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1 |
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2 theory PDerivs |
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3 imports Spec |
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4 begin |
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5 |
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6 |
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7 |
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8 abbreviation |
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9 "SEQs rs r \<equiv> (\<Union>r' \<in> rs. {SEQ r' r})" |
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10 |
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11 lemma SEQs_eq_image: |
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12 "SEQs rs r = (\<lambda>r'. SEQ r' r) ` rs" |
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13 by auto |
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14 |
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15 primrec |
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16 pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set" |
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17 where |
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18 "pder c ZERO = {}" |
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19 | "pder c ONE = {}" |
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20 | "pder c (CHAR d) = (if c = d then {ONE} else {})" |
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21 | "pder c (ALT r1 r2) = (pder c r1) \<union> (pder c r2)" |
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22 | "pder c (SEQ r1 r2) = |
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23 (if nullable r1 then SEQs (pder c r1) r2 \<union> pder c r2 else SEQs (pder c r1) r2)" |
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24 | "pder c (STAR r) = SEQs (pder c r) (STAR r)" |
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25 |
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26 primrec |
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27 pders :: "char list \<Rightarrow> rexp \<Rightarrow> rexp set" |
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28 where |
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29 "pders [] r = {r}" |
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30 | "pders (c # s) r = \<Union> (pders s ` pder c r)" |
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31 |
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32 abbreviation |
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33 pder_set :: "char \<Rightarrow> rexp set \<Rightarrow> rexp set" |
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34 where |
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35 "pder_set c rs \<equiv> \<Union> (pder c ` rs)" |
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36 |
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37 abbreviation |
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38 pders_set :: "char list \<Rightarrow> rexp set \<Rightarrow> rexp set" |
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39 where |
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40 "pders_set s rs \<equiv> \<Union> (pders s ` rs)" |
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41 |
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42 lemma pders_append: |
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43 "pders (s1 @ s2) r = \<Union> (pders s2 ` pders s1 r)" |
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44 by (induct s1 arbitrary: r) (simp_all) |
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45 |
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46 lemma pders_snoc: |
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47 shows "pders (s @ [c]) r = pder_set c (pders s r)" |
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48 by (simp add: pders_append) |
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49 |
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50 lemma pders_simps [simp]: |
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51 shows "pders s ZERO = (if s = [] then {ZERO} else {})" |
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52 and "pders s ONE = (if s = [] then {ONE} else {})" |
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53 and "pders s (ALT r1 r2) = (if s = [] then {ALT r1 r2} else (pders s r1) \<union> (pders s r2))" |
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54 by (induct s) (simp_all) |
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55 |
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56 lemma pders_CHAR: |
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57 shows "pders s (CHAR c) \<subseteq> {CHAR c, ONE}" |
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58 by (induct s) (simp_all) |
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59 |
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60 subsection \<open>Relating left-quotients and partial derivatives\<close> |
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61 |
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62 lemma Sequ_UNION_distrib: |
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63 shows "A ;; \<Union>(M ` I) = \<Union>((\<lambda>i. A ;; M i) ` I)" |
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64 and "\<Union>(M ` I) ;; A = \<Union>((\<lambda>i. M i ;; A) ` I)" |
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65 by (auto simp add: Sequ_def) |
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66 |
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67 |
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68 lemma Der_pder: |
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69 shows "Der c (L r) = \<Union> (L ` pder c r)" |
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70 by (induct r) (simp_all add: nullable_correctness Sequ_UNION_distrib) |
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71 |
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72 lemma Ders_pders: |
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73 shows "Ders s (L r) = \<Union> (L ` pders s r)" |
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74 proof (induct s arbitrary: r) |
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75 case (Cons c s) |
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76 have ih: "\<And>r. Ders s (L r) = \<Union> (L ` pders s r)" by fact |
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77 have "Ders (c # s) (L r) = Ders s (Der c (L r))" by (simp add: Ders_def Der_def) |
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78 also have "\<dots> = Ders s (\<Union> (L ` pder c r))" by (simp add: Der_pder) |
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79 also have "\<dots> = (\<Union>A\<in>(L ` (pder c r)). (Ders s A))" |
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80 by (auto simp add: Ders_def) |
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81 also have "\<dots> = \<Union> (L ` (pders_set s (pder c r)))" |
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82 using ih by auto |
