--- a/src/HOL/Data_Structures/AA_Map.thy Tue Mar 01 22:49:33 2016 +0100
+++ b/src/HOL/Data_Structures/AA_Map.thy Wed Mar 02 10:01:31 2016 +0100
@@ -1,6 +1,6 @@
(* Author: Tobias Nipkow *)
-section "AA Implementation of Maps"
+section "AA Tree Implementation of Maps"
theory AA_Map
imports
@@ -26,21 +26,190 @@
else let (l',ab') = del_max l in adjust (Node lv l' ab' r)))"
+subsection "Invariance"
+
+subsubsection "Proofs for insert"
+
+lemma lvl_update_aux:
+ "lvl (update x y t) = lvl t \<or> lvl (update x y t) = lvl t + 1 \<and> sngl (update x y t)"
+apply(induction t)
+apply (auto simp: lvl_skew)
+apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+
+done
+
+lemma lvl_update: obtains
+ (Same) "lvl (update x y t) = lvl t" |
+ (Incr) "lvl (update x y t) = lvl t + 1" "sngl (update x y t)"
+using lvl_update_aux by fastforce
+
+declare invar.simps(2)[simp]
+
+lemma lvl_update_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(update x y t) = lvl t"
+proof (induction t rule: update.induct)
+ case (2 x y lv t1 a b t2)
+ consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a"
+ using less_linear by blast
+ thus ?case proof cases
+ case LT
+ thus ?thesis using 2 by (auto simp add: skew_case split_case split: tree.splits)
+ next
+ case GT
+ thus ?thesis using 2 proof (cases t1)
+ case Node
+ thus ?thesis using 2 GT
+ apply (auto simp add: skew_case split_case split: tree.splits)
+ by (metis less_not_refl2 lvl.simps(2) lvl_update_aux n_not_Suc_n sngl.simps(3))+
+ qed (auto simp add: lvl_0_iff)
+ qed simp
+qed simp
+
+lemma lvl_update_incr_iff: "(lvl(update a b t) = lvl t + 1) \<longleftrightarrow>
+ (EX l x r. update a b t = Node (lvl t + 1) l x r \<and> lvl l = lvl r)"
+apply(cases t)
+apply(auto simp add: skew_case split_case split: if_splits)
+apply(auto split: tree.splits if_splits)
+done
+
+lemma invar_update: "invar t \<Longrightarrow> invar(update a b t)"
+proof(induction t)
+ case (Node n l xy r)
+ hence il: "invar l" and ir: "invar r" by auto
+ obtain x y where [simp]: "xy = (x,y)" by fastforce
+ note N = Node
+ let ?t = "Node n l xy r"
+ have "a < x \<or> a = x \<or> x < a" by auto
+ moreover
+ { assume "a < x"
+ note iil = Node.IH(1)[OF il]
+ have ?case
+ proof (cases rule: lvl_update[of a b l])
+ case (Same) thus ?thesis
+ using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
+ by (simp add: skew_invar split_invar del: invar.simps)
+ next
+ case (Incr)
+ then obtain t1 w t2 where ial[simp]: "update a b l = Node n t1 w t2"
+ using Node.prems by (auto simp: lvl_Suc_iff)
+ have l12: "lvl t1 = lvl t2"
+ by (metis Incr(1) ial lvl_update_incr_iff tree.inject)
+ have "update a b ?t = split(skew(Node n (update a b l) xy r))"
+ by(simp add: \<open>a<x\<close>)
+ also have "skew(Node n (update a b l) xy r) = Node n t1 w (Node n t2 xy r)"
+ by(simp)
+ also have "invar(split \<dots>)"
+ proof (cases r)
+ case Leaf
+ hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
+ thus ?thesis using Leaf ial by simp
+ next
+ case [simp]: (Node m t3 y t4)
+ show ?thesis (*using N(3) iil l12 by(auto)*)
+ proof cases
+ assume "m = n" thus ?thesis using N(3) iil by(auto)
+ next
+ assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ }
+ moreover
+ { assume "x < a"
+ note iir = Node.IH(2)[OF ir]
+ from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
+ hence ?case
+ proof
+ assume 0: "n = lvl r"
+ have "update a b ?t = split(skew(Node n l xy (update a b r)))"
+ using \<open>a>x\<close> by(auto)
+ also have "skew(Node n l xy (update a b r)) = Node n l xy (update a b r)"
+ using Node.prems by(simp add: skew_case split: tree.split)
+ also have "invar(split \<dots>)"
+ proof -
+ from lvl_update_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a b]
+ obtain t1 p t2 where iar: "update a b r = Node n t1 p t2"
+ using Node.prems 0 by (auto simp: lvl_Suc_iff)
+ from Node.prems iar 0 iir
+ show ?thesis by (auto simp: split_case split: tree.splits)
+ qed
+ finally show ?thesis .
