author | haftmann |
Thu, 29 Nov 2007 17:08:26 +0100 | |
changeset 25502 | 9200b36280c0 |
parent 23373 | ead82c82da9e |
child 25966 | 74f6817870f9 |
permissions | -rw-r--r-- |
22803 | 1 |
(* Title: HOL/Library/AssocList.thy |
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ID: $Id$ |
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Author: Norbert Schirmer, Tobias Nipkow, Martin Wildmoser |
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*) |
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header {* Map operations implemented on association lists*} |
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theory AssocList |
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imports Map |
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begin |
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text {* |
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The operations preserve distinctness of keys and |
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function @{term "clearjunk"} distributes over them. Since |
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@{term clearjunk} enforces distinctness of keys it can be used |
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to establish the invariant, e.g. for inductive proofs. |
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*} |
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|
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fun |
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delete :: "'key \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"delete k [] = []" |
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| "delete k (p#ps) = (if fst p = k then delete k ps else p # delete k ps)" |
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|
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fun |
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update :: "'key \<Rightarrow> 'val \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"update k v [] = [(k, v)]" |
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| "update k v (p#ps) = (if fst p = k then (k, v) # ps else p # update k v ps)" |
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|
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function |
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updates :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"updates [] vs ps = ps" |
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| "updates (k#ks) vs ps = (case vs |
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of [] \<Rightarrow> ps |
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| (v#vs') \<Rightarrow> updates ks vs' (update k v ps))" |
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by pat_completeness auto |
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termination by lexicographic_order |
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fun |
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merge :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"merge qs [] = qs" |
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| "merge qs (p#ps) = update (fst p) (snd p) (merge qs ps)" |
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lemma length_delete_le: "length (delete k al) \<le> length al" |
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proof (induct al) |
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case Nil thus ?case by simp |
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next |
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case (Cons a al) |
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note length_filter_le [of "\<lambda>p. fst p \<noteq> fst a" al] |
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also have "\<And>n. n \<le> Suc n" |
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by simp |
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finally have "length [p\<leftarrow>al . fst p \<noteq> fst a] \<le> Suc (length al)" . |
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with Cons show ?case |
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by auto |
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qed |
59 |
||
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lemma compose_hint [simp]: |
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"length (delete k al) < Suc (length al)" |
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proof - |
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note length_delete_le |
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also have "\<And>n. n < Suc n" |
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by simp |
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finally show ?thesis . |
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qed |
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||
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function |
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compose :: "('key \<times> 'a) list \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('key \<times> 'b) list" |
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where |
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"compose [] ys = []" |