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83 also have "\<dots> = \<Union> (L ` (pders (c # s) r))" by simp |
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84 finally show "Ders (c # s) (L r) = \<Union> (L ` pders (c # s) r)" . |
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85 qed (simp add: Ders_def) |
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86 |
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87 subsection \<open>Relating derivatives and partial derivatives\<close> |
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88 |
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89 lemma der_pder: |
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90 shows "\<Union> (L ` (pder c r)) = L (der c r)" |
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91 unfolding der_correctness Der_pder by simp |
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92 |
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93 lemma ders_pders: |
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94 shows "\<Union> (L ` (pders s r)) = L (ders s r)" |
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95 unfolding der_correctness ders_correctness Ders_pders by simp |
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96 |
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97 |
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98 subsection \<open>Finiteness property of partial derivatives\<close> |
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99 |
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100 definition |
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101 pders_Set :: "string set \<Rightarrow> rexp \<Rightarrow> rexp set" |
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102 where |
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103 "pders_Set A r \<equiv> \<Union>x \<in> A. pders x r" |
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104 |
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105 lemma pders_Set_subsetI: |
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106 assumes "\<And>s. s \<in> A \<Longrightarrow> pders s r \<subseteq> C" |
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107 shows "pders_Set A r \<subseteq> C" |
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108 using assms unfolding pders_Set_def by (rule UN_least) |
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109 |
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110 lemma pders_Set_union: |
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111 shows "pders_Set (A \<union> B) r = (pders_Set A r \<union> pders_Set B r)" |
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112 by (simp add: pders_Set_def) |
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113 |
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114 lemma pders_Set_subset: |
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115 shows "A \<subseteq> B \<Longrightarrow> pders_Set A r \<subseteq> pders_Set B r" |
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116 by (auto simp add: pders_Set_def) |
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117 |
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118 definition |
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119 "UNIV1 \<equiv> UNIV - {[]}" |
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120 |
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121 lemma pders_Set_ZERO [simp]: |
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122 shows "pders_Set UNIV1 ZERO = {}" |
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123 unfolding UNIV1_def pders_Set_def by auto |
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124 |
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125 lemma pders_Set_ONE [simp]: |
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126 shows "pders_Set UNIV1 ONE = {}" |
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127 unfolding UNIV1_def pders_Set_def by (auto split: if_splits) |
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128 |
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129 lemma pders_Set_CHAR [simp]: |
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130 shows "pders_Set UNIV1 (CHAR c) = {ONE}" |
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131 unfolding UNIV1_def pders_Set_def |
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132 apply(auto) |
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133 apply(frule rev_subsetD) |
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134 apply(rule pders_CHAR) |
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135 apply(simp) |
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136 apply(case_tac xa) |
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137 apply(auto split: if_splits) |
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138 done |
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139 |
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140 lemma pders_Set_ALT [simp]: |
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141 shows "pders_Set UNIV1 (ALT r1 r2) = pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2" |
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142 unfolding UNIV1_def pders_Set_def by auto |
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143 |
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144 |
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145 text \<open>Non-empty suffixes of a string (needed for the cases of @{const SEQ} and @{const STAR} below)\<close> |
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146 |
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147 definition |
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148 "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}" |
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149 |
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150 lemma PSuf_snoc: |
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151 shows "PSuf (s @ [c]) = (PSuf s) ;; {[c]} \<union> {[c]}" |
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152 unfolding PSuf_def Sequ_def |
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153 by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv) |
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154 |
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155 lemma PSuf_Union: |
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156 shows "(\<Union>v \<in> PSuf s ;; {[c]}. f v) = (\<Union>v \<in> PSuf s. f (v @ [c]))" |
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157 by (auto simp add: Sequ_def) |
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158 |
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159 lemma pders_Set_snoc: |
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160 shows "pders_Set (PSuf s ;; {[c]}) r = (pder_set c (pders_Set (PSuf s) r))" |
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161 unfolding pders_Set_def |
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162 by (simp add: PSuf_Union pders_snoc) |
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163 |
