Theory LongDiv

(*
    Experimental theory: long division of polynomials
    $Id: LongDiv.thy,v 1.8 2005/09/06 14:59:48 wenzelm Exp $
    Author: Clemens Ballarin, started 23 June 1999
*)

theory LongDiv imports PolyHomo begin

consts
  lcoeff :: "'a::ring up => 'a"
  eucl_size :: "'a::ring => nat"

defs
  lcoeff_def: "lcoeff p == coeff p (deg p)"
  eucl_size_def: "eucl_size p == (if p = 0 then 0 else deg p+1)"

lemma SUM_shrink_below_lemma:
  "!! f::(nat=>'a::ring). (ALL i. i < m --> f i = 0) --> 
  setsum (%i. f (i+m)) {..d} = setsum f {..m+d}"
  apply (induct_tac d)
   apply (induct_tac m)
    apply (simp)
   apply (force)
  apply (simp add: ab_semigroup_add_class.add_commute[of m]) 
  done

end

lemma SUM_shrink_below_lemma:

  (∀i<m. f i = (0::'a)) --> (∑i≤d. f (i + m)) = setsum f {..m + d}

theorem SUM_extend_below:

  [| m ≤ n; !!i. i < m ==> f i = (0::'a); P (∑i≤n - m. f (i + m)) |]
  ==> P (setsum f {..n})

theorem up_repr2D:

  [| deg p ≤ n; P (∑i≤n. coeff p i*X^i) |] ==> P p

theorem deg_lcoeff_cancel:

  [| deg p ≤ deg r; deg q ≤ deg r; coeff p (deg r) = - coeff q (deg r);
     deg r ≠ 0 |]
  ==> deg (p + q) < deg r

theorem deg_lcoeff_cancel2:

  [| deg p ≤ deg r; deg q ≤ deg r; p ≠ - q; coeff p (deg r) = - coeff q (deg r) |]
  ==> deg (p + q) < deg r

theorem long_div_eucl_size:

  g ≠ 0
  ==> Ex (%(q, r, k). lcoeff g ^ k *s f = q * g + r ∧ eucl_size r < eucl_size g)

theorem long_div_ring:

  g ≠ 0
  ==> Ex (%(q, r, k). lcoeff g ^ k *s f = q * g + r ∧ (r = 0 ∨ deg r < deg g))

theorem long_div_unit:

  [| g ≠ 0; lcoeff g dvd (1::'a) |]
  ==> Ex (%(q, r). f = q * g + r ∧ (r = 0 ∨ deg r < deg g))

theorem long_div_theorem:

  g ≠ 0 ==> Ex (%(q, r). f = q * g + r ∧ (r = 0 ∨ deg r < deg g))

theorem long_div_quo_unique:

  [| g ≠ 0; f = q1.0 * g + r1.0; r1.0 = 0 ∨ deg r1.0 < deg g; f = q2.0 * g + r2.0;
     r2.0 = 0 ∨ deg r2.0 < deg g |]
  ==> q1.0 = q2.0

theorem long_div_rem_unique:

  [| g ≠ 0; f = q1.0 * g + r1.0; r1.0 = 0 ∨ deg r1.0 < deg g; f = q2.0 * g + r2.0;
     r2.0 = 0 ∨ deg r2.0 < deg g |]
  ==> r1.0 = r2.0