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| "compose (x#xs) ys = (case map_of ys (snd x) |
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of None \<Rightarrow> compose (delete (fst x) xs) ys |
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| Some v \<Rightarrow> (fst x, v) # compose xs ys)" |
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by pat_completeness auto |
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termination by lexicographic_order |
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fun |
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restrict :: "'key set \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"restrict A [] = []" |
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| "restrict A (p#ps) = (if fst p \<in> A then p#restrict A ps else restrict A ps)" |
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22740 | 85 |
fun |
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map_ran :: "('key \<Rightarrow> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"map_ran f [] = []" |
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| "map_ran f (p#ps) = (fst p, f (fst p) (snd p)) # map_ran f ps" |
90 |
||
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fun |
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clearjunk :: "('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list" |
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where |
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"clearjunk [] = []" |
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| "clearjunk (p#ps) = p # clearjunk (delete (fst p) ps)" |
96 |
||
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lemmas [simp del] = compose_hint |
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||
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subsection {* Lookup *} |
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||
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lemma lookup_simps [code func]: |
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"map_of [] k = None" |
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"map_of (p#ps) k = (if fst p = k then Some (snd p) else map_of ps k)" |
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by simp_all |
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||
22740 | 107 |
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subsection {* @{const delete} *} |
109 |
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lemma delete_def: |
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"delete k xs = filter (\<lambda>p. fst p \<noteq> k) xs" |
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by (induct xs) auto |
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lemma delete_id [simp]: "k \<notin> fst ` set al \<Longrightarrow> delete k al = al" |
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by (induct al) auto |
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lemma delete_conv: "map_of (delete k al) k' = ((map_of al)(k := None)) k'" |
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by (induct al) auto |
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lemma delete_conv': "map_of (delete k al) = ((map_of al)(k := None))" |
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by (rule ext) (rule delete_conv) |
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lemma delete_idem: "delete k (delete k al) = delete k al" |
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by (induct al) auto |
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lemma map_of_delete [simp]: |
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"k' \<noteq> k \<Longrightarrow> map_of (delete k al) k' = map_of al k'" |
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by (induct al) auto |
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lemma delete_notin_dom: "k \<notin> fst ` set (delete k al)" |
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by (induct al) auto |
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lemma dom_delete_subset: "fst ` set (delete k al) \<subseteq> fst ` set al" |
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by (induct al) auto |
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lemma distinct_delete: |
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assumes "distinct (map fst al)" |
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shows "distinct (map fst (delete k al))" |
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23373 | 139 |
using assms |
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proof (induct al) |
141 |
case Nil thus ?case by simp |
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next |
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case (Cons a al) |
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from Cons.prems obtain |
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a_notin_al: "fst a \<notin> fst ` set al" and |
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dist_al: "distinct (map fst al)" |
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by auto |
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show ?case |
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proof (cases "fst a = k") |