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164 lemma pders_SEQ: |
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165 shows "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)" |
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166 proof (induct s rule: rev_induct) |
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167 case (snoc c s) |
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168 have ih: "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)" |
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169 by fact |
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170 have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))" |
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171 by (simp add: pders_snoc) |
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172 also have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2))" |
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173 using ih by fastforce |
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174 also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pder_set c (pders_Set (PSuf s) r2)" |
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175 by (simp) |
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176 also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pders_Set (PSuf s ;; {[c]}) r2" |
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177 by (simp add: pders_Set_snoc) |
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178 also |
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179 have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2) \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2" |
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180 by auto |
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181 also |
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182 have "\<dots> \<subseteq> SEQs (pder_set c (pders s r1)) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2" |
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183 by (auto simp add: if_splits) |
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184 also have "\<dots> = SEQs (pders (s @ [c]) r1) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2" |
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185 by (simp add: pders_snoc) |
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186 also have "\<dots> \<subseteq> SEQs (pders (s @ [c]) r1) r2 \<union> pders_Set (PSuf (s @ [c])) r2" |
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187 unfolding pders_Set_def by (auto simp add: PSuf_snoc) |
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188 finally show ?case . |
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189 qed (simp) |
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190 |
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191 lemma pders_Set_SEQ_aux1: |
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192 assumes a: "s \<in> UNIV1" |
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193 shows "pders_Set (PSuf s) r \<subseteq> pders_Set UNIV1 r" |
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194 using a unfolding UNIV1_def PSuf_def pders_Set_def by auto |
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195 |
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196 lemma pders_Set_SEQ_aux2: |
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197 assumes a: "s \<in> UNIV1" |
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198 shows "SEQs (pders s r1) r2 \<subseteq> SEQs (pders_Set UNIV1 r1) r2" |
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199 using a unfolding pders_Set_def by auto |
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200 |
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201 lemma pders_Set_SEQ: |
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202 shows "pders_Set UNIV1 (SEQ r1 r2) \<subseteq> SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2" |
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203 apply(rule pders_Set_subsetI) |
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204 apply(rule subset_trans) |
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205 apply(rule pders_SEQ) |
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206 using pders_Set_SEQ_aux1 pders_Set_SEQ_aux2 |
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207 apply auto |
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208 apply blast |
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209 done |
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210 |
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211 lemma pders_STAR: |
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212 assumes a: "s \<noteq> []" |
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213 shows "pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) r) (STAR r)" |
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214 using a |
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215 proof (induct s rule: rev_induct) |
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216 case (snoc c s) |
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217 have ih: "s \<noteq> [] \<Longrightarrow> pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) r) (STAR r)" by fact |
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218 { assume asm: "s \<noteq> []" |
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219 have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by (simp add: pders_snoc) |
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220 also have "\<dots> \<subseteq> pder_set c (SEQs (pders_Set (PSuf s) r) (STAR r))" |
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221 using ih[OF asm] by fast |
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222 also have "\<dots> \<subseteq> SEQs (pder_set c (pders_Set (PSuf s) r)) (STAR r) \<union> pder c (STAR r)" |
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223 by (auto split: if_splits) |
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224 also have "\<dots> \<subseteq> SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r) \<union> (SEQs (pder c r) (STAR r))" |
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225 by (simp only: PSuf_snoc pders_Set_snoc pders_Set_union) |
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226 (auto simp add: pders_Set_def) |
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227 also have "\<dots> = SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r)" |
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228 by (auto simp add: PSuf_snoc PSuf_Union pders_snoc pders_Set_def) |
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229 finally have ?case . |
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230 } |
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231 moreover |
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232 { assume asm: "s = []" |
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233 then have ?case by (auto simp add: pders_Set_def pders_snoc PSuf_def) |
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234 } |
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235 ultimately show ?case by blast |
