24 GT \<Rightarrow> adjust (Node lv l (a,b) (delete x r)) | |
24 GT \<Rightarrow> adjust (Node lv l (a,b) (delete x r)) | |
25 EQ \<Rightarrow> (if l = Leaf then r |
25 EQ \<Rightarrow> (if l = Leaf then r |
26 else let (l',ab') = del_max l in adjust (Node lv l' ab' r)))" |
26 else let (l',ab') = del_max l in adjust (Node lv l' ab' r)))" |
27 |
27 |
28 |
28 |
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29 subsection "Invariance" |
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30 |
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31 subsubsection "Proofs for insert" |
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32 |
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33 lemma lvl_update_aux: |
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34 "lvl (update x y t) = lvl t \<or> lvl (update x y t) = lvl t + 1 \<and> sngl (update x y t)" |
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35 apply(induction t) |
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36 apply (auto simp: lvl_skew) |
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37 apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+ |
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38 done |
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39 |
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40 lemma lvl_update: obtains |
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41 (Same) "lvl (update x y t) = lvl t" | |
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42 (Incr) "lvl (update x y t) = lvl t + 1" "sngl (update x y t)" |
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43 using lvl_update_aux by fastforce |
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44 |
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45 declare invar.simps(2)[simp] |
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46 |
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47 lemma lvl_update_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(update x y t) = lvl t" |
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48 proof (induction t rule: update.induct) |
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49 case (2 x y lv t1 a b t2) |
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50 consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a" |
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51 using less_linear by blast |
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52 thus ?case proof cases |
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53 case LT |
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54 thus ?thesis using 2 by (auto simp add: skew_case split_case split: tree.splits) |
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55 next |
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56 case GT |
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57 thus ?thesis using 2 proof (cases t1) |
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58 case Node |
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59 thus ?thesis using 2 GT |
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60 apply (auto simp add: skew_case split_case split: tree.splits) |
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61 by (metis less_not_refl2 lvl.simps(2) lvl_update_aux n_not_Suc_n sngl.simps(3))+ |
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62 qed (auto simp add: lvl_0_iff) |
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63 qed simp |
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64 qed simp |
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65 |
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66 lemma lvl_update_incr_iff: "(lvl(update a b t) = lvl t + 1) \<longleftrightarrow> |
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67 (EX l x r. update a b t = Node (lvl t + 1) l x r \<and> lvl l = lvl r)" |
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68 apply(cases t) |
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69 apply(auto simp add: skew_case split_case split: if_splits) |
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70 apply(auto split: tree.splits if_splits) |
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71 done |
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72 |
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73 lemma invar_update: "invar t \<Longrightarrow> invar(update a b t)" |
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74 proof(induction t) |
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75 case (Node n l xy r) |
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76 hence il: "invar l" and ir: "invar r" by auto |
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77 obtain x y where [simp]: "xy = (x,y)" by fastforce |
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78 note N = Node |
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79 let ?t = "Node n l xy r" |
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80 have "a < x \<or> a = x \<or> x < a" by auto |
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81 moreover |
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82 { assume "a < x" |
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83 note iil = Node.IH(1)[OF il] |
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84 have ?case |
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85 proof (cases rule: lvl_update[of a b l]) |
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86 case (Same) thus ?thesis |
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87 using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same] |
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88 by (simp add: skew_invar split_invar del: invar.simps) |
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89 next |
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90 case (Incr) |
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91 then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2" |
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92 using Node.prems by (auto simp: lvl_Suc_iff) |
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93 have l12: "lvl t1 = lvl t2" |
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94 by (metis Incr(1) ial lvl_update_incr_iff tree.inject) |
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95 have "update a b ?t = split(skew(Node n (update a b l) xy r))" |
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96 by(simp add: \<open>a<x\<close>) |
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97 also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)" |
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98 by(simp) |
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99 also have "invar(split \<dots>)" |
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100 proof (cases r) |
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101 case Leaf |
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102 hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff) |
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103 thus ?thesis using Leaf ial by simp |
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104 next |
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105 case [simp]: (Node m t3 y t4) |
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106 show ?thesis (*using N(3) iil l12 by(auto)*) |
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107 proof cases |
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108 assume "m = n" thus ?thesis using N(3) iil by(auto) |
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109 next |
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110 assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto) |
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111 qed |
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112 qed |
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113 finally show ?thesis . |
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114 qed |
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115 } |
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116 moreover |
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117 { assume "x < a" |
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118 note iir = Node.IH(2)[OF ir] |
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119 from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto |
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120 hence ?case |
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121 proof |
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122 assume 0: "n = lvl r" |
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123 have "update a b ?t = split(skew(Node n l xy (update a b r)))" |
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124 using \<open>a>x\<close> by(auto) |
