vectors_cos[n: posnat]: THEORY
BEGIN

%                                      
%                                      .
%                                     / \
%                                   /    \
%                                 /       \
%                               /          \
%                      vc     /             \
%                           /                \    vb
%                         /                   \
%                       /                      \
%                     /                         \
%                   /                            \
%                 /                           ab  \
%               /                                  \
%               ------------------------------------ 
%                               va                   

  IMPORTING vectors, trig@trig_basic

  va,vb,vc,v0,v1,v2: VAR Vector[n]

  ab: VAR real

  cosines_law   : LEMMA
                     LET a = norm(va),
                         b = norm(vb),
                         c = norm(vc)
                     IN
                      vc = va - vb AND
                      a*b*cos(ab) = va*vb
                        IMPLIES
                           sq(c) = sq(a) + sq(b) - 2*a*b*cos(ab)


  IMPORTING trig@trig_inverses

  angle_exists: LEMMA (EXISTS ab:  LET  a = norm(va),
                                        b = norm(vb) IN
                              a*b*cos(ab) = va*vb)



  angle_between(u,v:Nz_vector[n]): real = arccos(u*v/(norm(u)*norm(v)))


  cosines_law_bnd : LEMMA
                     LET a = norm(va),
                         b = norm(vb),
                         c = norm(vc)
                     IN
                         vc = va - vb IMPLIES
                            sq(c) >= sq(a-b)

  cosines_law_ge : LEMMA
                     LET a = norm(va),
                         b = norm(vb),
                         c = norm(vc)
                     IN
                         vc = va - vb IMPLIES
                            c >= abs(a-b)


  cosines_law_le : LEMMA 
                     LET a = norm(va),
                         b = norm(vb),
                         c = norm(vc)
                     IN
                         vc = va - vb IMPLIES
                                 c <= a + b
 


END vectors_cos