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case True |
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with Cons dist_al show ?thesis by simp |
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next |
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case False |
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from dist_al |
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have "distinct (map fst (delete k al))" |
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by (rule Cons.hyps) |
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moreover from a_notin_al dom_delete_subset [of k al] |
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have "fst a \<notin> fst ` set (delete k al)" |
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by blast |
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ultimately show ?thesis using False by simp |
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qed |
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qed |
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lemma delete_twist: "delete x (delete y al) = delete y (delete x al)" |
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by (induct al) auto |
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lemma clearjunk_delete: "clearjunk (delete x al) = delete x (clearjunk al)" |
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by (induct al rule: clearjunk.induct) (auto simp add: delete_idem delete_twist) |
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||
23373 | 170 |
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subsection {* @{const clearjunk} *} |
172 |
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lemma insert_fst_filter: |
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"insert a(fst ` {x \<in> set ps. fst x \<noteq> a}) = insert a (fst ` set ps)" |
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by (induct ps) auto |
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lemma dom_clearjunk: "fst ` set (clearjunk al) = fst ` set al" |
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by (induct al rule: clearjunk.induct) (simp_all add: insert_fst_filter delete_def) |
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lemma notin_filter_fst: "a \<notin> fst ` {x \<in> set ps. fst x \<noteq> a}" |
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by (induct ps) auto |
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lemma distinct_clearjunk [simp]: "distinct (map fst (clearjunk al))" |
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by (induct al rule: clearjunk.induct) |
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(simp_all add: dom_clearjunk notin_filter_fst delete_def) |
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23281 | 187 |
lemma map_of_filter: "k \<noteq> a \<Longrightarrow> map_of [q\<leftarrow>ps . fst q \<noteq> a] k = map_of ps k" |
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by (induct ps) auto |
189 |
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lemma map_of_clearjunk: "map_of (clearjunk al) = map_of al" |
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apply (rule ext) |
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apply (induct al rule: clearjunk.induct) |
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apply simp |
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apply (simp add: map_of_filter) |
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done |
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lemma length_clearjunk: "length (clearjunk al) \<le> length al" |
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proof (induct al rule: clearjunk.induct [case_names Nil Cons]) |
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case Nil thus ?case by simp |
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200 |
next |
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22740 | 201 |
case (Cons p ps) |
23281 | 202 |
from Cons have "length (clearjunk [q\<leftarrow>ps . fst q \<noteq> fst p]) \<le> length [q\<leftarrow>ps . fst q \<noteq> fst p]" |
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by (simp add: delete_def) |
204 |
also have "\<dots> \<le> length ps" |
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by simp |
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finally show ?case |
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by (simp add: delete_def) |
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qed |
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lemma notin_fst_filter: "a \<notin> fst ` set ps \<Longrightarrow> [q\<leftarrow>ps . fst q \<noteq> a] = ps" |
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by (induct ps) auto |
212 |
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213 |
lemma distinct_clearjunk_id [simp]: "distinct (map fst al) \<Longrightarrow> clearjunk al = al" |
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by (induct al rule: clearjunk.induct) (auto simp add: notin_fst_filter) |
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215 |
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216 |
lemma clearjunk_idem: "clearjunk (clearjunk al) = clearjunk al" |
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by simp |
