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%------------------------------------------------------------------------------
% definitions.pvs
% ACCoRD v1.0
%------------------------------------------------------------------------------
definitions : THEORY
BEGIN
IMPORTING vectors@vectors_2D,
vectors@det_2D,
vectors@closest_approach_2D,
reals@quad_minmax
s,v,w,
vo,vi,
vo1,vo2 : VAR Vect2
dir : VAR Sign
nzs,nzv : VAR Nz_vect2
t,k,j : VAR real
nzj : VAR nzreal
t1,t2 : VAR nnreal
sz,vz : VAR real
nzvz : VAR nzreal
Entry : MACRO Sign = -1
Exit : MACRO Sign = 1
%------------------------------------------------------------------------------
% Horizontal Definitions
%------------------------------------------------------------------------------
%% horizontal_dir?(s,v,Entry) if s,v points towards (0,0)
%% horizontal_dir?(s,v,Exit) if s,v, points away from (0,0)
horizontal_dir?(s,v,dir) : MACRO bool =
dir*(s*v) >= 0
horizontal_dir_at?(s,v,t,dir) : MACRO bool =
horizontal_dir?(s+t*v,v,dir)
horizontal_entry?(s,v) : MACRO bool =
horizontal_dir?(s,v,Entry)
horizontal_entry_at?(s,v,t) : MACRO bool =
horizontal_entry?(s+t*v,v)
horizontal_exit?(s,v) : MACRO bool =
horizontal_dir?(s,v,Exit)
horizontal_exit_at?(s,v,t) : MACRO bool =
horizontal_exit?(s+t*v,v)
horizontal_tangent?(s,v) : MACRO bool =
s*v = 0
horizontal_diverging?(s,v) : MACRO bool =
s*v > 0
horizontal_converging?(s,v) : MACRO bool =
s*v < 0
% Returns the eps that determines the closest horizontal tangent to relative
% velocity vector v
horizontalCoordination(s,v) : Sign =
sign(det(v,s))
% HorizontalCoordinationConflict is an horizontal strategy
horizontalCoordination_symm : LEMMA
horizontalCoordination(-s,-v) = horizontalCoordination(s,v)
Horizontal_Strategy : TYPE+ = {f:[[Vect2,Vect2]->Sign]| FORALL (s,v):f(-s,-v)=f(s,v)}
horizontalCoordination_strategy : JUDGEMENT
horizontalCoordination HAS_TYPE Horizontal_Strategy
% Time of closest approach
horizontal_tca(s,nzv) : real =
-(s*nzv)/sqv(nzv)
horizontal_tca_gt_0 : LEMMA
horizontal_converging?(s,v) IMPLIES horizontal_tca(s,v) > 0
horizontal_tca_nzv_gt_0 : LEMMA
horizontal_converging?(s,nzv) IFF horizontal_tca(s,nzv) > 0
horizontal_tca_lt_0 : LEMMA
horizontal_diverging?(s,v) IMPLIES horizontal_tca(s,v) < 0
horizontal_tca_nzv_lt_0 : LEMMA
horizontal_diverging?(s,nzv) IFF horizontal_tca(s,nzv) < 0
horizontal_tca_dot_zero: LEMMA
(s+horizontal_tca(s,nzv)*nzv)*nzv = 0
% Square of distance at time of closest approach
horizontal_sq_dtca(s,nzv) : real =
sq(det(s,nzv))/sqv(nzv)
horizontal_sq_dtca_eq : LEMMA
horizontal_sq_dtca(s,nzv) = sqv(s+horizontal_tca(s,nzv)*nzv)
horizontal_tca_min : LEMMA
horizontal_sq_dtca(s,nzv) <= sqv(s+t*nzv)
sqv_decreasing_before_horizontal_tca: LEMMA
LET htca = horizontal_tca(s,nzv) IN
FORALL (tr1,tr2:real): (tr1<=htca AND tr2<=htca) IMPLIES
(tr1<tr2 IFF sqv(s+tr1*nzv)>sqv(s+tr2*nzv))
sqv_increasing_after_horizontal_tca: LEMMA
LET htca = horizontal_tca(s,nzv) IN
FORALL (tr1,tr2:real): (tr1>=htca AND tr2>=htca) IMPLIES
(tr1>tr2 IFF sqv(s+tr1*nzv)>sqv(s+tr2*nzv))