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236 qed (simp) |
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237 |
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238 lemma pders_Set_STAR: |
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239 shows "pders_Set UNIV1 (STAR r) \<subseteq> SEQs (pders_Set UNIV1 r) (STAR r)" |
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240 apply(rule pders_Set_subsetI) |
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241 apply(rule subset_trans) |
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242 apply(rule pders_STAR) |
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243 apply(simp add: UNIV1_def) |
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244 apply(simp add: UNIV1_def PSuf_def) |
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245 apply(auto simp add: pders_Set_def) |
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246 done |
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247 |
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248 lemma finite_SEQs [simp]: |
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249 assumes a: "finite A" |
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250 shows "finite (SEQs A r)" |
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251 using a by auto |
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252 |
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253 thm finite.intros |
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254 |
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255 lemma finite_pders_Set_UNIV1: |
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256 shows "finite (pders_Set UNIV1 r)" |
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257 apply(induct r) |
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258 apply(simp_all add: |
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259 finite_subset[OF pders_Set_SEQ] |
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260 finite_subset[OF pders_Set_STAR]) |
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261 done |
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262 |
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263 lemma pders_Set_UNIV: |
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264 shows "pders_Set UNIV r = pders [] r \<union> pders_Set UNIV1 r" |
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265 unfolding UNIV1_def pders_Set_def |
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266 by blast |
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267 |
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268 lemma finite_pders_Set_UNIV: |
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269 shows "finite (pders_Set UNIV r)" |
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270 unfolding pders_Set_UNIV |
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271 by (simp add: finite_pders_Set_UNIV1) |
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272 |
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273 lemma finite_pders_set: |
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274 shows "finite (pders_Set A r)" |
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275 by (metis finite_pders_Set_UNIV pders_Set_subset rev_finite_subset subset_UNIV) |
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276 |
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277 |
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278 text \<open>The following relationship between the alphabetic width of regular expressions |
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279 (called \<open>awidth\<close> below) and the number of partial derivatives was proved |
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280 by Antimirov~\cite{Antimirov95} and formalized by Max Haslbeck.\<close> |
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281 |
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282 fun awidth :: "rexp \<Rightarrow> nat" where |
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283 "awidth ZERO = 0" | |
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284 "awidth ONE = 0" | |
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285 "awidth (CHAR a) = 1" | |
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286 "awidth (ALT r1 r2) = awidth r1 + awidth r2" | |
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287 "awidth (SEQ r1 r2) = awidth r1 + awidth r2" | |
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288 "awidth (STAR r1) = awidth r1" |
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289 |
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290 lemma card_SEQs_pders_Set_le: |
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291 shows "card (SEQs (pders_Set A r) s) \<le> card (pders_Set A r)" |
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292 using finite_pders_set |
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293 unfolding SEQs_eq_image |
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294 by (rule card_image_le) |
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295 |
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296 lemma card_pders_set_UNIV1_le_awidth: |
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297 shows "card (pders_Set UNIV1 r) \<le> awidth r" |
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298 proof (induction r) |
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299 case (ALT r1 r2) |
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300 have "card (pders_Set UNIV1 (ALT r1 r2)) = card (pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2)" by simp |
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301 also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)" |
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302 by(simp add: card_Un_le) |
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303 also have "\<dots> \<le> awidth (ALT r1 r2)" using ALT.IH by simp |
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304 finally show ?case . |
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305 next |
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306 case (SEQ r1 r2) |
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307 have "card (pders_Set UNIV1 (SEQ r1 r2)) \<le> card (SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2)" |
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308 by (simp add: card_mono finite_pders_set pders_Set_SEQ) |
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309 also have "\<dots> \<le> card (SEQs (pders_Set UNIV1 r1) r2) + card (pders_Set UNIV1 r2)" |
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310 by (simp add: card_Un_le) |
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311 also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)" |
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312 by (simp add: card_SEQs_pders_Set_le) |
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313 also have "\<dots> \<le> awidth (SEQ r1 r2)" using SEQ.IH by simp |
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314 finally show ?case . |