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||
23373 | 219 |
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19234 | 220 |
subsection {* @{const dom} and @{term "ran"} *} |
221 |
||
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lemma dom_map_of': "fst ` set al = dom (map_of al)" |
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by (induct al) auto |
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||
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lemmas dom_map_of = dom_map_of' [symmetric] |
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||
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lemma ran_clearjunk: "ran (map_of (clearjunk al)) = ran (map_of al)" |
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by (simp add: map_of_clearjunk) |
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229 |
||
230 |
lemma ran_distinct: |
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assumes dist: "distinct (map fst al)" |
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shows "ran (map_of al) = snd ` set al" |
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using dist |
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proof (induct al) |
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235 |
case Nil |
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thus ?case by simp |
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237 |
next |
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238 |
case (Cons a al) |
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hence hyp: "snd ` set al = ran (map_of al)" |
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240 |
by simp |
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241 |
||
242 |
have "ran (map_of (a # al)) = {snd a} \<union> ran (map_of al)" |
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proof |
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show "ran (map_of (a # al)) \<subseteq> {snd a} \<union> ran (map_of al)" |
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proof |
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fix v |
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assume "v \<in> ran (map_of (a#al))" |
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then obtain x where "map_of (a#al) x = Some v" |
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by (auto simp add: ran_def) |
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then show "v \<in> {snd a} \<union> ran (map_of al)" |
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by (auto split: split_if_asm simp add: ran_def) |
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qed |
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next |
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show "{snd a} \<union> ran (map_of al) \<subseteq> ran (map_of (a # al))" |
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proof |
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256 |
fix v |
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257 |
assume v_in: "v \<in> {snd a} \<union> ran (map_of al)" |
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258 |
show "v \<in> ran (map_of (a#al))" |
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259 |
proof (cases "v=snd a") |
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260 |
case True |
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with v_in show ?thesis |
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by (auto simp add: ran_def) |
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next |
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case False |
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with v_in have "v \<in> ran (map_of al)" by auto |
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then obtain x where al_x: "map_of al x = Some v" |
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by (auto simp add: ran_def) |
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from map_of_SomeD [OF this] |
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have "x \<in> fst ` set al" |
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by (force simp add: image_def) |
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with Cons.prems have "x\<noteq>fst a" |
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by - (rule ccontr,simp) |
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273 |
with al_x |
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274 |
show ?thesis |
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by (auto simp add: ran_def) |
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276 |
qed |
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277 |
qed |
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278 |
qed |
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279 |
with hyp show ?case |
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280 |
by (simp only:) auto |
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281 |
qed |
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282 |
||
283 |
lemma ran_map_of: "ran (map_of al) = snd ` set (clearjunk al)" |
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284 |
proof - |
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285 |