% Velocity vector comparison with respect to a vector
le?(v)(vo1,vo2): bool =
norm(vo1-v) <= norm(vo2-v)
le_rel : LEMMA
le?(vo-vi)(vo1-vi,vo2-vi) IFF le?(vo)(vo1,vo2)
% Vdir is a vector perpendicular to nzs that points in the same
% direction as v
Vdir(nzs,v) : Nz_vect2 =
LET ps = perpR(nzs) IN
sign(ps*v)*ps
% Vdir(s,v) is perpendicular to s
Vdir_perp : LEMMA
nzs*Vdir(nzs,v) = 0
% W0(s,j) is a vector such that s*W0(s,j) = j
W0(nzs,j) : Vect2 =
(j/sqv(nzs))*nzs
W0_dot : LEMMA
nzs*W0(nzs,j) = j
nz_W0 : JUDGEMENT
W0(nzs,nzj) HAS_TYPE Nz_vect2
% W is a parametric vector that represents the equation: nzs*W=j
% if k is positive W points in the same direction as v
W(nzs,v,k)(j) : MACRO Vect2 =
W0(nzs,j)+k*Vdir(nzs,v)
nz_W : JUDGEMENT
W(nzs,v,k)(nzj) HAS_TYPE Nz_vect2
W_dot : LEMMA
nzs*W(nzs,v,k)(j) = j
dot_W : LEMMA
nzs*w = j IFF
EXISTS (k) : w = W(nzs,v,k)(j)
s_dot_Vdir_eq_0: LEMMA
nzs*Vdir(nzs,v) = 0
W0_dot_Vdir_eq_0: LEMMA
W0(nzs,j)*Vdir(nzs,v) = 0
norm_W0_ge: LEMMA
t1 >= t2 AND
j >= 0 IMPLIES
norm(nzs+t1*W0(nzs,j)) >= norm(nzs+t2*W0(nzs,j))
divergent?(s,v): MACRO bool =
divergent?(s,zero,v,zero)
dot_pos_divergent : LEMMA
s * v > 0 IMPLIES divergent?(s,v)
dot_nneg_divergent: LEMMA
s*nzv >= 0 IFF divergent?(s,nzv)
gs(v) : nnreal = norm(v)
gs_nz : JUDGEMENT
gs(nzv) HAS_TYPE posreal
%%
%% Ground Speed Only Solutions
%%
gs_only?(vo1)(vo2) : bool =
EXISTS (l:posreal): vo2 = l*vo1
gs_only_id : LEMMA
gs_only?(v)(v)
gs_only_scal : LEMMA
FORALL (l:posreal):
gs_only?(v)(l*v)
gs_only_symm : LEMMA
gs_only?(vo1)(vo2) IFF gs_only?(vo2)(vo1)
gs_only_trans : LEMMA
gs_only?(vo)(vo1) AND gs_only?(vo1)(vo2) IMPLIES
gs_only?(vo)(vo2)
gs_only_normalize : LEMMA
gs_only?(nzv)(v) IFF
EXISTS (k:posreal): v = k*^(nzv)
gs_only_zero_left : LEMMA
gs_only?(zero)(v) IFF v = zero
gs_only_zero_right : LEMMA
gs_only?(v)(zero) IFF v = zero
%%
%% Track Only Solutions
%%
trk_only?(vo1)(vo2) : bool =
norm(vo2) = norm(vo1)
trk_only_id : LEMMA
trk_only?(v)(v)
trk_only_symm : LEMMA
trk_only?(vo1)(vo2) IFF trk_only?(vo2)(vo1)
trk_only_zero_left : LEMMA
trk_only?(zero)(v) IFF v = zero
trk_only_zero_right : LEMMA
trk_only?(v)(zero) IFF v = zero
%%
%% Optimal Solutions
%%
optimal?(vo,nzv)(vo1) : bool =
FORALL (vo2) : parallel?(vo2-vo1,nzv) IMPLIES le?(vo)(vo1,vo2)
optimal_id : LEMMA
optimal?(vo,nzv)(vo)
%------------------------------------------------------------------------------
% Vertical Definitions
%------------------------------------------------------------------------------
%% vertical_dir?(sz,vz,Entry) if sz,vz points towards 0
%% vertical_dir?(sz,vz,Exit) if sz,vz points away from 0
vertical_dir?(sz,vz,dir) : MACRO bool =
dir*(sz*vz) >= 0
vertical_dir_at?(sz,vz,t,dir) : MACRO bool =
vertical_dir?(sz+t*vz,vz,dir)
vertical_entry?(sz,vz) : MACRO bool = vertical_dir?(sz,vz,Entry)
vertical_exit?(sz,vz) : MACRO bool = vertical_dir?(sz,vz,Exit)
vertical_tau(sz,nzvz) : real =
-sz/nzvz
z_at_vertical_tau_eq_0 : LEMMA
sz + vertical_tau(sz,nzvz)*nzvz = 0
END definitions
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