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315 next |
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316 case (STAR r) |
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317 have "card (pders_Set UNIV1 (STAR r)) \<le> card (SEQs (pders_Set UNIV1 r) (STAR r))" |
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318 by (simp add: card_mono finite_pders_set pders_Set_STAR) |
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319 also have "\<dots> \<le> card (pders_Set UNIV1 r)" by (rule card_SEQs_pders_Set_le) |
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320 also have "\<dots> \<le> awidth (STAR r)" by (simp add: STAR.IH) |
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321 finally show ?case . |
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322 qed (auto) |
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323 |
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324 text\<open>Antimirov's Theorem 3.4:\<close> |
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325 |
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326 theorem card_pders_set_UNIV_le_awidth: |
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327 shows "card (pders_Set UNIV r) \<le> awidth r + 1" |
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328 proof - |
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329 have "card (insert r (pders_Set UNIV1 r)) \<le> Suc (card (pders_Set UNIV1 r))" |
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330 by(auto simp: card_insert_if[OF finite_pders_Set_UNIV1]) |
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331 also have "\<dots> \<le> Suc (awidth r)" by(simp add: card_pders_set_UNIV1_le_awidth) |
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332 finally show ?thesis by(simp add: pders_Set_UNIV) |
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333 qed |
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334 |
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335 text\<open>Antimirov's Corollary 3.5:\<close> |
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336 |
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337 corollary card_pders_set_le_awidth: |
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338 shows "card (pders_Set A r) \<le> awidth r + 1" |
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339 proof - |
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340 have "card (pders_Set A r) \<le> card (pders_Set UNIV r)" |
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341 by (simp add: card_mono finite_pders_set pders_Set_subset) |
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342 also have "... \<le> awidth r + 1" |
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343 by (rule card_pders_set_UNIV_le_awidth) |
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344 finally show "card (pders_Set A r) \<le> awidth r + 1" by simp |
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345 qed |
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346 |
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347 (* other result by antimirov *) |
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348 |
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349 lemma card_pders_awidth: |
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350 shows "card (pders s r) \<le> awidth r + 1" |
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351 proof - |
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352 have "pders s r \<subseteq> pders_Set UNIV r" |
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353 using pders_Set_def by auto |
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354 then have "card (pders s r) \<le> card (pders_Set UNIV r)" |
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355 by (simp add: card_mono finite_pders_set) |
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356 then show "card (pders s r) \<le> awidth r + 1" |
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357 using card_pders_set_le_awidth order_trans by blast |
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358 qed |
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359 |
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360 |
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361 |
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362 |
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363 |
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364 fun subs :: "rexp \<Rightarrow> rexp set" where |
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365 "subs ZERO = {ZERO}" | |
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366 "subs ONE = {ONE}" | |
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367 "subs (CHAR a) = {CHAR a, ONE}" | |
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368 "subs (ALT r1 r2) = (subs r1 \<union> subs r2 \<union> {ALT r1 r2})" | |
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369 "subs (SEQ r1 r2) = (subs r1 \<union> subs r2 \<union> {SEQ r1 r2} \<union> SEQs (subs r1) r2)" | |
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370 "subs (STAR r1) = (subs r1 \<union> {STAR r1} \<union> SEQs (subs r1) (STAR r1))" |
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371 |
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372 lemma subs_finite: |
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373 shows "finite (subs r)" |
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374 apply(induct r) |
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375 apply(simp_all) |
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376 done |
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377 |
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378 |
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379 |
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380 lemma pders_subs: |
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381 shows "pders s r \<subseteq> subs r" |
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382 apply(induct r arbitrary: s) |
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383 apply(simp) |
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384 apply(simp) |
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385 apply(simp add: pders_CHAR) |
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386 (* SEQ case *) |
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387 apply(simp) |
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388 apply(rule subset_trans) |
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389 apply(rule pders_SEQ) |
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390 defer |
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391 (* ALT case *) |
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392 apply(simp) |
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393 apply(rule impI) |
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394 apply(rule conjI) |
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395 apply blast |
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396 apply blast |
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397 (* STAR case *) |
