author | wenzelm |
Sat, 28 Nov 2020 15:15:53 +0100 | |
changeset 72755 | 8dffbe01a3e1 |
parent 70755 | 3fb16bed5d6c |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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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::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) aa_tree \<Rightarrow> ('a*'b) aa_tree" where |
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"update x y Leaf = Node Leaf ((x,y), 1) Leaf" | |
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"update x y (Node t1 ((a,b), lv) t2) = |
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(case cmp x a of |
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LT \<Rightarrow> split (skew (Node (update x y t1) ((a,b), lv) t2)) | |
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GT \<Rightarrow> split (skew (Node t1 ((a,b), lv) (update x y t2))) | |
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EQ \<Rightarrow> Node t1 ((x,y), lv) t2)" |
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fun delete :: "'a::linorder \<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 l ((a,b), lv) r) = |
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(case cmp x a of |
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LT \<Rightarrow> adjust (Node (delete x l) ((a,b), lv) r) | |
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GT \<Rightarrow> adjust (Node l ((a,b), lv) (delete x r)) | |
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EQ \<Rightarrow> (if l = Leaf then r |
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else let (l',ab') = split_max l in adjust (Node l' (ab', lv) 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 t1 a b lv 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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(\<exists>l x r. update a b t = Node l (x,lvl t + 1) 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 rule: tree2_induct) |
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case N: (Node l xy n r) |
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hence il: "invar l" and ir: "invar r" by auto |
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note iil = N.IH(1)[OF il] |
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note iir = N.IH(2)[OF ir] |
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obtain x y where [simp]: "xy = (x,y)" by fastforce |
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let ?t = "Node l (xy, n) 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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have ?case if "a < x" |
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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 N.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 t1 (w, n) t2" |
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using N.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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replaced new type ('a,'b) tree by old type ('a*'b) tree.
nipkow
parents:
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changeset
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have "update a b ?t = split(skew(Node (update a b l) (xy, n) r))" |
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by(simp add: \<open>a<x\<close>) |
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replaced new type ('a,'b) tree by old type ('a*'b) tree.
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parents:
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also have "skew(Node (update a b l) (xy, n) r) = Node t1 (w, n) (Node t2 (xy, n) r)" |
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by(simp) |
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also have "invar(split \<dots>)" |
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replaced new type ('a,'b) tree by old type ('a*'b) tree.
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parents:
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proof (cases r rule: tree2_cases) |
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case Leaf |
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hence "l = Leaf" using N.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 t3 y m 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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moreover |
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have ?case if "x < a" |
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proof - |
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from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto |
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thus ?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 l (xy, n) (update a b r)))" |
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using \<open>a>x\<close> by(auto) |
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also have "skew(Node l (xy, n) (update a b r)) = Node l (xy, n) (update a b r)" |
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using N.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 t1 (p, n) t2" |
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using N.prems 0 by (auto simp: lvl_Suc_iff) |
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from N.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 N.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 \<open>invar ?t\<close> 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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qed |
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moreover |
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have "a = x \<Longrightarrow> ?case" using N.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 rule: tree2_induct) |
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case (Node l ab lv 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 l (ab, lv) 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_split_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 \<open>Functional Correctness Proofs\<close> |
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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_split_max post_delete split_maxD split: prod.splits) |
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interpretation I: Map_by_Ordered |
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where empty = empty 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 add: empty_def) |
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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 add: empty_def) |
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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 |