1 (* |
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2 ID: $Id$ |
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3 Author: Amine Chaieb, TU Muenchen |
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4 *) |
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5 |
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6 header {* Dense linear order without endpoints |
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7 and a quantifier elimination procedure in Ferrante and Rackoff style *} |
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8 |
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9 theory Dense_Linear_Order |
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10 imports Finite_Set |
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11 uses |
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12 "Tools/Qelim/qelim.ML" |
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13 "Tools/Qelim/langford_data.ML" |
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14 "Tools/Qelim/ferrante_rackoff_data.ML" |
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15 ("Tools/Qelim/langford.ML") |
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16 ("Tools/Qelim/ferrante_rackoff.ML") |
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17 begin |
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18 |
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19 setup Langford_Data.setup |
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20 setup Ferrante_Rackoff_Data.setup |
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21 |
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22 context linorder |
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23 begin |
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24 |
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25 lemma less_not_permute: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear) |
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26 |
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27 lemma gather_simps: |
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28 shows |
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29 "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)" |
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30 and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)" |
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31 "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))" |
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32 and "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))" by auto |
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33 |
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34 lemma |
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35 gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" |
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36 by simp |
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37 |
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38 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*} |
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39 lemma minf_lt: "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto |
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40 lemma minf_gt: "\<exists>z . \<forall>x. x < z \<longrightarrow> (t < x \<longleftrightarrow> False)" |
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41 by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) |
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42 |
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43 lemma minf_le: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le) |
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44 lemma minf_ge: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)" |
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45 by (auto simp add: less_le not_less not_le) |
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46 lemma minf_eq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto |
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47 lemma minf_neq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto |
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48 lemma minf_P: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast |
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49 |
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50 text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*} |
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51 lemma pinf_gt: "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto |
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52 lemma pinf_lt: "\<exists>z . \<forall>x. z < x \<longrightarrow> (x < t \<longleftrightarrow> False)" |
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53 by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) |
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54 |
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55 lemma pinf_ge: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le) |
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56 lemma pinf_le: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)" |
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57 by (auto simp add: less_le not_less not_le) |
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58 lemma pinf_eq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto |
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59 lemma pinf_neq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto |
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60 lemma pinf_P: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast |
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61 |
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62 lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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63 lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" |
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64 by (auto simp add: le_less) |
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65 lemma nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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66 lemma nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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67 lemma nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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68 lemma nmi_neq: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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69 lemma nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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70 lemma nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x) ; |
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71 \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow> |
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72 \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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73 lemma nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x) ; |
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74 \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow> |
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75 \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto |
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76 |
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77 lemma npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x < t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by (auto simp add: le_less) |
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78 lemma npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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79 lemma npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x \<le> t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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80 lemma npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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81 lemma npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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82 lemma npi_neq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists> u\<in> U. x \<le> u )" by auto |
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83 lemma npi_P: "\<forall> x. ~P \<and> P \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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84 lemma npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk> |
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85 \<Longrightarrow> \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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86 lemma npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk> |
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87 \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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88 |
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89 lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x < t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y < t)" |
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90 proof(clarsimp) |
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91 fix x l u y assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" |
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92 and xu: "x<u" and px: "x < t" and ly: "l<y" and yu:"y < u" |
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93 from tU noU ly yu have tny: "t\<noteq>y" by auto |
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94 {assume H: "t < y" |
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95 from less_trans[OF lx px] less_trans[OF H yu] |
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96 have "l < t \<and> t < u" by simp |
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97 with tU noU have "False" by auto} |
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98 hence "\<not> t < y" by auto hence "y \<le> t" by (simp add: not_less) |
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99 thus "y < t" using tny by (simp add: less_le) |
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100 qed |
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101 |
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102 lemma lin_dense_gt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t < y)" |
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103 proof(clarsimp) |
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104 fix x l u y |
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105 assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u" |