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398 apply(case_tac s) |
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399 apply(simp) |
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400 apply(rule subset_trans) |
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401 thm pders_STAR |
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402 apply(rule pders_STAR) |
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403 apply(simp) |
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404 apply(auto simp add: pders_Set_def)[1] |
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405 apply(simp) |
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406 apply(rule conjI) |
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407 apply blast |
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408 apply(auto simp add: pders_Set_def)[1] |
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409 done |
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410 |
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411 fun size2 :: "rexp \<Rightarrow> nat" where |
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412 "size2 ZERO = 1" | |
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413 "size2 ONE = 1" | |
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414 "size2 (CHAR c) = 1" | |
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415 "size2 (ALT r1 r2) = Suc (size2 r1 + size2 r2)" | |
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416 "size2 (SEQ r1 r2) = Suc (size2 r1 + size2 r2)" | |
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417 "size2 (STAR r1) = Suc (size2 r1)" |
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418 |
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419 |
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420 lemma size_rexp: |
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421 fixes r :: rexp |
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422 shows "1 \<le> size2 r" |
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423 apply(induct r) |
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424 apply(simp) |
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425 apply(simp_all) |
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426 done |
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427 |
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428 lemma subs_card: |
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429 shows "card (subs r) \<le> Suc (size2 r + size2 r)" |
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430 apply(induct r) |
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431 apply(auto) |
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432 apply(subst card_insert) |
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433 apply(simp add: subs_finite) |
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434 apply(simp add: subs_finite) |
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435 oops |
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436 |
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437 lemma subs_size2: |
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438 shows "\<forall>r1 \<in> subs r. size2 r1 \<le> Suc (size2 r * size2 r)" |
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439 apply(induct r) |
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440 apply(simp) |
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441 apply(simp) |
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442 apply(simp) |
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443 (* SEQ case *) |
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444 apply(simp) |
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445 apply(auto)[1] |
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446 apply (smt Suc_n_not_le_n add.commute distrib_left le_Suc_eq left_add_mult_distrib nat_le_linear trans_le_add1) |
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447 apply (smt Suc_le_mono Suc_n_not_le_n le_trans nat_le_linear power2_eq_square power2_sum semiring_normalization_rules(23) trans_le_add2) |
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448 apply (smt Groups.add_ac(3) Suc_n_not_le_n distrib_left le_Suc_eq left_add_mult_distrib nat_le_linear trans_le_add1) |
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449 (* ALT case *) |
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450 apply(simp) |
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451 apply(auto)[1] |
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452 apply (smt Groups.add_ac(2) Suc_le_mono Suc_n_not_le_n le_add2 linear order_trans power2_eq_square power2_sum) |
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453 apply (smt Groups.add_ac(2) Suc_le_mono Suc_n_not_le_n left_add_mult_distrib linear mult.commute order.trans trans_le_add1) |
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454 (* STAR case *) |
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455 apply(auto)[1] |
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456 apply(drule_tac x="r'" in bspec) |
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457 apply(simp) |
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458 apply(rule le_trans) |
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459 apply(assumption) |
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460 apply(simp) |
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461 using size_rexp |
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462 apply(simp) |
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463 done |
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464 |
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465 lemma awidth_size: |
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466 shows "awidth r \<le> size2 r" |
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467 apply(induct r) |
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468 apply(simp_all) |
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469 done |
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470 |
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471 lemma Sum1: |
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472 fixes A B :: "nat set" |
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473 assumes "A \<subseteq> B" "finite A" "finite B" |
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474 shows "\<Sum>A \<le> \<Sum>B" |
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475 using assms |
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476 by (simp add: sum_mono2) |
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477 |
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478 lemma Sum2: |
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479 fixes A :: "rexp set" |
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480 and f g :: "rexp \<Rightarrow> nat" |
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481 assumes "finite A" "\<forall>x \<in> A. f x \<le> g x" |
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482 shows "sum f A \<le> sum g A" |
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483 using assms |
