Theory Groebner_Basis

(*  Title:      HOL/Groebner_Basis.thy
    Author:     Amine Chaieb, TU Muenchen

section ‹Groebner bases›

theory Groebner_Basis
imports Semiring_Normalization Parity

subsection ‹Groebner Bases›

lemmas bool_simps = simp_thms(1-34) ― ‹FIXME move to theoryHOL.HOL

lemma nnf_simps: ― ‹FIXME shadows fact binding in theoryHOL.HOL
  "(¬(P  Q)) = (¬P  ¬Q)" "(¬(P  Q)) = (¬P  ¬Q)"
  "(P  Q) = (¬P  Q)"
  "(P = Q) = ((P  Q)  (¬P  ¬ Q))" "(¬ ¬(P)) = P"
  by blast+

lemma dnf:
  "(P  (Q  R)) = ((PQ)  (PR))"
  "((Q  R)  P) = ((QP)  (RP))"
  "(P  Q) = (Q  P)"
  "(P  Q) = (Q  P)"
  by blast+

lemmas weak_dnf_simps = dnf bool_simps

lemma PFalse:
    "P  False  ¬ P"
    "¬ P  (P  False)"
  by auto

named_theorems algebra "pre-simplification rules for algebraic methods"
ML_file ‹Tools/groebner.ML›

method_setup algebra = let
    fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
    val addN = "add"
    val delN = "del"
    val any_keyword = keyword addN || keyword delN
    val thms = Scan.repeats (Scan.unless any_keyword Attrib.multi_thm);
    Scan.optional (keyword addN |-- thms) [] --
     Scan.optional (keyword delN |-- thms) [] >>
    (fn (add_ths, del_ths) => fn ctxt =>
      SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
  end "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"

declare dvd_def[algebra]
declare mod_eq_0_iff_dvd[algebra]
declare mod_div_trivial[algebra]
declare mod_mod_trivial[algebra]
declare div_by_0[algebra]
declare mod_by_0[algebra]
declare mult_div_mod_eq[algebra]
declare div_minus_minus[algebra]
declare mod_minus_minus[algebra]
declare div_minus_right[algebra]
declare mod_minus_right[algebra]
declare div_0[algebra]
declare mod_0[algebra]
declare mod_by_1[algebra]
declare div_by_1[algebra]
declare mod_minus1_right[algebra]
declare div_minus1_right[algebra]
declare mod_mult_self2_is_0[algebra]
declare mod_mult_self1_is_0[algebra]

lemma zmod_eq_0_iff [algebra]:
  m mod d = 0  (q. m = d * q) for m d :: int
  by (auto simp add: mod_eq_0_iff_dvd)

declare dvd_0_left_iff[algebra]
declare zdvd1_eq[algebra]
declare mod_eq_dvd_iff[algebra]
declare nat_mod_eq_iff[algebra]

context semiring_parity

declare even_mult_iff [algebra]
declare even_power [algebra]


context ring_parity

declare even_minus [algebra]


declare even_Suc [algebra]
declare even_diff_nat [algebra]