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106 and px: "t < x" and ly: "l<y" and yu:"y < u" |
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107 from tU noU ly yu have tny: "t\<noteq>y" by auto |
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108 {assume H: "y< t" |
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109 from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp |
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110 with tU noU have "False" by auto} |
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111 hence "\<not> y<t" by auto hence "t \<le> y" by (auto simp add: not_less) |
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112 thus "t < y" using tny by (simp add:less_le) |
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113 qed |
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114 |
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115 lemma lin_dense_le: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<le> t)" |
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116 proof(clarsimp) |
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117 fix x l u y |
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118 assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u" |
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119 and px: "x \<le> t" and ly: "l<y" and yu:"y < u" |
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120 from tU noU ly yu have tny: "t\<noteq>y" by auto |
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121 {assume H: "t < y" |
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122 from less_le_trans[OF lx px] less_trans[OF H yu] |
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123 have "l < t \<and> t < u" by simp |
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124 with tU noU have "False" by auto} |
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125 hence "\<not> t < y" by auto thus "y \<le> t" by (simp add: not_less) |
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126 qed |
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127 |
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128 lemma lin_dense_ge: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t \<le> y)" |
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129 proof(clarsimp) |
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130 fix x l u y |
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131 assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" and xu: "x<u" |
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132 and px: "t \<le> x" and ly: "l<y" and yu:"y < u" |
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133 from tU noU ly yu have tny: "t\<noteq>y" by auto |
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134 {assume H: "y< t" |
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135 from less_trans[OF ly H] le_less_trans[OF px xu] |
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136 have "l < t \<and> t < u" by simp |
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137 with tU noU have "False" by auto} |
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138 hence "\<not> y<t" by auto thus "t \<le> y" by (simp add: not_less) |
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139 qed |
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140 lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x = t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y= t)" by auto |
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141 lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<noteq> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<noteq> t)" by auto |
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142 lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P)" by auto |
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143 |
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144 lemma lin_dense_conj: |
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145 "\<lbrakk>\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P1 x |
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146 \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ; |
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147 \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x |
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148 \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow> |
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149 \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> (P1 x \<and> P2 x) |
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150 \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))" |
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151 by blast |
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152 lemma lin_dense_disj: |
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153 "\<lbrakk>\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P1 x |
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154 \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ; |
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155 \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x |
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156 \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow> |
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157 \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> (P1 x \<or> P2 x) |
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158 \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))" |
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159 by blast |
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160 |
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161 lemma npmibnd: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk> |
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162 \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<le> x \<and> x \<le> u')" |
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163 by auto |
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164 |
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165 lemma finite_set_intervals: |
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166 assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S" |
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167 and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" |
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168 shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x" |
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169 proof- |
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170 let ?Mx = "{y. y\<in> S \<and> y \<le> x}" |
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171 let ?xM = "{y. y\<in> S \<and> x \<le> y}" |
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172 let ?a = "Max ?Mx" |
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173 let ?b = "Min ?xM" |
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174 have MxS: "?Mx \<subseteq> S" by blast |
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175 hence fMx: "finite ?Mx" using fS finite_subset by auto |
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176 from lx linS have linMx: "l \<in> ?Mx" by blast |
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177 hence Mxne: "?Mx \<noteq> {}" by blast |
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178 have xMS: "?xM \<subseteq> S" by blast |
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179 hence fxM: "finite ?xM" using fS finite_subset by auto |
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180 from xu uinS have linxM: "u \<in> ?xM" by blast |
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181 hence xMne: "?xM \<noteq> {}" by blast |
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182 have ax:"?a \<le> x" using Mxne fMx by auto |
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183 have xb:"x \<le> ?b" using xMne fxM by auto |
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184 have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast |
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185 have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast |
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186 have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S" |
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187 proof(clarsimp) |
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188 fix y assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S" |
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189 from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear) |
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190 moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])} |
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191 moreover {assume "y \<in> ?xM" hence "?b \<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])} |
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192 ultimately show "False" by blast |
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193 qed |
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194 from ainS binS noy ax xb px show ?thesis by blast |
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195 qed |
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196 |
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197 lemma finite_set_intervals2: |
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198 assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S" |
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199 and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" |
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200 shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)" |
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201 proof- |
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202 from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] |
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203 obtain a and b where |
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204 as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" |
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205 and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto |
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206 from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by (auto simp add: le_less) |
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207 thus ?thesis using px as bs noS by blast |
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208 qed |
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209 |
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210 end |
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211 |
