@*Matrices of Approximate Complex $1$-Jets.
This section implements some operations on matrices of |ACJ|s.
It is used to model intervals of isometries of $H^3$.

An |SL2ACJ| $g$ is a tuple |(a,b,c,d)| of |ACJ|s,
which represents the class $\reps{g}$ of functions
$h : \cube \ra \PSL_2(\Complex)$
such that $h_{0,0} \in \reps{a}$, $h_{0,1} \in \reps{b}$,
$h_{1,0} \in \reps{c}$, and $h_{1,1} \in \reps{d}$.
@<Definition of |SL2ACJ|@>=
struct SL2ACJ {
  SL2ACJ() :a(1),b(0),c(0),d(1) { }
  SL2ACJ(const ACJ& aa, const ACJ& bb, const ACJ& cc, const ACJ& dd)
     :a(aa),b(bb),c(cc),d(dd) { }
  ACJ a,b,c,d;
};

@ \proposition{M*J}
If |x| and |y| are |SL2ACJ|, then
$\reps{|x * y|} \supset \reps{x} \reps{y}$.
\endproposition
@<Definition of |x*y| for |SL2ACJ x,y|@>=
  return SL2ACJ(
    x.a*y.a+x.b*y.c, x.a*y.b+x.b*y.d, 
    x.c*y.a+x.d*y.c, x.c*y.b+x.d*y.d
  );

@ \proposition{inverse(M)}
If |x| is |SL2ACJ|, then
\endproposition
$\reps{|inverse(x)|} = \reps{x}^{-1}$.
@<Definition of |inverse(x)| for |SL2ACJ x|@>=
  return SL2ACJ(x.d, -x.b, -x.c, x.a);

@ \proposition{orthodist(M)}
If |x| is |SL2ACJ|, then
\endproposition
$\reps{|orthodist(x)|} \supset \olength{\reps{x}}$.
@<Definition of |orthodist(x)| for |SL2ACJ x|@>=
  ACJ t = x.a*x.d + x.b*x.c;
  ACJ r = ACJ(sqrt(t*t - 1));
  ACJ r1 = t+r;
  if (r1.f.re*r1.f.re + r1.f.im*r1.f.im >= 1)
    return t + r;
  else
    return t - r;

@ \proposition{length(M)}
If |x| is |SL2ACJ|, then
$\reps{|length(x)|} \supset \tlength{\reps{x}}$.
\endproposition
@<Definition of |length(x)| for |SL2ACJ x|@>=
  ACJ t = (x.a + x.d)*0.5;
  ACJ r = ACJ(sqrt(t*t-1));
  ACJ r1 = t+r;
  if (r1.f.re*r1.f.re + r1.f.im*r1.f.im >= 1)
    return (t+r)*(t+r);
  else
    return (t-r)*(t-r);

@ \proposition{notIdentity(M)}
If |x| is |SL2ACJ|, then
|notIdentity(x)| returns $1$ implies $\pm I \notin \reps{x}$
\endproposition
@<Definition of |notIdentity(x)| for |SL2ACJ x|@>=
  return absLB(x.b) > 0
      || absLB(x.c) > 0
      || (absLB(x.a-1) > 0 && absLB(x.a+1) > 0)
      || (absLB(x.d-1) > 0 && absLB(x.d+1) > 0);

@ \proposition{notFPower(M)}
If |x| is |SL2ACJ|, and |notFPower(x)| returns 1,
then for all $k$, $\pm f^k \notin \reps{M}$.
\endproposition
We actually test a stronger condition:
$M \neq \pm shortGenerator(x)$ for all $x$.
@<Definition of |notFPower(x)| for |SL2ACJ x|@>=
  return absLB(x.b) > 0 || absLB(x.c) > 0;

@ \proposition{shortGenerator(J)}
If |z1| is |ACJ|, then
$\reps{|shortGenerator(z)|}   \supset\shortGen{\reps{z}}$.
\endproposition
@<Definition of |shortGenerator(z)| for |ACJ z|@>=
  ACJ sz = sqrt(z);
  ACJ zero(0);
  return SL2ACJ(sz,zero,zero,1/sz);


@ \proposition{closeGenerator(J,J)}
If |x| and |z| are |ACJ|, then
$\reps{|closeGenerator(x,z)|} \supset\closeGen{\reps{x}}{\reps{z}}$.
\endproposition
@<Definition of |closeGenerator(x,z)| for |ACJ x,z|@>=
  ACJ sx = sqrt(x), sz = sqrt(z);
  ACJ sh = (sx-1/sx)*0.5;
  ACJ ch = (sx+1/sx)*0.5;
  return SL2ACJ(ch*sz, sh/sz, sh*sz, ch/sz);