src/HOL/Groebner_Basis.thy
author hoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 51474 1e9e68247ad1
parent 48891 c0eafbd55de3
child 54251 adea9f6986b2
permissions -rw-r--r--
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
     1 (*  Title:      HOL/Groebner_Basis.thy
     2     Author:     Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 header {* Groebner bases *}
     6 
     7 theory Groebner_Basis
     8 imports Semiring_Normalization
     9 begin
    10 
    11 subsection {* Groebner Bases *}
    12 
    13 lemmas bool_simps = simp_thms(1-34)
    14 
    15 lemma dnf:
    16     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
    17     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
    18   by blast+
    19 
    20 lemmas weak_dnf_simps = dnf bool_simps
    21 
    22 lemma nnf_simps:
    23     "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
    24     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
    25   by blast+
    26 
    27 lemma PFalse:
    28     "P \<equiv> False \<Longrightarrow> \<not> P"
    29     "\<not> P \<Longrightarrow> (P \<equiv> False)"
    30   by auto
    31 
    32 ML {*
    33 structure Algebra_Simplification = Named_Thms
    34 (
    35   val name = @{binding algebra}
    36   val description = "pre-simplification rules for algebraic methods"
    37 )
    38 *}
    39 
    40 setup Algebra_Simplification.setup
    41 
    42 ML_file "Tools/groebner.ML"
    43 
    44 method_setup algebra = {*
    45   let
    46     fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
    47     val addN = "add"
    48     val delN = "del"
    49     val any_keyword = keyword addN || keyword delN
    50     val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
    51   in
    52     Scan.optional (keyword addN |-- thms) [] --
    53      Scan.optional (keyword delN |-- thms) [] >>
    54     (fn (add_ths, del_ths) => fn ctxt =>
    55       SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
    56   end
    57 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
    58 
    59 declare dvd_def[algebra]
    60 declare dvd_eq_mod_eq_0[symmetric, algebra]
    61 declare mod_div_trivial[algebra]
    62 declare mod_mod_trivial[algebra]
    63 declare div_by_0[algebra]
    64 declare mod_by_0[algebra]
    65 declare zmod_zdiv_equality[symmetric,algebra]
    66 declare div_mod_equality2[symmetric, algebra]
    67 declare div_minus_minus[algebra]
    68 declare mod_minus_minus[algebra]
    69 declare div_minus_right[algebra]
    70 declare mod_minus_right[algebra]
    71 declare div_0[algebra]
    72 declare mod_0[algebra]
    73 declare mod_by_1[algebra]
    74 declare div_by_1[algebra]
    75 declare mod_minus1_right[algebra]
    76 declare div_minus1_right[algebra]
    77 declare mod_mult_self2_is_0[algebra]
    78 declare mod_mult_self1_is_0[algebra]
    79 declare zmod_eq_0_iff[algebra]
    80 declare dvd_0_left_iff[algebra]
    81 declare zdvd1_eq[algebra]
    82 declare zmod_eq_dvd_iff[algebra]
    83 declare nat_mod_eq_iff[algebra]
    84 
    85 end