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212 section {* The classical QE after Langford for dense linear orders *} |
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213 |
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214 context dense_linear_order |
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215 begin |
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216 |
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217 lemma dlo_qe_bnds: |
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218 assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U" |
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219 shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l < u)" |
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220 proof (simp only: atomize_eq, rule iffI) |
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221 assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" |
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222 then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast |
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223 {fix l u assume l: "l \<in> L" and u: "u \<in> U" |
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224 have "l < x" using xL l by blast |
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225 also have "x < u" using xU u by blast |
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226 finally (less_trans) have "l < u" .} |
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227 thus "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast |
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228 next |
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229 assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u" |
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230 let ?ML = "Max L" |
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231 let ?MU = "Min U" |
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232 from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto |
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233 from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto |
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234 from th1 th2 H have "?ML < ?MU" by auto |
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235 with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast |
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236 from th3 th1' have "\<forall>l \<in> L. l < w" by auto |
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237 moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto |
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238 ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto |
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239 qed |
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240 |
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241 lemma dlo_qe_noub: |
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242 assumes ne: "L \<noteq> {}" and fL: "finite L" |
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243 shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True" |
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244 proof(simp add: atomize_eq) |
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245 from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast |
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246 from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp |
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247 with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans) |
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248 thus "\<exists>x. \<forall>y\<in>L. y < x" by blast |
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249 qed |
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250 |
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251 lemma dlo_qe_nolb: |
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252 assumes ne: "U \<noteq> {}" and fU: "finite U" |
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253 shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True" |
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254 proof(simp add: atomize_eq) |
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255 from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast |
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256 from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp |
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257 with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans) |
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258 thus "\<exists>x. \<forall>y\<in>U. x < y" by blast |
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259 qed |
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260 |
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261 lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" |
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262 using gt_ex[of t] by auto |
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263 |
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264 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq |
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265 le_less neq_iff linear less_not_permute |
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266 |
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267 lemma axiom: "dense_linear_order (op \<le>) (op <)" by (rule dense_linear_order_axioms) |
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268 lemma atoms: |
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269 includes meta_term_syntax |
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270 shows "TERM (less :: 'a \<Rightarrow> _)" |
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271 and "TERM (less_eq :: 'a \<Rightarrow> _)" |
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272 and "TERM (op = :: 'a \<Rightarrow> _)" . |
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273 |
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274 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms] |
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275 declare dlo_simps[langfordsimp] |
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276 |
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277 end |
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278 |
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279 (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *) |
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280 lemma dnf: |
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281 "(P & (Q | R)) = ((P&Q) | (P&R))" |
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282 "((Q | R) & P) = ((Q&P) | (R&P))" |
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283 by blast+ |
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284 |
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285 lemmas weak_dnf_simps = simp_thms dnf |
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286 |
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287 lemma nnf_simps: |
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288 "(\<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)" |
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289 "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P" |
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290 by blast+ |
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291 |
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292 lemma ex_distrib: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast |
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293 |
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294 lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib |
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295 |
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296 use "Tools/Qelim/langford.ML" |
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297 method_setup dlo = {* |
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298 Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac) |
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299 *} "Langford's algorithm for quantifier elimination in dense linear orders" |
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300 |
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301 |
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302 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *} |
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303 |
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304 text {* Linear order without upper bounds *} |
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305 |
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306 locale linorder_stupid_syntax = linorder |
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307 begin |
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308 notation |
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309 less_eq ("op \<sqsubseteq>") and |
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310 less_eq ("(_/ \<sqsubseteq> _)" [51, 51] 50) and |
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311 less ("op \<sqsubset>") and |
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312 less ("(_/ \<sqsubset> _)" [51, 51] 50) |
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313 |
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314 end |
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315 |
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316 locale linorder_no_ub = linorder_stupid_syntax + |
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317 assumes gt_ex: "\<exists>y. less x y" |
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318 begin |
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319 lemma ge_ex: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto |
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320 |
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321 text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *} |
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322 lemma pinf_conj: |
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323 assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
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324 and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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325 shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))" |
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326 proof- |
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327 from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
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328 and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast |