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125 also have "skew(Node n l xy (update a b r)) = Node n l xy (update a b r)" |
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126 using Node.prems by(simp add: skew_case split: tree.split) |
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127 also have "invar(split \<dots>)" |
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128 proof - |
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129 from lvl_update_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a b] |
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130 obtain t1 p t2 where iar: "update a b r = Node n t1 p t2" |
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131 using Node.prems 0 by (auto simp: lvl_Suc_iff) |
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132 from Node.prems iar 0 iir |
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133 show ?thesis by (auto simp: split_case split: tree.splits) |
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134 qed |
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135 finally show ?thesis . |
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136 next |
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137 assume 1: "n = lvl r + 1" |
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138 hence "sngl ?t" by(cases r) auto |
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139 show ?thesis |
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140 proof (cases rule: lvl_update[of a b r]) |
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141 case (Same) |
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142 show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same] |
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143 by (auto simp add: skew_invar split_invar) |
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144 next |
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145 case (Incr) |
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146 thus ?thesis using invar_NodeR2[OF `invar ?t` Incr(2) 1 iir] 1 \<open>x < a\<close> |
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147 by (auto simp add: skew_invar split_invar split: if_splits) |
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148 qed |
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149 qed |
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150 } |
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151 moreover { assume "a = x" hence ?case using Node.prems by auto } |
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152 ultimately show ?case by blast |
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153 qed simp |
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154 |
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155 subsubsection "Proofs for delete" |
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156 |
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157 declare invar.simps(2)[simp del] |
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158 |
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159 theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)" |
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160 proof (induction t) |
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161 case (Node lv l ab r) |
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162 |
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163 obtain a b where [simp]: "ab = (a,b)" by fastforce |
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164 |
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165 let ?l' = "delete x l" and ?r' = "delete x r" |
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166 let ?t = "Node lv l ab r" let ?t' = "delete x ?t" |
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167 |
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168 from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto) |
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169 |
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170 note post_l' = Node.IH(1)[OF inv_l] |
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171 note preL = pre_adj_if_postL[OF Node.prems post_l'] |
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172 |
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173 note post_r' = Node.IH(2)[OF inv_r] |
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174 note preR = pre_adj_if_postR[OF Node.prems post_r'] |
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175 |
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176 show ?case |
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177 proof (cases rule: linorder_cases[of x a]) |
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178 case less |
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179 thus ?thesis using Node.prems by (simp add: post_del_adjL preL) |
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180 next |
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181 case greater |
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182 thus ?thesis using Node.prems preR by (simp add: post_del_adjR post_r') |
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183 next |
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184 case equal |
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185 show ?thesis |
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186 proof cases |
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187 assume "l = Leaf" thus ?thesis using equal Node.prems |
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188 by(auto simp: post_del_def invar.simps(2)) |
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189 next |
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190 assume "l \<noteq> Leaf" thus ?thesis using equal Node.prems |
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191 by simp (metis inv_l post_del_adjL post_del_max pre_adj_if_postL) |
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192 qed |
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193 qed |
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194 qed (simp add: post_del_def) |
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195 |
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196 |
29 subsection {* Functional Correctness Proofs *} |
197 subsection {* Functional Correctness Proofs *} |
30 |
198 |
31 theorem inorder_update: |
199 theorem inorder_update: |
32 "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)" |
200 "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)" |
33 by (induct t) (auto simp: upd_list_simps inorder_split inorder_skew) |
201 by (induct t) (auto simp: upd_list_simps inorder_split inorder_skew) |
34 |
202 |
35 |
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36 theorem inorder_delete: |
203 theorem inorder_delete: |
37 "sorted1(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)" |
204 "\<lbrakk>invar t; sorted1(inorder t)\<rbrakk> \<Longrightarrow> |
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205 inorder (delete x t) = del_list x (inorder t)" |
38 by(induction t) |
206 by(induction t) |
39 (auto simp: del_list_simps inorder_adjust del_maxD split: prod.splits) |
207 (auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR |
40 |
208 post_del_max post_delete del_maxD split: prod.splits) |
41 interpretation Map_by_Ordered |
209 |
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210 interpretation I: Map_by_Ordered |
42 where empty = Leaf and lookup = lookup and update = update and delete = delete |
211 where empty = Leaf and lookup = lookup and update = update and delete = delete |
43 and inorder = inorder and inv = "\<lambda>_. True" |
212 and inorder = inorder and inv = invar |
44 proof (standard, goal_cases) |
213 proof (standard, goal_cases) |
45 case 1 show ?case by simp |
214 case 1 show ?case by simp |
46 next |
215 next |
47 case 2 thus ?case by(simp add: lookup_map_of) |
216 case 2 thus ?case by(simp add: lookup_map_of) |
48 next |
217 next |
49 case 3 thus ?case by(simp add: inorder_update) |
218 case 3 thus ?case by(simp add: inorder_update) |
50 next |
219 next |
51 case 4 thus ?case by(simp add: inorder_delete) |
220 case 4 thus ?case by(simp add: inorder_delete) |
52 qed auto |
221 next |
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222 case 5 thus ?case by(simp) |
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223 next |
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224 case 6 thus ?case by(simp add: invar_update) |
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225 next |
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226 case 7 thus ?case using post_delete by(auto simp: post_del_def) |
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227 qed |
53 |
228 |
54 end |
229 end |