have "ran (map_of al) = ran (map_of (clearjunk al))" |
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286 |
by (simp add: ran_clearjunk) |
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also have "\<dots> = snd ` set (clearjunk al)" |
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288 |
by (simp add: ran_distinct) |
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289 |
finally show ?thesis . |
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290 |
qed |
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291 |
||
23373 | 292 |
|
19234 | 293 |
subsection {* @{const update} *} |
294 |
||
295 |
lemma update_conv: "map_of (update k v al) k' = ((map_of al)(k\<mapsto>v)) k'" |
|
296 |
by (induct al) auto |
|
297 |
||
298 |
lemma update_conv': "map_of (update k v al) = ((map_of al)(k\<mapsto>v))" |
|
299 |
by (rule ext) (rule update_conv) |
|
300 |
||
301 |
lemma dom_update: "fst ` set (update k v al) = {k} \<union> fst ` set al" |
|
302 |
by (induct al) auto |
|
303 |
||
304 |
lemma distinct_update: |
|
305 |
assumes "distinct (map fst al)" |
|
306 |
shows "distinct (map fst (update k v al))" |
|
23373 | 307 |
using assms |
19234 | 308 |
proof (induct al) |
309 |
case Nil thus ?case by simp |
|
310 |
next |
|
311 |
case (Cons a al) |
|
312 |
from Cons.prems obtain |
|
313 |
a_notin_al: "fst a \<notin> fst ` set al" and |
|
314 |
dist_al: "distinct (map fst al)" |
|
315 |
by auto |
|
316 |
show ?case |
|
317 |
proof (cases "fst a = k") |
|
318 |
case True |
|
319 |
from True dist_al a_notin_al show ?thesis by simp |
|
320 |
next |
|
321 |
case False |
|
322 |
from dist_al |
|
323 |
have "distinct (map fst (update k v al))" |
|
324 |
by (rule Cons.hyps) |
|
325 |
with False a_notin_al show ?thesis by (simp add: dom_update) |
|
326 |
qed |
|
327 |
qed |
|
328 |
||
329 |
lemma update_filter: |
|
23281 | 330 |
"a\<noteq>k \<Longrightarrow> update k v [q\<leftarrow>ps . fst q \<noteq> a] = [q\<leftarrow>update k v ps . fst q \<noteq> a]" |
19234 | 331 |
by (induct ps) auto |
332 |
||
333 |
lemma clearjunk_update: "clearjunk (update k v al) = update k v (clearjunk al)" |
|
334 |
by (induct al rule: clearjunk.induct) (auto simp add: update_filter delete_def) |
|
335 |
||
336 |
lemma update_triv: "map_of al k = Some v \<Longrightarrow> update k v al = al" |
|
337 |
by (induct al) auto |
|
338 |
||
339 |
lemma update_nonempty [simp]: "update k v al \<noteq> []" |
|
340 |
by (induct al) auto |
|
341 |
||
342 |
lemma update_eqD: "update k v al = update k v' al' \<Longrightarrow> v=v'" |
|
20503 | 343 |
proof (induct al arbitrary: al') |
19234 | 344 |
case Nil thus ?case |
345 |
by (cases al') (auto split: split_if_asm) |
|
346 |
next |
|
347 |
case Cons thus ?case |
|
348 |
by (cases al') (auto split: split_if_asm) |
|
349 |
qed |
|
350 |
||
351 |
lemma update_last [simp]: "update k v (update k v' al) = update k v al" |
|
352 |
by (induct al) auto |
|
353 |
||
354 |
text {* Note that the lists are not necessarily the same: |
|
355 |
@{term "update k v (update k' v' []) = [(k',v'),(k,v)]"} and |
|
356 |
@{term "update k' v' (update k v []) = [(k,v),(k',v')]"}.*} |
|
357 |
lemma update_swap: "k\<noteq>k' |
|
358 |
\<Longrightarrow> map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))" |
|
359 |
by (auto simp add: update_conv' intro: ext) |
|
360 |
||
361 |
lemma update_Some_unfold: |
|
362 |
"(map_of (update k v al) x = Some y) = |
|
363 |
(x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y)" |
|
364 |
by (simp add: update_conv' map_upd_Some_unfold) |
|
365 |
||
366 |
lemma image_update[simp]: "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A" |
|
367 |
by (simp add: update_conv' image_map_upd) |
|
368 |
||
369 |
||
370 |
subsection {* @{const updates} *} |
|
371 |
||
372 |
lemma updates_conv: "map_of (updates ks vs al) k = ((map_of al)(ks[\<mapsto>]vs)) k" |
|
20503 | 373 |
proof (induct ks arbitrary: vs al) |
19234 | 374 |
case Nil |
375 |
thus ?case by simp |
|
376 |
next |
|
377 |
case (Cons k ks) |
|
378 |
show ?case |
|
379 |
proof (cases vs) |
|
380 |
case Nil |
|
381 |
with Cons show ?thesis by simp |
|
382 |
next |
|
383 |
case (Cons k ks') |
|
384 |
with Cons.hyps show ?thesis |
|
385 |
by (simp add: update_conv fun_upd_def) |
|
386 |
qed |
|
387 |
qed |
|
388 |
||
389 |
lemma updates_conv': "map_of (updates ks vs al) = ((map_of al)(ks[\<mapsto>]vs))" |
|
390 |
by (rule ext) (rule updates_conv) |
|
391 |
||
392 |
lemma distinct_updates: |
|
393 |
assumes "distinct (map fst al)" |
|
394 |
shows "distinct (map fst (updates ks vs al))" |
|
23373 | 395 |
using assms |
22740 | 396 |
by (induct ks arbitrary: vs al) |
397 |
(auto simp add: distinct_update split: list.splits) |
|
19234 | 398 |
|
399 |
lemma clearjunk_updates: |
|
400 |
"clearjunk (updates ks vs al) = updates ks vs (clearjunk al)" |
|
20503 | 401 |
by (induct ks arbitrary: vs al) (auto simp add: clearjunk_update split: list.splits) |
19234 | 402 |
|
403 |
lemma updates_empty[simp]: "updates vs [] al = al" |
|
404 |
by (induct vs) auto |
|
405 |
||
406 |
lemma updates_Cons: "updates (k#ks) (v#vs) al = updates ks vs (update k v al)" |
|
407 |
by simp |
|
408 |
||
409 |
lemma updates_append1[simp]: "size ks < size vs \<Longrightarrow> |
|
410 |