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329 from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast |
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330 from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all |
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331 {fix x assume H: "z \<sqsubset> x" |
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332 from less_trans[OF zz1 H] less_trans[OF zz2 H] |
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333 have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" using z1 zz1 z2 zz2 by auto |
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334 } |
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335 thus ?thesis by blast |
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336 qed |
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337 |
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338 lemma pinf_disj: |
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339 assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
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340 and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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341 shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))" |
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342 proof- |
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343 from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
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344 and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast |
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345 from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast |
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346 from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all |
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347 {fix x assume H: "z \<sqsubset> x" |
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348 from less_trans[OF zz1 H] less_trans[OF zz2 H] |
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349 have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" using z1 zz1 z2 zz2 by auto |
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350 } |
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351 thus ?thesis by blast |
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352 qed |
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353 |
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354 lemma pinf_ex: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x" |
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355 proof- |
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356 from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast |
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357 from gt_ex obtain x where x: "z \<sqsubset> x" by blast |
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358 from z x p1 show ?thesis by blast |
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359 qed |
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360 |
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361 end |
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362 |
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363 text {* Linear order without upper bounds *} |
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364 |
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365 locale linorder_no_lb = linorder_stupid_syntax + |
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366 assumes lt_ex: "\<exists>y. less y x" |
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367 begin |
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368 lemma le_ex: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto |
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369 |
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370 |
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371 text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *} |
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372 lemma minf_conj: |
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373 assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
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374 and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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375 shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))" |
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376 proof- |
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377 from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast |
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378 from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast |
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379 from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all |
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380 {fix x assume H: "x \<sqsubset> z" |
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381 from less_trans[OF H zz1] less_trans[OF H zz2] |
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382 have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" using z1 zz1 z2 zz2 by auto |
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383 } |
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384 thus ?thesis by blast |
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385 qed |
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386 |
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387 lemma minf_disj: |
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388 assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
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389 and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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390 shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))" |
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391 proof- |
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392 from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast |
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393 from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast |
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394 from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all |
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395 {fix x assume H: "x \<sqsubset> z" |
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396 from less_trans[OF H zz1] less_trans[OF H zz2] |
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397 have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" using z1 zz1 z2 zz2 by auto |
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398 } |
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399 thus ?thesis by blast |
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400 qed |
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401 |
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402 lemma minf_ex: assumes ex:"\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x" |
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403 proof- |
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404 from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast |
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405 from lt_ex obtain x where x: "x \<sqsubset> z" by blast |
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406 from z x p1 show ?thesis by blast |
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407 qed |
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408 |
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409 end |
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410 |
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411 |
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412 locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub + |
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413 fixes between |
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414 assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y" |
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415 and between_same: "between x x = x" |
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416 |
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417 interpretation constr_dense_linear_order < dense_linear_order |
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418 apply unfold_locales |
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419 using gt_ex lt_ex between_less |
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420 by (auto, rule_tac x="between x y" in exI, simp) |
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421 |
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422 context constr_dense_linear_order |
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423 begin |
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424 |
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425 lemma rinf_U: |
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426 assumes fU: "finite U" |
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427 and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x |
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428 \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )" |
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429 and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" |
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430 and nmi: "\<not> MP" and npi: "\<not> PP" and ex: "\<exists> x. P x" |
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431 shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')" |
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432 proof- |
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433 from ex obtain x where px: "P x" by blast |
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434 from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto |
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435 then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto |
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436 from uU have Une: "U \<noteq> {}" by auto |
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437 term "linorder.Min less_eq" |
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438 let ?l = "linorder.Min less_eq U" |
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439 let ?u = "linorder.Max less_eq U" |
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440 have linM: "?l \<in> U" using fU Une by simp |
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441 have uinM: "?u \<in> U" using fU Une by simp |
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442 have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto |
