Theory HOL.Groebner_Basis
section ‹Groebner bases›
theory Groebner_Basis
imports Semiring_Normalization Parity
begin
subsection ‹Groebner Bases›
lemmas bool_simps = simp_thms(1-34)
lemma nnf_simps:
"(¬(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]
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
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