src/HOL/Groebner_Basis.thy
 author haftmann Sun Sep 21 16:56:11 2014 +0200 (2014-09-21) changeset 58410 6d46ad54a2ab parent 57951 7896762b638b child 58777 6ba2f1fa243b permissions -rw-r--r--
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
```     1 (*  Title:      HOL/Groebner_Basis.thy
```
```     2     Author:     Amine Chaieb, TU Muenchen
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```     3 *)
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```     4
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```     5 header {* Groebner bases *}
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```     6
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```     7 theory Groebner_Basis
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```     8 imports Semiring_Normalization
```
```     9 keywords "try0" :: diag
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```    10 begin
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```    11
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```    12 subsection {* Groebner Bases *}
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```    13
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```    14 lemmas bool_simps = simp_thms(1-34) -- {* FIXME move to @{theory HOL} *}
```
```    15
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```    16 lemma nnf_simps: -- {* FIXME shadows fact binding in @{theory HOL} *}
```
```    17   "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)"
```
```    18   "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
```
```    19   "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```    20   by blast+
```
```    21
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```    22 lemma dnf:
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```    23   "(P & (Q | R)) = ((P&Q) | (P&R))"
```
```    24   "((Q | R) & P) = ((Q&P) | (R&P))"
```
```    25   "(P \<and> Q) = (Q \<and> P)"
```
```    26   "(P \<or> Q) = (Q \<or> P)"
```
```    27   by blast+
```
```    28
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```    29 lemmas weak_dnf_simps = dnf bool_simps
```
```    30
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```    31 lemma PFalse:
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```    32     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```    33     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```    34   by auto
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```    35
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```    36 named_theorems algebra "pre-simplification rules for algebraic methods"
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```    37 ML_file "Tools/groebner.ML"
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```    38
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```    39 method_setup algebra = {*
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```    40   let
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```    41     fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```    42     val addN = "add"
```
```    43     val delN = "del"
```
```    44     val any_keyword = keyword addN || keyword delN
```
```    45     val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
```
```    46   in
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```    47     Scan.optional (keyword addN |-- thms) [] --
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```    48      Scan.optional (keyword delN |-- thms) [] >>
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```    49     (fn (add_ths, del_ths) => fn ctxt =>
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```    50       SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
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```    51   end
```
```    52 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
```
```    53
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```    54 declare dvd_def[algebra]
```
```    55 declare dvd_eq_mod_eq_0[symmetric, algebra]
```
```    56 declare mod_div_trivial[algebra]
```
```    57 declare mod_mod_trivial[algebra]
```
```    58 declare div_by_0[algebra]
```
```    59 declare mod_by_0[algebra]
```
```    60 declare zmod_zdiv_equality[symmetric,algebra]
```
```    61 declare div_mod_equality2[symmetric, algebra]
```
```    62 declare div_minus_minus[algebra]
```
```    63 declare mod_minus_minus[algebra]
```
```    64 declare div_minus_right[algebra]
```
```    65 declare mod_minus_right[algebra]
```
```    66 declare div_0[algebra]
```
```    67 declare mod_0[algebra]
```
```    68 declare mod_by_1[algebra]
```
```    69 declare div_by_1[algebra]
```
```    70 declare mod_minus1_right[algebra]
```
```    71 declare div_minus1_right[algebra]
```
```    72 declare mod_mult_self2_is_0[algebra]
```
```    73 declare mod_mult_self1_is_0[algebra]
```
```    74 declare zmod_eq_0_iff[algebra]
```
```    75 declare dvd_0_left_iff[algebra]
```
```    76 declare zdvd1_eq[algebra]
```
```    77 declare zmod_eq_dvd_iff[algebra]
```
```    78 declare nat_mod_eq_iff[algebra]
```
```    79
```
```    80 end
```