Theory Berlekamp_Zassenhaus

    Authors:      Jose Divasón
                  Sebastiaan Joosten
                  René Thiemann
                  Akihisa Yamada
section ‹The Polynomial Factorization Algorithm›

subsection ‹Factoring Square-Free Integer Polynomials›

text ‹We combine all previous results, i.e., Berlekamp's algorithm, Hensel-lifting, the reconstruction
  of Zassenhaus, Mignotte-bounds, etc., to eventually assemble the factorization algorithm for 
  integer polynomials.›

theory Berlekamp_Zassenhaus

private partial_function (tailrec) find_exponent_main :: "int  int  nat  int  nat" where
  [code]: "find_exponent_main p pm m bnd = (if pm > bnd then m
    else find_exponent_main p (pm * p) (Suc m) bnd)"

definition find_exponent :: "int  int  nat" where
  "find_exponent p bnd = find_exponent_main p p 1 bnd"
lemma find_exponent: assumes p: "p > 1" 
  shows "p ^ find_exponent p bnd > bnd" "find_exponent p bnd  0" 
proof -
    fix m and n
    assume "n = nat (1 + bnd - p^m)" and "m  1" 
    hence "bnd < p ^ find_exponent_main p (p^m) m bnd  find_exponent_main p (p^m) m bnd  1" 
    proof (induct n arbitrary: m rule: less_induct)
      case (less n m)
      note simp = find_exponent_main.simps[of p "p^m"]
      show ?case
      proof (cases "bnd < p ^ m")
        case True
        thus ?thesis using less unfolding simp by simp
        case False
        hence id: "find_exponent_main p (p ^ m) m bnd = find_exponent_main p (p ^ Suc m) (Suc m) bnd" 
          unfolding simp by (simp add: ac_simps)
        show ?thesis unfolding id 
          by (rule less(1)[OF _ refl], unfold less(2), insert False p, auto)
  from this[OF refl, of 1]
  show "p ^ find_exponent p bnd > bnd" "find_exponent p bnd  0"
    unfolding find_exponent_def by auto


definition berlekamp_zassenhaus_factorization :: "int poly  int poly list" where
  "berlekamp_zassenhaus_factorization f = (let 
     ― ‹find suitable prime›
     p = suitable_prime_bz f;
     ― ‹compute finite field factorization›
     (_, fs) = finite_field_factorization_int p f; 
     ― ‹determine maximal degree that we can build by multiplying at most half of the factors›
     max_deg = degree_bound fs;
     ― ‹determine a number large enough to represent all coefficients of every›
     ― ‹factor of lc * f› that has at most degree most max_deg›
     bnd = 2 * ¦lead_coeff f¦ * factor_bound f max_deg;
     ― ‹determine k› such that p^k > bnd›
     k = find_exponent p bnd;
     ― ‹perform hensel lifting to lift factorization to mod p^k›
     vs = hensel_lifting p k f fs
     ― ‹reconstruct integer factors›
   in zassenhaus_reconstruction vs p k f)" 
theorem berlekamp_zassenhaus_factorization_irreducibled:  
  assumes res: "berlekamp_zassenhaus_factorization f = fs" 
  and sf: "square_free f"
  and deg: "degree f > 0" 
  shows "f = prod_list fs  ( fi  set fs. irreducibled fi)" 
proof -
  let ?lc = "lead_coeff f" 
  define p where "p  suitable_prime_bz f" 
  obtain c gs where berl: "finite_field_factorization_int p f = (c,gs)" by force
  let ?degs = "map degree gs" 
  note res = res[unfolded berlekamp_zassenhaus_factorization_def Let_def, folded p_def,
    unfolded berl split, folded]
  from suitable_prime_bz[OF sf refl]
  have prime: "prime p" and cop: "coprime ?lc p" and sf: "poly_mod.square_free_m p f" 
    unfolding p_def by auto
  from prime interpret poly_mod_prime p by unfold_locales
  define n where "n = find_exponent p (2 * abs ?lc * factor_bound f (degree_bound gs))" 
  note n = find_exponent[OF m1, of "2 * abs ?lc * factor_bound f (degree_bound gs)",
    folded n_def]
  note bh = berlekamp_and_hensel_separated[OF cop sf refl berl n(2)]
  have db: "degree_bound (berlekamp_hensel p n f) = degree_bound gs" unfolding bh
    degree_bound_def max_factor_degree_def by simp
  note res = res[folded n_def bh(1)]
  show ?thesis 
    by (rule zassenhaus_reconstruction_irreducibled[OF prime cop sf deg refl _ res], insert n db, auto)

corollary berlekamp_zassenhaus_factorization_irreducible:
  assumes res: "berlekamp_zassenhaus_factorization f = fs" 
    and sf: "square_free f"
    and pr: "primitive f"
    and deg: "degree f > 0"
  shows "f = prod_list fs  ( fi  set fs. irreducible fi)" 
  using pr irreducible_primitive_connect[OF primitive_prod_list]
    berlekamp_zassenhaus_factorization_irreducibled[OF res sf deg] by auto