updates (ks@[k]) vs al = update k (vs!size ks) (updates ks vs al)" |
|
20503 | 411 |
by (induct ks arbitrary: vs al) (auto split: list.splits) |
19234 | 412 |
|
413 |
lemma updates_list_update_drop[simp]: |
|
414 |
"\<lbrakk>size ks \<le> i; i < size vs\<rbrakk> |
|
415 |
\<Longrightarrow> updates ks (vs[i:=v]) al = updates ks vs al" |
|
20503 | 416 |
by (induct ks arbitrary: al vs i) (auto split:list.splits nat.splits) |
19234 | 417 |
|
418 |
lemma update_updates_conv_if: " |
|
419 |
map_of (updates xs ys (update x y al)) = |
|
420 |
map_of (if x \<in> set(take (length ys) xs) then updates xs ys al |
|
421 |
else (update x y (updates xs ys al)))" |
|
422 |
by (simp add: updates_conv' update_conv' map_upd_upds_conv_if) |
|
423 |
||
424 |
lemma updates_twist [simp]: |
|
425 |
"k \<notin> set ks \<Longrightarrow> |
|
426 |
map_of (updates ks vs (update k v al)) = map_of (update k v (updates ks vs al))" |
|
427 |
by (simp add: updates_conv' update_conv' map_upds_twist) |
|
428 |
||
429 |
lemma updates_apply_notin[simp]: |
|
430 |
"k \<notin> set ks ==> map_of (updates ks vs al) k = map_of al k" |
|
431 |
by (simp add: updates_conv) |
|
432 |
||
433 |
lemma updates_append_drop[simp]: |
|
434 |
"size xs = size ys \<Longrightarrow> updates (xs@zs) ys al = updates xs ys al" |
|
20503 | 435 |
by (induct xs arbitrary: ys al) (auto split: list.splits) |
19234 | 436 |
|
437 |
lemma updates_append2_drop[simp]: |
|
438 |
"size xs = size ys \<Longrightarrow> updates xs (ys@zs) al = updates xs ys al" |
|
20503 | 439 |
by (induct xs arbitrary: ys al) (auto split: list.splits) |
19234 | 440 |
|
23373 | 441 |
|
19333 | 442 |
subsection {* @{const map_ran} *} |
19234 | 443 |
|
19333 | 444 |
lemma map_ran_conv: "map_of (map_ran f al) k = option_map (f k) (map_of al k)" |
19234 | 445 |
by (induct al) auto |
446 |
||
19333 | 447 |
lemma dom_map_ran: "fst ` set (map_ran f al) = fst ` set al" |
19234 | 448 |
by (induct al) auto |
449 |
||
19333 | 450 |
lemma distinct_map_ran: "distinct (map fst al) \<Longrightarrow> distinct (map fst (map_ran f al))" |
451 |
by (induct al) (auto simp add: dom_map_ran) |
|
19234 | 452 |
|
23281 | 453 |
lemma map_ran_filter: "map_ran f [p\<leftarrow>ps. fst p \<noteq> a] = [p\<leftarrow>map_ran f ps. fst p \<noteq> a]" |
19234 | 454 |
by (induct ps) auto |
455 |
||
19333 | 456 |
lemma clearjunk_map_ran: "clearjunk (map_ran f al) = map_ran f (clearjunk al)" |
457 |
by (induct al rule: clearjunk.induct) (auto simp add: delete_def map_ran_filter) |
|
19234 | 458 |
|
23373 | 459 |
|
19234 | 460 |
subsection {* @{const merge} *} |
461 |
||
462 |
lemma dom_merge: "fst ` set (merge xs ys) = fst ` set xs \<union> fst ` set ys" |
|
20503 | 463 |
by (induct ys arbitrary: xs) (auto simp add: dom_update) |
19234 | 464 |
|
465 |
lemma distinct_merge: |
|
466 |
assumes "distinct (map fst xs)" |
|
467 |
shows "distinct (map fst (merge xs ys))" |
|
23373 | 468 |
using assms |
20503 | 469 |
by (induct ys arbitrary: xs) (auto simp add: dom_merge distinct_update) |
19234 | 470 |
|
471 |
lemma clearjunk_merge: |
|
472 |
"clearjunk (merge xs ys) = merge (clearjunk xs) ys" |
|
473 |
by (induct ys) (auto simp add: clearjunk_update) |
|
474 |
||
475 |
lemma merge_conv: "map_of (merge xs ys) k = (map_of xs ++ map_of ys) k" |
|
476 |
proof (induct ys) |
|
477 |
case Nil thus ?case by simp |
|
478 |
next |
|
479 |
case (Cons y ys) |
|
480 |
show ?case |
|
481 |
proof (cases "k = fst y") |
|
482 |
case True |
|
483 |
from True show ?thesis |
|
484 |
by (simp add: update_conv) |
|
485 |
next |
|
486 |
case False |
|
487 |
from False show ?thesis |
|
488 |
by (auto simp add: update_conv Cons.hyps map_add_def) |
|
489 |
qed |
|
490 |
qed |
|
491 |
||
492 |
lemma merge_conv': "map_of (merge xs ys) = (map_of xs ++ map_of ys)" |
|
493 |
by (rule ext) (rule merge_conv) |
|
494 |
||
495 |
lemma merge_emty: "map_of (merge [] ys) = map_of ys" |
|
496 |
by (simp add: merge_conv') |
|
497 |
||
498 |
lemma merge_assoc[simp]: "map_of (merge m1 (merge m2 m3)) = |
|
499 |
map_of (merge (merge m1 m2) m3)" |
|
500 |
by (simp add: merge_conv') |
|
501 |
||
502 |
lemma merge_Some_iff: |
|
503 |
"(map_of (merge m n) k = Some x) = |
|
504 |
(map_of n k = Some x \<or> map_of n k = None \<and> map_of m k = Some x)" |
|
505 |
by (simp add: merge_conv' map_add_Some_iff) |
|
506 |
||
507 |
lemmas merge_SomeD = merge_Some_iff [THEN iffD1, standard] |
|
508 |
declare merge_SomeD [dest!] |
|
509 |
||
510 |
lemma merge_find_right[simp]: "map_of n k = Some v \<Longrightarrow> map_of (merge m n) k = Some v" |
|
511 |
by (simp add: merge_conv') |
|
512 |
||
513 |
lemma merge_None [iff]: |
|
514 |
"(map_of (merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)" |
|
515 |
by (simp add: merge_conv') |
|
516 |
||
517 |
lemma merge_upd[simp]: |
|
518 |
"map_of (merge m (update k v n)) = map_of (update k v (merge m n))" |
|
519 |
by (simp add: update_conv' merge_conv') |
|
520 |
||
521 |
lemma merge_updatess[simp]: |
|
522 |
"map_of (merge m (updates xs ys n)) = map_of (updates xs ys (merge m n))" |
|
523 |
by (simp add: updates_conv' merge_conv') |
|
524 |
||
525 |
lemma merge_append: "map_of (xs@ys) = map_of (merge ys xs)" |
|
526 |
by (simp add: merge_conv') |
|
527 |
||
23373 | 528 |
|
19234 | 529 |
subsection {* @{const compose} *} |
