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1 (* Author: Tobias Nipkow *) |
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2 |
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3 section {* Unbalanced Tree as Map *} |
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4 |
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5 theory Tree_Map |
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6 imports |
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7 "~~/src/HOL/Library/Tree" |
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8 Map_by_Ordered |
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9 begin |
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10 |
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11 fun lookup :: "('a::linorder*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where |
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12 "lookup Leaf x = None" | |
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13 "lookup (Node l (a,b) r) x = (if x < a then lookup l x else |
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14 if x > a then lookup r x else Some b)" |
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15 |
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16 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where |
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17 "update a b Leaf = Node Leaf (a,b) Leaf" | |
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18 "update a b (Node l (x,y) r) = |
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19 (if a < x then Node (update a b l) (x,y) r |
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20 else if a=x then Node l (a,b) r |
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21 else Node l (x,y) (update a b r))" |
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22 |
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23 fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where |
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24 "del_min (Node Leaf a r) = (a, r)" | |
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25 "del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))" |
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26 |
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27 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where |
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28 "delete k Leaf = Leaf" | |
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29 "delete k (Node l (a,b) r) = (if k<a then Node (delete k l) (a,b) r else |
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30 if k > a then Node l (a,b) (delete k r) else |
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31 if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')" |
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32 |
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33 |
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34 subsection "Functional Correctness Proofs" |
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35 |
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36 lemma lookup_eq: "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x" |
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37 apply (induction t) |
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38 apply (auto simp: sorted_lems map_of_append map_of_sorteds split: option.split) |
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39 done |
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40 |
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41 |
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42 lemma inorder_update: |
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43 "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)" |
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44 by(induction t) (auto simp: upd_list_sorteds sorted_lems) |
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45 |
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46 |
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47 lemma del_minD: |
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48 "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow> |
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49 x # inorder t' = inorder t" |
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50 by(induction t arbitrary: t' rule: del_min.induct) |
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51 (auto simp: sorted_lems split: prod.splits) |
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52 |
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53 lemma inorder_delete: |
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54 "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)" |
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55 by(induction t) |
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56 (auto simp: del_list_sorted sorted_lems dest!: del_minD split: prod.splits) |
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57 |
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58 |
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59 interpretation Map_by_Ordered |
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60 where empty = Leaf and lookup = lookup and update = update and delete = delete |
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61 and inorder = inorder and wf = "\<lambda>_. True" |
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62 proof (standard, goal_cases) |
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63 case 1 show ?case by simp |
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64 next |
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65 case 2 thus ?case by(simp add: lookup_eq) |
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66 next |
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67 case 3 thus ?case by(simp add: inorder_update) |
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68 next |
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69 case 4 thus ?case by(simp add: inorder_delete) |
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70 qed (rule TrueI)+ |
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71 |
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72 end |