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(* Author: Tobias Nipkow *)
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section {* Implementing Ordered Maps *}
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theory Map_by_Ordered
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imports AList_Upd_Del
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begin
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locale Map =
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fixes empty :: "'m"
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fixes update :: "'a \<Rightarrow> 'b \<Rightarrow> 'm \<Rightarrow> 'm"
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fixes delete :: "'a \<Rightarrow> 'm \<Rightarrow> 'm"
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fixes map_of :: "'m \<Rightarrow> 'a \<Rightarrow> 'b option"
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fixes invar :: "'m \<Rightarrow> bool"
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assumes "map_of empty = (\<lambda>_. None)"
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assumes "invar m \<Longrightarrow> map_of(update a b m) = (map_of m)(a := Some b)"
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assumes "invar m \<Longrightarrow> map_of(delete a m) = (map_of m)(a := None)"
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assumes "invar empty"
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assumes "invar m \<Longrightarrow> invar(update a b m)"
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assumes "invar m \<Longrightarrow> invar(delete a m)"
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locale Map_by_Ordered =
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fixes empty :: "'t"
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fixes update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> 't \<Rightarrow> 't"
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fixes delete :: "'a \<Rightarrow> 't \<Rightarrow> 't"
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fixes lookup :: "'t \<Rightarrow> 'a \<Rightarrow> 'b option"
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fixes inorder :: "'t \<Rightarrow> ('a * 'b) list"
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fixes wf :: "'t \<Rightarrow> bool"
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assumes empty: "inorder empty = []"
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assumes lookup: "wf t \<and> sorted1 (inorder t) \<Longrightarrow>
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lookup t a = map_of (inorder t) a"
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assumes update: "wf t \<and> sorted1 (inorder t) \<Longrightarrow>
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inorder(update a b t) = upd_list a b (inorder t)"
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assumes delete: "wf t \<and> sorted1 (inorder t) \<Longrightarrow>
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inorder(delete a t) = del_list a (inorder t)"
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assumes wf_empty: "wf empty"
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assumes wf_insert: "wf t \<and> sorted1 (inorder t) \<Longrightarrow> wf(update a b t)"
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assumes wf_delete: "wf t \<and> sorted1 (inorder t) \<Longrightarrow> wf(delete a t)"
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begin
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sublocale Map
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empty update delete "map_of o inorder" "\<lambda>t. wf t \<and> sorted1 (inorder t)"
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proof(standard, goal_cases)
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case 1 show ?case by (auto simp: empty)
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next
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case 2 thus ?case by(simp add: update map_of_ins_list)
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next
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case 3 thus ?case by(simp add: delete map_of_del_list)
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next
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case 4 thus ?case by(simp add: empty wf_empty)
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next
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case 5 thus ?case by(simp add: update wf_insert sorted_upd_list)
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next
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case 6 thus ?case by (auto simp: delete wf_delete sorted_del_list)
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qed
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end
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end
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