# Theory Groebner_Basis

```(*  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.HOL››

lemma nnf_simps: ― ‹FIXME shadows fact binding in \<^theory>‹HOL.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 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 =>
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

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_mult_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
```