+ next
+ assume 1: "n = lvl r + 1"
+ hence "sngl ?t" by(cases r) auto
+ show ?thesis
+ proof (cases rule: lvl_update[of a b r])
+ case (Same)
+ show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
+ by (auto simp add: skew_invar split_invar)
+ next
+ case (Incr)
+ thus ?thesis using invar_NodeR2[OF `invar ?t` Incr(2) 1 iir] 1 \<open>x < a\<close>
+ by (auto simp add: skew_invar split_invar split: if_splits)
+ qed
+ qed
+ }
+ moreover { assume "a = x" hence ?case using Node.prems by auto }
+ ultimately show ?case by blast
+qed simp
+
+subsubsection "Proofs for delete"
+
+declare invar.simps(2)[simp del]
+
+theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
+proof (induction t)
+ case (Node lv l ab r)
+
+ obtain a b where [simp]: "ab = (a,b)" by fastforce
+
+ let ?l' = "delete x l" and ?r' = "delete x r"
+ let ?t = "Node lv l ab r" let ?t' = "delete x ?t"
+
+ from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)
+
+ note post_l' = Node.IH(1)[OF inv_l]
+ note preL = pre_adj_if_postL[OF Node.prems post_l']
+
+ note post_r' = Node.IH(2)[OF inv_r]
+ note preR = pre_adj_if_postR[OF Node.prems post_r']
+
+ show ?case
+ proof (cases rule: linorder_cases[of x a])
+ case less
+ thus ?thesis using Node.prems by (simp add: post_del_adjL preL)
+ next
+ case greater
+ thus ?thesis using Node.prems preR by (simp add: post_del_adjR post_r')
+ next
+ case equal
+ show ?thesis
+ proof cases
+ assume "l = Leaf" thus ?thesis using equal Node.prems
+ by(auto simp: post_del_def invar.simps(2))
+ next
+ assume "l \<noteq> Leaf" thus ?thesis using equal Node.prems
+ by simp (metis inv_l post_del_adjL post_del_max pre_adj_if_postL)
+ qed
+ qed
+qed (simp add: post_del_def)
+
+
subsection {* Functional Correctness Proofs *}
theorem inorder_update:
"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
by (induct t) (auto simp: upd_list_simps inorder_split inorder_skew)
-
theorem inorder_delete:
- "sorted1(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
+ "\<lbrakk>invar t; sorted1(inorder t)\<rbrakk> \<Longrightarrow>
+ inorder (delete x t) = del_list x (inorder t)"
by(induction t)
- (auto simp: del_list_simps inorder_adjust del_maxD split: prod.splits)
+ (auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR
+ post_del_max post_delete del_maxD split: prod.splits)
-interpretation Map_by_Ordered
+interpretation I: Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
-and inorder = inorder and inv = "\<lambda>_. True"
+and inorder = inorder and inv = invar
proof (standard, goal_cases)
case 1 show ?case by simp
next
@@ -49,6 +218,12 @@
case 3 thus ?case by(simp add: inorder_update)
next
case 4 thus ?case by(simp add: inorder_delete)
-qed auto
+next
+ case 5 thus ?case by(simp)
+next
+ case 6 thus ?case by(simp add: invar_update)
+next
+ case 7 thus ?case using post_delete by(auto simp: post_del_def)
+qed
end