rminus, Rdiv in |- *;
rewrite (Rplus_comm (xi intr 0)); rewrite Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite Rmult_assoc;
rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | assumption ].
Qed.
intros; rewrite Rsy; unfold Rminus, Rdiv in |- *;
rewrite (Rplus_comm (yi intr 0)); rewrite Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite Rmult_assoc;
rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | assumption ].
Qed.

(* With the two formulas above, you can prove the following result *)
(* Using the cos_sin_0's lemma                                     *)
Lemma Rs_derivable : derivable Rs.