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(* Author: Tobias Nipkow *)
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62496
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section "AA Tree Implementation of Maps"
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theory AA_Map
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imports
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AA_Set
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Lookup2
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begin
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fun update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) aa_tree \<Rightarrow> ('a*'b) aa_tree" where
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"update x y Leaf = Node 1 Leaf (x,y) Leaf" |
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"update x y (Node lv t1 (a,b) t2) =
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(case cmp x a of
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LT \<Rightarrow> split (skew (Node lv (update x y t1) (a,b) t2)) |
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GT \<Rightarrow> split (skew (Node lv t1 (a,b) (update x y t2))) |
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EQ \<Rightarrow> Node lv t1 (x,y) t2)"
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fun delete :: "'a::cmp \<Rightarrow> ('a*'b) aa_tree \<Rightarrow> ('a*'b) aa_tree" where
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"delete _ Leaf = Leaf" |
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"delete x (Node lv l (a,b) r) =
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(case cmp x a of
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LT \<Rightarrow> adjust (Node lv (delete x l) (a,b) r) |
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GT \<Rightarrow> adjust (Node lv l (a,b) (delete x r)) |
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EQ \<Rightarrow> (if l = Leaf then r
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else let (l',ab') = del_max l in adjust (Node lv l' ab' r)))"
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subsection "Invariance"
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subsubsection "Proofs for insert"
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lemma lvl_update_aux:
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"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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apply(induction t)
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apply (auto simp: lvl_skew)
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apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+
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done
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lemma lvl_update: obtains
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(Same) "lvl (update x y t) = lvl t" |
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(Incr) "lvl (update x y t) = lvl t + 1" "sngl (update x y t)"
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using lvl_update_aux by fastforce
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declare invar.simps(2)[simp]
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lemma lvl_update_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(update x y t) = lvl t"
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proof (induction t rule: update.induct)
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case (2 x y lv t1 a b t2)
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consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a"
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using less_linear by blast
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thus ?case proof cases
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case LT
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thus ?thesis using 2 by (auto simp add: skew_case split_case split: tree.splits)
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next
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case GT
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thus ?thesis using 2 proof (cases t1)
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case Node
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thus ?thesis using 2 GT
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apply (auto simp add: skew_case split_case split: tree.splits)
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by (metis less_not_refl2 lvl.simps(2) lvl_update_aux n_not_Suc_n sngl.simps(3))+
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qed (auto simp add: lvl_0_iff)
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qed simp
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qed simp
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lemma lvl_update_incr_iff: "(lvl(update a b t) = lvl t + 1) \<longleftrightarrow>
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(EX l x r. update a b t = Node (lvl t + 1) l x r \<and> lvl l = lvl r)"
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apply(cases t)
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apply(auto simp add: skew_case split_case split: if_splits)
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apply(auto split: tree.splits if_splits)
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done
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lemma invar_update: "invar t \<Longrightarrow> invar(update a b t)"
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proof(induction t)
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case (Node n l xy r)
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hence il: "invar l" and ir: "invar r" by auto
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obtain x y where [simp]: "xy = (x,y)" by fastforce
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note N = Node
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let ?t = "Node n l xy r"
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have "a < x \<or> a = x \<or> x < a" by auto
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moreover
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{ assume "a < x"
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note iil = Node.IH(1)[OF il]
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have ?case
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proof (cases rule: lvl_update[of a b l])
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case (Same) thus ?thesis
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using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
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by (simp add: skew_invar split_invar del: invar.simps)
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next
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case (Incr)
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then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2"
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using Node.prems by (auto simp: lvl_Suc_iff)
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have l12: "lvl t1 = lvl t2"
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by (metis Incr(1) ial lvl_update_incr_iff tree.inject)
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have "update a b ?t = split(skew(Node n (update a b l) xy r))"
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by(simp add: \<open>a<x\<close>)
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also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)"
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by(simp)
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also have "invar(split \<dots>)"
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proof (cases r)
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case Leaf
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hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
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thus ?thesis using Leaf ial by simp
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next
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case [simp]: (Node m t3 y t4)
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show ?thesis (*using N(3) iil l12 by(auto)*)
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proof cases
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assume "m = n" thus ?thesis using N(3) iil by(auto)
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next
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assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
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qed
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qed
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finally show ?thesis .
