Theory Extended_Interval_Division

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chapter‹Extended Division on Intervals›
theory
  Extended_Interval_Division
imports
  Interval_Division_Non_Zero
begin

text‹
  In this theory, we define an extended division operation on intervals. This definition is inspired
  by the definition given in~\<^cite>‹"moore.ea:introduction:2009"›, but we use an over-approximation
  for the case in which zero is an element of the divisor interval. By this, we avoid the need for multi-intervals.
 ›

instantiation "interval" :: ("{infinity, linordered_field, real_normed_algebra,linear_continuum_topology}") inverse
begin
  definition inverse_interval :: "'a interval ⇒ 'a interval"
    where "inverse_interval a = (
                                  if (¬ 0 ∈⇩i a) then mk_interval ( 1 / (upper a), 1 / (lower a))
                                  else if lower a = 0 ∧ 0 < upper a then mk_interval (1/ upper a, ∞)
                                  else if lower a < 0 ∧ 0 < upper a then mk_interval (-∞, ∞)
                                  else if lower a < upper a ∧ upper a = 0 then mk_interval(-∞, 1 / lower a)
                                  else undefined 
                                )"

  definition divide_interval :: "'a interval ⇒ 'a interval ⇒ 'a interval"
    where "divide_interval a b = inverse b * a"
  instance ..
end

interpretation interval_division_inverse divide inverse
  apply(unfold_locales)
  subgoal
    by (simp add: inverse_interval_def)
  subgoal
    by(simp add: divide_interval_def)
  done

end