530 |
||
531 |
lemma compose_first_None [simp]: |
|
532 |
assumes "map_of xs k = None" |
|
533 |
shows "map_of (compose xs ys) k = None" |
|
23373 | 534 |
using assms by (induct xs ys rule: compose.induct) |
22916 | 535 |
(auto split: option.splits split_if_asm) |
19234 | 536 |
|
537 |
lemma compose_conv: |
|
538 |
shows "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" |
|
22916 | 539 |
proof (induct xs ys rule: compose.induct) |
540 |
case 1 then show ?case by simp |
|
19234 | 541 |
next |
22916 | 542 |
case (2 x xs ys) show ?case |
19234 | 543 |
proof (cases "map_of ys (snd x)") |
22916 | 544 |
case None with 2 |
19234 | 545 |
have hyp: "map_of (compose (delete (fst x) xs) ys) k = |
546 |
(map_of ys \<circ>\<^sub>m map_of (delete (fst x) xs)) k" |
|
547 |
by simp |
|
548 |
show ?thesis |
|
549 |
proof (cases "fst x = k") |
|
550 |
case True |
|
551 |
from True delete_notin_dom [of k xs] |
|
552 |
have "map_of (delete (fst x) xs) k = None" |
|
553 |
by (simp add: map_of_eq_None_iff) |
|
554 |
with hyp show ?thesis |
|
555 |
using True None |
|
556 |
by simp |
|
557 |
next |
|
558 |
case False |
|
559 |
from False have "map_of (delete (fst x) xs) k = map_of xs k" |
|
560 |
by simp |
|
561 |
with hyp show ?thesis |
|
562 |
using False None |
|
563 |
by (simp add: map_comp_def) |
|
564 |
qed |
|
565 |
next |
|
566 |
case (Some v) |
|
22916 | 567 |
with 2 |
19234 | 568 |
have "map_of (compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k" |
569 |
by simp |
|
570 |
with Some show ?thesis |
|
571 |
by (auto simp add: map_comp_def) |
|
572 |
qed |
|
573 |
qed |
|
574 |
||
575 |
lemma compose_conv': |
|
576 |
shows "map_of (compose xs ys) = (map_of ys \<circ>\<^sub>m map_of xs)" |
|
577 |
by (rule ext) (rule compose_conv) |
|
578 |
||
579 |
lemma compose_first_Some [simp]: |
|
580 |
assumes "map_of xs k = Some v" |
|
581 |
shows "map_of (compose xs ys) k = map_of ys v" |
|
23373 | 582 |
using assms by (simp add: compose_conv) |
19234 | 583 |
|
584 |
lemma dom_compose: "fst ` set (compose xs ys) \<subseteq> fst ` set xs" |
|
22916 | 585 |
proof (induct xs ys rule: compose.induct) |
586 |
case 1 thus ?case by simp |
|
19234 | 587 |
next |
22916 | 588 |
case (2 x xs ys) |
19234 | 589 |
show ?case |
590 |
proof (cases "map_of ys (snd x)") |
|
591 |
case None |
|
22916 | 592 |
with "2.hyps" |
19234 | 593 |
have "fst ` set (compose (delete (fst x) xs) ys) \<subseteq> fst ` set (delete (fst x) xs)" |
594 |
by simp |
|
595 |
also |
|
596 |
have "\<dots> \<subseteq> fst ` set xs" |
|
597 |
by (rule dom_delete_subset) |
|
598 |
finally show ?thesis |
|
599 |
using None |
|
600 |
by auto |
|
601 |
next |
|
602 |
case (Some v) |
|
22916 | 603 |
with "2.hyps" |
19234 | 604 |
have "fst ` set (compose xs ys) \<subseteq> fst ` set xs" |
605 |
by simp |
|
606 |
with Some show ?thesis |
|
607 |
by auto |
|
608 |
qed |
|
609 |
qed |
|
610 |
||
611 |
lemma distinct_compose: |
|
612 |
assumes "distinct (map fst xs)" |
|
613 |
shows "distinct (map fst (compose xs ys))" |
|
23373 | 614 |
using assms |
22916 | 615 |
proof (induct xs ys rule: compose.induct) |
616 |
case 1 thus ?case by simp |
|
19234 | 617 |
next |
22916 | 618 |
case (2 x xs ys) |
19234 | 619 |
show ?case |
620 |
proof (cases "map_of ys (snd x)") |
|
621 |
case None |
|
22916 | 622 |
with 2 show ?thesis by simp |
19234 | 623 |
next |
624 |
case (Some v) |
|
22916 | 625 |
with 2 dom_compose [of xs ys] show ?thesis |
19234 | 626 |
by (auto) |
627 |
qed |
|
628 |
qed |
|
629 |
||
630 |
lemma compose_delete_twist: "(compose (delete k xs) ys) = delete k (compose xs ys)" |
|
22916 | 631 |
proof (induct xs ys rule: compose.induct) |
632 |
case 1 thus ?case by simp |
|
19234 | 633 |
next |
22916 | 634 |
case (2 x xs ys) |
19234 | 635 |
show ?case |
636 |
proof (cases "map_of ys (snd x)") |
|
637 |
case None |
|
22916 | 638 |
with 2 have |
19234 | 639 |
hyp: "compose (delete k (delete (fst x) xs)) ys = |
640 |
delete k (compose (delete (fst x) xs) ys)" |
|
641 |
by simp |
|
642 |
show ?thesis |
|
643 |
proof (cases "fst x = k") |
|
644 |
case True |
|
645 |
with None hyp |
|
646 |
show ?thesis |
|
647 |
by (simp add: delete_idem) |
|
648 |
next |
|
649 |
case False |
|
650 |
from None False hyp |
|
651 |
show ?thesis |
|
652 |
by (simp add: delete_twist) |
|
653 |
qed |
|
654 |
next |
|
655 |
case (Some v) |
|
22916 | 656 |
with 2 have hyp: "compose (delete k xs) ys = delete k (compose xs ys)" by simp |
19234 | 657 |
with Some show ?thesis |
658 |
by simp |
|
659 |
qed |
|
660 |
qed |
|
661 |
||
662 |
lemma compose_clearjunk: "compose xs (clearjunk ys) = compose xs ys" |
|
22916 | 663 |
by (induct xs ys rule: compose.induct) |
19234 | 664 |
(auto simp add: map_of_clearjunk split: option.splits) |
665 |
||
666 |
lemma clearjunk_compose: "clearjunk (compose xs ys) = compose (clearjunk xs) ys" |
|
667 |
by (induct xs rule: clearjunk.induct) |
|
668 |
(auto split: option.splits simp add: clearjunk_delete delete_idem |
|
669 |
compose_delete_twist) |
|
670 |
||
671 |
lemma compose_empty [simp]: |
|
672 |
"compose xs [] = []" |
|
22916 | 673 |
by (induct xs) (auto simp add: compose_delete_twist) |
19234 | 674 |
|
675 |
lemma compose_Some_iff: |
|
676 |
"(map_of (compose xs ys) k = Some v) = |
|
677 |