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qed
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}
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moreover
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{ assume "x < a"
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note iir = Node.IH(2)[OF ir]
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from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
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hence ?case
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proof
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assume 0: "n = lvl r"
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have "update a b ?t = split(skew(Node n l xy (update a b r)))"
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using \<open>a>x\<close> by(auto)
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also have "skew(Node n l xy (update a b r)) = Node n l xy (update a b r)"
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using Node.prems by(simp add: skew_case split: tree.split)
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also have "invar(split \<dots>)"
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proof -
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from lvl_update_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a b]
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obtain t1 p t2 where iar: "update a b r = Node n t1 p t2"
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using Node.prems 0 by (auto simp: lvl_Suc_iff)
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from Node.prems iar 0 iir
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show ?thesis by (auto simp: split_case split: tree.splits)
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qed
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finally show ?thesis .
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next
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assume 1: "n = lvl r + 1"
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hence "sngl ?t" by(cases r) auto
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show ?thesis
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proof (cases rule: lvl_update[of a b r])
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case (Same)
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show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
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by (auto simp add: skew_invar split_invar)
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next
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case (Incr)
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thus ?thesis using invar_NodeR2[OF `invar ?t` Incr(2) 1 iir] 1 \<open>x < a\<close>
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by (auto simp add: skew_invar split_invar split: if_splits)
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qed
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qed
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}
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moreover { assume "a = x" hence ?case using Node.prems by auto }
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ultimately show ?case by blast
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qed simp
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subsubsection "Proofs for delete"
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declare invar.simps(2)[simp del]
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theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
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proof (induction t)
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case (Node lv l ab r)
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obtain a b where [simp]: "ab = (a,b)" by fastforce
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let ?l' = "delete x l" and ?r' = "delete x r"
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let ?t = "Node lv l ab r" let ?t' = "delete x ?t"
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from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)
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note post_l' = Node.IH(1)[OF inv_l]
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note preL = pre_adj_if_postL[OF Node.prems post_l']
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note post_r' = Node.IH(2)[OF inv_r]
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note preR = pre_adj_if_postR[OF Node.prems post_r']
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show ?case
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proof (cases rule: linorder_cases[of x a])
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case less
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thus ?thesis using Node.prems by (simp add: post_del_adjL preL)
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next
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case greater
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thus ?thesis using Node.prems preR by (simp add: post_del_adjR post_r')
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next
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case equal
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show ?thesis
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proof cases
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assume "l = Leaf" thus ?thesis using equal Node.prems
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by(auto simp: post_del_def invar.simps(2))
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next
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assume "l \<noteq> Leaf" thus ?thesis using equal Node.prems
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by simp (metis inv_l post_del_adjL post_del_max pre_adj_if_postL)
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qed
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qed
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qed (simp add: post_del_def)
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subsection {* Functional Correctness Proofs *}
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theorem inorder_update:
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"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
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by (induct t) (auto simp: upd_list_simps inorder_split inorder_skew)
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theorem inorder_delete:
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"\<lbrakk>invar t; sorted1(inorder t)\<rbrakk> \<Longrightarrow>
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inorder (delete x t) = del_list x (inorder t)"
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by(induction t)
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(auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR
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post_del_max post_delete del_maxD split: prod.splits)
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interpretation I: Map_by_Ordered
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where empty = Leaf and lookup = lookup and update = update and delete = delete
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and inorder = inorder and inv = invar
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proof (standard, goal_cases)
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case 1 show ?case by simp
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next
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case 2 thus ?case by(simp add: lookup_map_of)
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next
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case 3 thus ?case by(simp add: inorder_update)
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next
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case 4 thus ?case by(simp add: inorder_delete)
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next
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case 5 thus ?case by(simp)
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next
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case 6 thus ?case by(simp add: invar_update)
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next
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case 7 thus ?case using post_delete by(auto simp: post_del_def)
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qed
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end
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