src/HOL/Groebner_Basis.thy
 author wenzelm Fri Nov 07 15:19:30 2014 +0100 (2014-11-07) changeset 58925 1b655309617c parent 58889 5b7a9633cfa8 child 60758 d8d85a8172b5 permissions -rw-r--r--
more accurate keywords;
```     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 section {* Groebner bases *}
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```     6
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```     7 theory Groebner_Basis
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```     8 imports Semiring_Normalization Parity
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```     9 begin
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```    10
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```    11 subsection {* Groebner Bases *}
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```    12
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```    13 lemmas bool_simps = simp_thms(1-34) -- {* FIXME move to @{theory HOL} *}
```
```    14
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```    15 lemma nnf_simps: -- {* FIXME shadows fact binding in @{theory HOL} *}
```
```    16   "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)"
```
```    17   "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
```
```    18   "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```    19   by blast+
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```    20
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```    21 lemma dnf:
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```    22   "(P & (Q | R)) = ((P&Q) | (P&R))"
```
```    23   "((Q | R) & P) = ((Q&P) | (R&P))"
```
```    24   "(P \<and> Q) = (Q \<and> P)"
```
```    25   "(P \<or> Q) = (Q \<or> P)"
```
```    26   by blast+
```
```    27
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```    28 lemmas weak_dnf_simps = dnf bool_simps
```
```    29
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```    30 lemma PFalse:
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```    31     "P \<equiv> False \<Longrightarrow> \<not> P"
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```    32     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```    33   by auto
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```    34
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```    35 named_theorems algebra "pre-simplification rules for algebraic methods"
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```    36 ML_file "Tools/groebner.ML"
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```    37
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```    38 method_setup algebra = {*
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```    39   let
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```    40     fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```    41     val addN = "add"
```
```    42     val delN = "del"
```
```    43     val any_keyword = keyword addN || keyword delN
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```    44     val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
```
```    45   in
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```    46     Scan.optional (keyword addN |-- thms) [] --
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```    47      Scan.optional (keyword delN |-- thms) [] >>
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```    48     (fn (add_ths, del_ths) => fn ctxt =>
```
```    49       SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
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```    50   end
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```    51 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
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```    52
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```    53 declare dvd_def[algebra]
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```    54 declare dvd_eq_mod_eq_0[symmetric, algebra]
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```    55 declare mod_div_trivial[algebra]
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```    56 declare mod_mod_trivial[algebra]
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```    57 declare div_by_0[algebra]
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```    58 declare mod_by_0[algebra]
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```    59 declare zmod_zdiv_equality[symmetric,algebra]
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```    60 declare div_mod_equality2[symmetric, algebra]
```
```    61 declare div_minus_minus[algebra]
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```    62 declare mod_minus_minus[algebra]
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```    63 declare div_minus_right[algebra]
```
```    64 declare mod_minus_right[algebra]
```
```    65 declare div_0[algebra]
```
```    66 declare mod_0[algebra]
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```    67 declare mod_by_1[algebra]
```
```    68 declare div_by_1[algebra]
```
```    69 declare mod_minus1_right[algebra]
```
```    70 declare div_minus1_right[algebra]
```
```    71 declare mod_mult_self2_is_0[algebra]
```
```    72 declare mod_mult_self1_is_0[algebra]
```
```    73 declare zmod_eq_0_iff[algebra]
```
```    74 declare dvd_0_left_iff[algebra]
```
```    75 declare zdvd1_eq[algebra]
```
```    76 declare zmod_eq_dvd_iff[algebra]
```
```    77 declare nat_mod_eq_iff[algebra]
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```    78
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```    79 context semiring_parity
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```    80 begin
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```    81
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```    82 declare even_times_iff [algebra]
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```    83 declare even_power [algebra]
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```    84
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```    85 end
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```    86
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```    87 context ring_parity
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```    88 begin
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```    89
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```    90 declare even_minus [algebra]
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```    91
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```    92 end
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```    93
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```    94 declare even_Suc [algebra]
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```    95 declare even_diff_nat [algebra]
```
```    96
```
```    97 end
```