application "tropical";

# Let's start with some basic tropical arithmetic.
$a = new TropicalNumber<Max>(3); $b = new TropicalNumber<Max>(-4);
print $a + $b;
print $a * $b;

# We have basic tropical linear algebra
$m = new Matrix<TropicalNumber<Min> >([[0,1],["inf",-3]]); $v = new Vector<TropicalNumber<Min> >(1,0);
print $m * $v;
print $v * $v;
print tdet($m);

# Tropical polynomials are created just like ordinary polynomials:
$r = new Ring<TropicalNumber<Max> >(2); #Tropical polynomial ring in two variables
($x,$y) = $r->variables;
$p = (new TropicalNumber<Max>(3)) * $x*$x + $y;
print $p;

# There is an easier way to do this:
$p = toTropicalPolynomial("max(2x + 3,y)");
$p = toTropicalPolynomial("max(2x0 + 3, x1)",$r);
$p = toTropicalPolynomial("max(2x + 3,y)",$r,1);

#We can create tropical convex hulls
$c = new Polytope<Max>(POINTS=>[[0,0,"-inf"],[0,1,1],[0,0,2]]);
$c->VISUAL;
$c->VISUAL_SUBDIVISION;
$c->POLYTOPE_COVECTOR_DECOMPOSITION->VISUAL;
$d = cyclic<Min>(3,5);
$d->VISUAL;
$e = new Polytope<Min>(POINTS=>[[0,0,"inf","inf"],[0,"inf",0,"inf"],[0,"inf","inf",0],["inf",0,0,"inf"],["inf",0,"inf",0],["inf","inf",0,0]]);
$e->VISUAL;

#Parametrized linear programming
$monomial=new UniMonomial<Rational,Rational>(1);
$t=new PuiseuxFraction<Min>($monomial);
$c = polytope::cube<PuiseuxFraction<Min> >(3,$t,0);
print $c->VERTICES;
print new Matrix<PuiseuxFraction<Min> >($c->FACETS);
$c->LP = new polytope::LinearProgram<PuiseuxFraction<Min> >(LINEAR_OBJECTIVE=>[1,1,-$t,1]);
print $c->LP->MAXIMAL_VALUE;
print $c->LP->MAXIMAL_VERTEX;
print $c->VOLUME;


# The basic object of tropical geometry: The tropical cycle
# Note that everything is in projective coordinates and has an additional leading coordinate.
$c = new Cycle<Min>(PROJECTIVE_VERTICES=>[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],MAXIMAL_POLYTOPES=>[[0,1],[0,2],[0,3]],WEIGHTS=>[1,1,1]);
print is_balanced($c);
$c = new Cycle<Min>(PROJECTIVE_VERTICES=>thomog([[1,0,0],[0,1,0],[0,0,1],[0,-1,-1]]),MAXIMAL_POLYTOPES=>[[0,1],[0,2],[0,3]],WEIGHTS=>[1,1,1]);
$d = new Cycle<Max>(PROJECTIVE_VERTICES=>thomog([[1,0,0],[0,1,0],[0,0,1],[0,-1,-1]]),MAXIMAL_POLYTOPES=>[[0,1],[0,2],[0,3]],WEIGHTS=>[1,1,1]);
print $d->DEGREE;

# We can construct hypersurfaces from homogeneous polynomials!
$p = toTropicalPolynomial("min(a,b,c)");
$h = new Hypersurface<Min>(POLYNOMIAL=>$p);
print $h->VERTICES;
print $h->MAXIMAL_POLYTOPES;
print $h->WEIGHTS;
$h->VISUAL;

# Polymake can compute arbitrary tropical linear spaces

$p=new Vector<Int>( [0,0,3,1,2,1,0,1,0,2,2,0,3,0,4,1,2,2,0,0]);
#This is an element of the (min-) tropical Grassmannian!
$I=ideal::pluecker_ideal(3,6);
#Initial ideals always use max and require nonnegative weight vectors.
$pp=new Vector<Int>(5*ones_vector(20)-$p); 
$J=new ideal::Ideal(GENERATORS=> $I->GROEBNER(ORDER_VECTOR=>$pp)->INITIAL_FORMS);
print $J->contains_monomial();
#Let's compute the corresponding tropical linear space.
$mat = new matroid::ValuatedMatroid<Min>( BASES=>matroid::uniform_matroid(3,6)->BASES, VALUATION_ON_BASES=>$p, N_ELEMENTS=>6);
$tl = linear_space($mat);
#Let's compute its bounded cells.
@bounded = map { grep { ($_ * $tl->FAR_VERTICES)->size == 0} @{rows($_)} } @{$tl->CONES}; 
print join(",",@bounded);

# Now for some intersection theory
$F = toTropicalPolynomial("min(12+3*x0,-131+2*x0+x1, -67+2*x0+x2,-9+2*x0+x3,-131+x0+2*x1,-129+x0+x1+x2, -131+x0+x1+x3,-116+x0+2*x2,-76+x0+x2+x3,-24+x0+2*x3,-95+3*x1, -108+2*x1+x2,-92+2*x1+x3,-115+x1+2*x2,-117+x1+x2+x3, -83+x1+2*x3,-119+3*x2,-119+2*x2+x3,-82+x2+2*x3,-36+3*x3)");
$V = new Hypersurface<Min>(POLYNOMIAL=>$F);
print intersect(intersect($V,$V),$V)->DEGREE;
#The standard tropical line lies in this hypersurface.
$B = matroid_fan<Min>(matroid::uniform_matroid(2,4));
print intersect_in_smooth_surface($V,$B,$B)->DEGREE;

# Let's look at the lines in this cubic
$L = lines_in_cubic($F);
print $L->N_ISOLATED,", ",$L->N_FAMILIES,"\n";
@rp = map { $_->representative() } $L->all_families;
visualize_in_surface($V, @rp);

# There is an interface to gfan
$r = new Ring<Rational>(4);
($x,$y,$z,$w) = $r->variables;
$p = $x + $y +$z + $w;
$q = $x*$x - 3*$w*$z;
$ideal = new ideal::Ideal(GENERATORS=>[$p,$q]);
$trop = gfan_tropicalvariety_of_prime($ideal);
print $trop->VERTICES;