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484 apply(induct A) |
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485 apply(auto) |
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486 done |
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487 |
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488 |
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489 |
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490 |
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491 |
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492 lemma pders_max_size: |
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493 shows "(sum size2 (pders s r)) \<le> (Suc (size2 r)) ^ 3" |
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494 proof - |
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495 have "(sum size2 (pders s r)) \<le> sum (\<lambda>_. Suc (size2 r * size2 r)) (pders s r)" |
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496 apply(rule_tac Sum2) |
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497 apply (meson pders_subs rev_finite_subset subs_finite) |
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498 using pders_subs subs_size2 by blast |
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499 also have "... \<le> (Suc (size2 r * size2 r)) * (sum (\<lambda>_. 1) (pders s r))" |
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500 by simp |
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501 also have "... \<le> (Suc (size2 r * size2 r)) * card (pders s r)" |
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502 by simp |
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503 also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (awidth r))" |
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504 using Suc_eq_plus1 card_pders_awidth mult_le_mono2 by presburger |
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505 also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (size2 r))" |
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506 using Suc_le_mono awidth_size mult_le_mono2 by presburger |
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507 also have "... \<le> (Suc (size2 r)) ^ 3" |
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508 by (smt One_nat_def Suc_1 Suc_mult_le_cancel1 Suc_n_not_le_n antisym_conv le_Suc_eq mult.commute nat_le_linear numeral_3_eq_3 power2_eq_square power2_le_imp_le power_Suc size_rexp) |
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509 finally show ?thesis . |
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510 qed |
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511 |
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512 lemma pders_Set_max_size: |
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513 shows "(sum size2 (pders_Set A r)) \<le> (Suc (size2 r)) ^ 3" |
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514 proof - |
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515 have "(sum size2 (pders_Set A r)) \<le> sum (\<lambda>_. Suc (size2 r * size2 r)) (pders_Set A r)" |
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516 apply(rule_tac Sum2) |
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517 apply (simp add: finite_pders_set) |
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518 by (meson pders_Set_subsetI pders_subs subs_size2 subsetD) |
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519 also have "... \<le> (Suc (size2 r * size2 r)) * (sum (\<lambda>_. 1) (pders_Set A r))" |
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520 by simp |
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521 also have "... \<le> (Suc (size2 r * size2 r)) * card (pders_Set A r)" |
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522 by simp |
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523 also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (awidth r))" |
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524 using Suc_eq_plus1 card_pders_set_le_awidth mult_le_mono2 by presburger |
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525 also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (size2 r))" |
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526 using Suc_le_mono awidth_size mult_le_mono2 by presburger |
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527 also have "... \<le> (Suc (size2 r)) ^ 3" |
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528 by (smt One_nat_def Suc_1 Suc_mult_le_cancel1 Suc_n_not_le_n antisym_conv le_Suc_eq mult.commute nat_le_linear numeral_3_eq_3 power2_eq_square power2_le_imp_le power_Suc size_rexp) |
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529 finally show ?thesis . |
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530 qed |
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531 |
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532 fun height :: "rexp \<Rightarrow> nat" where |
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533 "height ZERO = 1" | |
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534 "height ONE = 1" | |
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535 "height (CHAR c) = 1" | |
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536 "height (ALT r1 r2) = Suc (max (height r1) (height r2))" | |
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537 "height (SEQ r1 r2) = Suc (max (height r1) (height r2))" | |
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538 "height (STAR r1) = Suc (height r1)" |
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539 |
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540 lemma height_size2: |
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541 shows "height r \<le> size2 r" |
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542 apply(induct r) |
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543 apply(simp_all) |
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544 done |
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545 |
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546 lemma height_rexp: |
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547 fixes r :: rexp |
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548 shows "1 \<le> height r" |
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549 apply(induct r) |
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550 apply(simp_all) |
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551 done |
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552 |
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553 lemma subs_height: |
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554 shows "\<forall>r1 \<in> subs r. height r1 \<le> Suc (height r)" |
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555 apply(induct r) |
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556 apply(auto)+ |
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557 done |
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558 |
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559 |
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560 |
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561 end |