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443 have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto |
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444 have th:"?l \<sqsubseteq> u" using uU Une lM by auto |
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445 from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" . |
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446 have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp |
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447 from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" . |
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448 from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu] |
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449 have "(\<exists> s\<in> U. P s) \<or> |
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450 (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x)" . |
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451 moreover { fix u assume um: "u\<in>U" and pu: "P u" |
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452 have "between u u = u" by (simp add: between_same) |
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453 with um pu have "P (between u u)" by simp |
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454 with um have ?thesis by blast} |
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455 moreover{ |
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456 assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x" |
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457 then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U" |
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458 and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x" |
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459 by blast |
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460 from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" . |
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461 let ?u = "between t1 t2" |
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462 from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto |
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463 from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast |
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464 with t1M t2M have ?thesis by blast} |
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465 ultimately show ?thesis by blast |
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466 qed |
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467 |
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468 theorem fr_eq: |
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469 assumes fU: "finite U" |
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470 and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x |
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471 \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )" |
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472 and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)" |
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473 and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)" |
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474 and mi: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x = MP)" and pi: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x = PP)" |
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475 shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))" |
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476 (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D") |
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477 proof- |
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478 { |
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479 assume px: "\<exists> x. P x" |
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480 have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast |
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481 moreover {assume "MP \<or> PP" hence "?D" by blast} |
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482 moreover {assume nmi: "\<not> MP" and npi: "\<not> PP" |
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483 from npmibnd[OF nmibnd npibnd] |
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484 have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" . |
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485 from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast} |
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486 ultimately have "?D" by blast} |
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487 moreover |
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488 { assume "?D" |
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489 moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .} |
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490 moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . } |
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491 moreover {assume f:"?F" hence "?E" by blast} |
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492 ultimately have "?E" by blast} |
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493 ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp |
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494 qed |
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495 |
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496 lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P |
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497 lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P |
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498 |
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499 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P |
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500 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P |
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501 lemmas lin_dense_thms = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P |
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502 |
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503 lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" |
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504 by (rule constr_dense_linear_order_axioms) |
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505 lemma atoms: |
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506 includes meta_term_syntax |
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507 shows "TERM (less :: 'a \<Rightarrow> _)" |
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508 and "TERM (less_eq :: 'a \<Rightarrow> _)" |
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509 and "TERM (op = :: 'a \<Rightarrow> _)" . |
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510 |
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511 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms |
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512 nmi: nmi_thms npi: npi_thms lindense: |
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513 lin_dense_thms qe: fr_eq atoms: atoms] |
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514 |
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515 declaration {* |
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516 let |
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517 fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}] |
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518 fun generic_whatis phi = |
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519 let |
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520 val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}] |
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521 fun h x t = |
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522 case term_of t of |
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523 Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq |
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524 else Ferrante_Rackoff_Data.Nox |
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525 | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq |
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526 else Ferrante_Rackoff_Data.Nox |
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527 | b$y$z => if Term.could_unify (b, lt) then |
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528 if term_of x aconv y then Ferrante_Rackoff_Data.Lt |
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529 else if term_of x aconv z then Ferrante_Rackoff_Data.Gt |
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530 else Ferrante_Rackoff_Data.Nox |
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531 else if Term.could_unify (b, le) then |
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532 if term_of x aconv y then Ferrante_Rackoff_Data.Le |
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533 else if term_of x aconv z then Ferrante_Rackoff_Data.Ge |
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534 else Ferrante_Rackoff_Data.Nox |
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535 else Ferrante_Rackoff_Data.Nox |
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536 | _ => Ferrante_Rackoff_Data.Nox |
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537 in h end |
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538 fun ss phi = HOL_ss addsimps (simps phi) |
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539 in |
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540 Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"} |
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541 {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss} |
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542 end |
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543 *} |
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544 |
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545 end |
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546 |
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547 use "Tools/Qelim/ferrante_rackoff.ML" |
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548 |
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549 method_setup ferrack = {* |
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550 Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac) |
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551 *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders" |
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552 |
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553 end |
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