Proof with trivial.
intros hyp_intr hyp_evad; intros...
unfold conflict in |- *...
cut
(l intr evad T * cos (beta intr evad T) <
(Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange) /
(2 * ve evad * T))%R...
cut
((thetat intr 0 + beta intr evad T < PI / 2)%R \/
(PI / 2 <= thetat intr 0 + beta intr evad T)%R /\
(3 * (PI / 2) < thetat intr 0 + beta intr evad T)%R)...
intros...
case (Rle_dec (Die intr evad T T) ConflictRange); intro...
unfold Die in r...
generalize (isometric_evader intr evad T T)...
unfold Rsqr_evader_distance in |- *...
unfold xe, ye in |- *...
intro...
rewrite H6 in r...
cut (0 <= T)%R...
cut (rho_vi intr * T <= PI / 2)%R...
intros...
generalize (ypt_PI2 intr evad T T H8 H7); intro...
generalize
(xpt_PI intr evad T T H8
(Rle_trans (rho_vi intr * T) (PI / 2) PI H7
(Rlt_le (PI / 2) PI PI2_Rlt_PI))); intro...
elim H4; intros...
rewrite Rplus_comm in H11...
cut (MinBeta <= beta intr evad T + thetat intr 0)%R...
intro...
elim H9; intros...
generalize
(Math_prop_no_conflict_1 (beta intr evad T + thetat intr 0)
(l intr evad T) (xp intr evad T T) (yp intr evad T T) T H0 H1 H12
(Rlt_le (beta intr evad T + thetat intr 0) (PI / 2) H11))...
unfold r_V, rho_V in |- *...
unfold r_vi, rho_vi in H13...
unfold r_vi, rho_vi in H10...
unfold vi in H10...
rewrite hyp_intr in H10...
unfold vi in H13...
rewrite hyp_intr in H13...
intro...
generalize (H15 H13 H10)...
intro...
cut (0 <= sqrt (Rsqr (xp intr evad T T) + Rsqr (yp intr evad T T)))%R...
cut (0 <= ConflictRange)%R...
intros...
generalize
(Rsqr_incr_1 (sqrt (Rsqr (xp intr evad T T) + Rsqr (yp intr evad T T)))
ConflictRange r H18 H17); intro...
rewrite Rsqr_sqrt in H19...
elim
(Rlt_irrefl (Rsqr ConflictRange)
(Rlt_le_trans (Rsqr ConflictRange)
(Rsqr (xp intr evad T T) + Rsqr (yp intr evad T T))
(Rsqr ConflictRange) H16 H19))...
apply Rplus_le_le_0_compat; apply Rle_0_sqr...
unfold ConflictRange in |- *; left; prove_sup...
apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr...
apply cos_decr_0...
generalize (beta_def intr evad T); intro...
decompose [and] H12...
left; apply Rlt_trans with (PI / 2)%R...
apply PI2_Rlt_PI...
left; apply MinBeta_pos...
unfold MinBeta in |- *...
left; apply Rlt_trans with 1%R...
unfold Rdiv in |- *...
apply Rmult_lt_reg_l with 1000%R...
prove_sup...
rewrite Rmult_1_r; rewrite (Rmult_comm 1000); rewrite Rmult_assoc...
rewrite <- Rinv_l_sym; [ prove_sup | discrR ]...
apply Rlt_trans with (PI / 2)%R...
apply Rlt_1_PI2...
apply PI2_Rlt_PI...
apply Rle_trans with (cos (beta intr evad T))...
apply cos_beta_NOT_Alpha...
elim H11; intros...
rewrite (Rplus_comm (thetat intr 0)) in H13...
cut (beta intr evad T + thetat intr 0 <= 2 * PI - MinBeta)%R...
intro...
generalize
(Math_prop_no_conflict_2 (beta intr evad T + thetat intr 0)
(l intr evad T) (xp intr evad T T) (yp intr evad T T) T H0 H1
(Rlt_le (3 * (PI / 2)) (beta intr evad T + thetat intr 0) H13) H14);
intro...
unfold r_V, rho_V in H15...
unfold r_vi, rho_vi in H10...
elim H9; intros...
unfold r_vi, rho_vi in H17...
unfold vi in H17...
rewrite hyp_intr in H17...
unfold vi in H10...
