Theory Groebner_Basis

theory Groebner_Basis
imports Semiring_Normalization
(*  Title:      HOL/Groebner_Basis.thy
    Author:     Amine Chaieb, TU Muenchen
*)

section ‹Groebner bases›

theory Groebner_Basis
imports Semiring_Normalization Parity
begin

subsection ‹Groebner Bases›

lemmas bool_simps = simp_thms(1-34)  ‹FIXME move to @{theory HOL}›

lemma nnf_simps:  ‹FIXME shadows fact binding in @{theory 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)) = ((P&Q) | (P&R))"
  "((Q | R) & P) = ((Q&P) | (R&P))"
  "(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);
  in
    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]
declare zmod_eq_0_iff[algebra]
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
begin

declare even_times_iff [algebra]
declare even_power [algebra]

end

context ring_parity
begin

declare even_minus [algebra]

end

declare even_Suc [algebra]
declare even_diff_nat [algebra]

end