(\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = Some v)" |
|
678 |
by (simp add: compose_conv map_comp_Some_iff) |
|
679 |
||
680 |
lemma map_comp_None_iff: |
|
681 |
"(map_of (compose xs ys) k = None) = |
|
682 |
(map_of xs k = None \<or> (\<exists>k'. map_of xs k = Some k' \<and> map_of ys k' = None)) " |
|
683 |
by (simp add: compose_conv map_comp_None_iff) |
|
684 |
||
685 |
||
686 |
subsection {* @{const restrict} *} |
|
687 |
||
22740 | 688 |
lemma restrict_def: |
689 |
"restrict A = filter (\<lambda>p. fst p \<in> A)" |
|
690 |
proof |
|
691 |
fix xs |
|
692 |
show "restrict A xs = filter (\<lambda>p. fst p \<in> A) xs" |
|
693 |
by (induct xs) auto |
|
694 |
qed |
|
19234 | 695 |
|
696 |
lemma distinct_restr: "distinct (map fst al) \<Longrightarrow> distinct (map fst (restrict A al))" |
|
697 |
by (induct al) (auto simp add: restrict_def) |
|
698 |
||
699 |
lemma restr_conv: "map_of (restrict A al) k = ((map_of al)|` A) k" |
|
700 |
apply (induct al) |
|
701 |
apply (simp add: restrict_def) |
|
702 |
apply (cases "k\<in>A") |
|
703 |
apply (auto simp add: restrict_def) |
|
704 |
done |
|
705 |
||
706 |
lemma restr_conv': "map_of (restrict A al) = ((map_of al)|` A)" |
|
707 |
by (rule ext) (rule restr_conv) |
|
708 |
||
709 |
lemma restr_empty [simp]: |
|
710 |
"restrict {} al = []" |
|
711 |
"restrict A [] = []" |
|
712 |
by (induct al) (auto simp add: restrict_def) |
|
713 |
||
714 |
lemma restr_in [simp]: "x \<in> A \<Longrightarrow> map_of (restrict A al) x = map_of al x" |
|
715 |
by (simp add: restr_conv') |
|
716 |
||
717 |
lemma restr_out [simp]: "x \<notin> A \<Longrightarrow> map_of (restrict A al) x = None" |
|
718 |
by (simp add: restr_conv') |
|
719 |
||
720 |
lemma dom_restr [simp]: "fst ` set (restrict A al) = fst ` set al \<inter> A" |
|
721 |
by (induct al) (auto simp add: restrict_def) |
|
722 |
||
723 |
lemma restr_upd_same [simp]: "restrict (-{x}) (update x y al) = restrict (-{x}) al" |
|
724 |
by (induct al) (auto simp add: restrict_def) |
|
725 |
||
726 |
lemma restr_restr [simp]: "restrict A (restrict B al) = restrict (A\<inter>B) al" |
|
727 |
by (induct al) (auto simp add: restrict_def) |
|
728 |
||
729 |
lemma restr_update[simp]: |
|
730 |
"map_of (restrict D (update x y al)) = |
|
731 |
map_of ((if x \<in> D then (update x y (restrict (D-{x}) al)) else restrict D al))" |
|
732 |
by (simp add: restr_conv' update_conv') |
|
733 |
||
734 |
lemma restr_delete [simp]: |
|
735 |
"(delete x (restrict D al)) = |
|
736 |
(if x\<in> D then restrict (D - {x}) al else restrict D al)" |
|
737 |
proof (induct al) |
|
738 |
case Nil thus ?case by simp |
|
739 |
next |
|
740 |
case (Cons a al) |
|
741 |
show ?case |
|
742 |
proof (cases "x \<in> D") |
|
743 |
case True |
|
744 |
note x_D = this |
|
745 |
with Cons have hyp: "delete x (restrict D al) = restrict (D - {x}) al" |
|
746 |
by simp |
|
747 |
show ?thesis |
|
748 |
proof (cases "fst a = x") |
|
749 |
case True |
|
750 |
from Cons.hyps |
|
751 |
show ?thesis |
|
752 |
using x_D True |
|
753 |
by simp |
|
754 |
next |
|
755 |
case False |
|
756 |
note not_fst_a_x = this |
|
757 |
show ?thesis |
|
758 |
proof (cases "fst a \<in> D") |
|
759 |
case True |
|
760 |
with not_fst_a_x |
|
761 |
have "delete x (restrict D (a#al)) = a#(delete x (restrict D al))" |
|
762 |
by (cases a) (simp add: restrict_def) |
|
763 |
also from not_fst_a_x True hyp have "\<dots> = restrict (D - {x}) (a # al)" |
|
764 |
by (cases a) (simp add: restrict_def) |
|
765 |
finally show ?thesis |
|
766 |
using x_D by simp |
|
767 |
next |
|
768 |
case False |
|
769 |
hence "delete x (restrict D (a#al)) = delete x (restrict D al)" |
|
770 |
by (cases a) (simp add: restrict_def) |
|
771 |
moreover from False not_fst_a_x |
|
772 |
have "restrict (D - {x}) (a # al) = restrict (D - {x}) al" |
|
773 |
by (cases a) (simp add: restrict_def) |
|
774 |
ultimately |
|
775 |
show ?thesis using x_D hyp by simp |
|
776 |
qed |
|
777 |
qed |
|
778 |
next |
|
779 |
case False |
|
780 |
from False Cons show ?thesis |
|
781 |
by simp |
|
782 |
qed |
|
783 |
qed |
|
784 |
||
785 |
lemma update_restr: |
|
786 |
"map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))" |
|
787 |
by (simp add: update_conv' restr_conv') (rule fun_upd_restrict) |
|
788 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20503
diff
changeset
|
789 |
lemma upate_restr_conv [simp]: |
19234 | 790 |
"x \<in> D \<Longrightarrow> |
791 |
map_of (update x y (restrict D al)) = map_of (update x y (restrict (D-{x}) al))" |
|
792 |
by (simp add: update_conv' restr_conv') |
|
793 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20503
diff
changeset
|
794 |
lemma restr_updates [simp]: " |
19234 | 795 |
\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
796 |
\<Longrightarrow> map_of (restrict D (updates xs ys al)) = |
|
797 |
map_of (updates xs ys (restrict (D - set xs) al))" |
|
798 |
by (simp add: updates_conv' restr_conv') |
|
799 |
||
800 |
lemma restr_delete_twist: "(restrict A (delete a ps)) = delete a (restrict A ps)" |
|
801 |
by (induct ps) auto |
|
802 |
||
803 |
lemma clearjunk_restrict: |
|
804 |
"clearjunk (restrict A al) = restrict A (clearjunk al)" |
|
805 |
by (induct al rule: clearjunk.induct) (auto simp add: restr_delete_twist) |
|
806 |
||
807 |
end |