rewrite hyp_intr in H10...
generalize (H15 H17 H10); intro...
cut (0 <= sqrt (Rsqr (xp intr evad T T) + Rsqr (yp intr evad T T)))%R...
cut (0 <= ConflictRange)%R...
intros...
generalize
(Rsqr_incr_1 (sqrt (Rsqr (xp intr evad T T) + Rsqr (yp intr evad T T)))
ConflictRange r H20 H19); intro...
rewrite Rsqr_sqrt in H21...
elim
(Rlt_irrefl (Rsqr ConflictRange)
(Rlt_le_trans (Rsqr ConflictRange)
(Rsqr (xp intr evad T T) + Rsqr (yp intr evad T T))
(Rsqr ConflictRange) H18 H21))...
apply Rplus_le_le_0_compat; apply Rle_0_sqr...
unfold ConflictRange in |- *; left; prove_sup...
apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr...
apply cos_incr_0...
left; apply Rlt_trans with (3 * (PI / 2))%R...
pattern PI at 1 in |- *; rewrite <- (Rplus_0_r PI)...
replace (3 * (PI / 2))%R with (PI + PI / 2)%R...
apply Rplus_lt_compat_l...
apply PI2_RGT_0...
pattern PI at 1 in |- *; rewrite double_var; ring...
generalize (beta_def intr evad T); intro...
decompose [and] H14...
left...
left...
apply Rplus_lt_reg_l with (MinBeta - PI)%R...
replace (MinBeta - PI + PI)%R with MinBeta...
replace (MinBeta - PI + (2 * PI - MinBeta))%R with PI...
unfold MinBeta in |- *...
apply Rlt_trans with 1%R...
unfold Rdiv in |- *...
apply Rmult_lt_reg_l with 1000%R...
prove_sup...
rewrite (Rmult_comm 1000); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
[ repeat rewrite Rmult_1_r; prove_sup | discrR ]...
apply Rlt_trans with (PI / 2)%R...
apply Rlt_1_PI2...
apply PI2_Rlt_PI...
ring...
ring...
unfold Rminus in |- *...
pattern (2 * PI)%R at 2 in |- *; rewrite <- (Rplus_0_r (2 * PI))...
apply Rplus_le_compat_l...
left; rewrite <- Ropp_0...
apply Ropp_lt_gt_contravar...
apply MinBeta_pos...
unfold Rminus in |- *...
rewrite (Rplus_comm (2 * PI))...
generalize (cos_period (- MinBeta) 1)...
unfold INR in |- *...
rewrite Rmult_1_r...
intro...
rewrite H14...
rewrite cos_neg...
apply Rle_trans with (cos (beta intr evad T))...
apply cos_beta_NOT_Alpha...
left; replace (rho_vi intr) with rho_V...
apply rho_t_PI2...
unfold rho_vi, rho_V in |- *...
unfold vi in |- *; rewrite hyp_intr...
left; apply Rlt_le_trans with MinT...
apply MinT_is_pos...
apply (cond_1 T)...
cut (Omega (thetat intr 0 + beta intr evad T) = false)...
unfold Omega in |- *...
case (Rle_dec (PI / 2) (thetat intr 0 + beta intr evad T)); intro...
case (Rle_dec (thetat intr 0 + beta intr evad T) (3 * (PI / 2))); intros...
elim diff_true_false...
right...
split...
auto with real...
intro...
left; auto with real...
cut (Alpha (beta intr evad T) = false)...
unfold Alpha in |- *...
case
(Rle_dec
((Rsqr (ve evad * T) + Rsqr (l intr evad T) - Rsqr AlertRange) /
(2 * ve evad * T)) (l intr evad T * cos (beta intr evad T)));
intros...
elim diff_true_false...
auto with real...
Qed.
(**************************************************************)
(*************************** THEOREME *************************)
(**************************************************************)

Theorem ails_no_conflict_tau_le0 :
h intr = V ->
h (tr evad) = V ->
(MinDistance T <= l intr evad T)%R ->
(l intr evad T <= MaxDistance T)%R ->
(AlertRange < d intr evad)%R ->
Omega (thetat intr 0 + beta intr evad T) = false ->
(tau (measure2state intr 0) (measure2state (tr evad) 0) 0 <= 0)%R ->
conflict intr evad T = false.