# HG changeset patch # User nipkow # Date 1447412810 -3600 # Node ID 5121b9a57cce70983cf931628bb66e14b902a84c # Parent 61995131cf2818484c9ec599ea8adfc682495cc0 tuned diff -r 61995131cf28 -r 5121b9a57cce src/HOL/Data_Structures/AVL_Set.thy --- a/src/HOL/Data_Structures/AVL_Set.thy Thu Nov 12 21:12:09 2015 +0100 +++ b/src/HOL/Data_Structures/AVL_Set.thy Fri Nov 13 12:06:50 2015 +0100 @@ -51,25 +51,23 @@ LT \ balL (insert x l) a r | GT \ balR l a (insert x r))" -fun delete_max :: "'a avl_tree \ 'a avl_tree * 'a" where -"delete_max (Node _ l a Leaf) = (l,a)" | -"delete_max (Node _ l a r) = - (let (r',a') = delete_max r in (balL l a r', a'))" +fun del_max :: "'a avl_tree \ 'a avl_tree * 'a" where +"del_max (Node _ l a r) = (if r = Leaf then (l,a) + else let (r',a') = del_max r in (balL l a r', a'))" -lemmas delete_max_induct = delete_max.induct[case_names Leaf Node] +lemmas del_max_induct = del_max.induct[case_names Node Leaf] -fun delete_root :: "'a avl_tree \ 'a avl_tree" where -"delete_root (Node h Leaf a r) = r" | -"delete_root (Node h l a Leaf) = l" | -"delete_root (Node h l a r) = - (let (l', a') = delete_max l in balR l' a' r)" +fun del_root :: "'a avl_tree \ 'a avl_tree" where +"del_root (Node h Leaf a r) = r" | +"del_root (Node h l a Leaf) = l" | +"del_root (Node h l a r) = (let (l', a') = del_max l in balR l' a' r)" -lemmas delete_root_cases = delete_root.cases[case_names Leaf_t Node_Leaf Node_Node] +lemmas del_root_cases = del_root.cases[case_names Leaf_t Node_Leaf Node_Node] fun delete :: "'a::cmp \ 'a avl_tree \ 'a avl_tree" where "delete _ Leaf = Leaf" | "delete x (Node h l a r) = (case cmp x a of - EQ \ delete_root (Node h l a r) | + EQ \ del_root (Node h l a r) | LT \ balR (delete x l) a r | GT \ balL l a (delete x r))" @@ -97,22 +95,22 @@ subsubsection "Proofs for delete" -lemma inorder_delete_maxD: - "\ delete_max t = (t',a); t \ Leaf \ \ +lemma inorder_del_maxD: + "\ del_max t = (t',a); t \ Leaf \ \ inorder t' @ [a] = inorder t" -by(induction t arbitrary: t' rule: delete_max.induct) - (auto simp: inorder_balL split: prod.splits tree.split) +by(induction t arbitrary: t' rule: del_max.induct) + (auto simp: inorder_balL split: if_splits prod.splits tree.split) -lemma inorder_delete_root: - "inorder (delete_root (Node h l a r)) = inorder l @ inorder r" -by(induction "Node h l a r" arbitrary: l a r h rule: delete_root.induct) - (auto simp: inorder_balR inorder_delete_maxD split: prod.splits) +lemma inorder_del_root: + "inorder (del_root (Node h l a r)) = inorder l @ inorder r" +by(induction "Node h l a r" arbitrary: l a r h rule: del_root.induct) + (auto simp: inorder_balL inorder_balR inorder_del_maxD split: if_splits prod.splits) theorem inorder_delete: "sorted(inorder t) \ inorder (delete x t) = del_list x (inorder t)" by(induction t) (auto simp: del_list_simps inorder_balL inorder_balR - inorder_delete_root inorder_delete_maxD split: prod.splits) + inorder_del_root inorder_del_maxD split: prod.splits) subsubsection "Overall functional correctness" @@ -295,65 +293,61 @@ subsubsection {* Deletion maintains AVL balance *} -lemma avl_delete_max: +lemma avl_del_max: assumes "avl x" and "x \ Leaf" - shows "avl (fst (delete_max x))" "height x = height(fst (delete_max x)) \ - height x = height(fst (delete_max x)) + 1" + shows "avl (fst (del_max x))" "height x = height(fst (del_max x)) \ + height x = height(fst (del_max x)) + 1" using assms -proof (induct x rule: delete_max_induct) - case (Node h l a rh rl b rr) +proof (induct x rule: del_max_induct) + case (Node h l a r) case 1 - with Node have "avl l" "avl (fst (delete_max (Node rh rl b rr)))" by auto - with 1 Node have "avl (balL l a (fst (delete_max (Node rh rl b rr))))" - by (intro avl_balL) fastforce+ - thus ?case - by (auto simp: height_balL height_balL2 + thus ?case using Node + by (auto simp: height_balL height_balL2 avl_balL linorder_class.max.absorb1 linorder_class.max.absorb2 split:prod.split) next - case (Node h l a rh rl b rr) + case (Node h l a r) case 2 - let ?r = "Node rh rl b rr" - let ?r' = "fst (delete_max ?r)" - from `avl x` Node 2 have "avl l" and "avl ?r" by simp_all + let ?r' = "fst (del_max r)" + from `avl x` Node 2 have "avl l" and "avl r" by simp_all thus ?case using Node 2 height_balL[of l ?r' a] height_balL2[of l ?r' a] apply (auto split:prod.splits simp del:avl.simps) by arith+ qed auto -lemma avl_delete_root: +lemma avl_del_root: assumes "avl t" and "t \ Leaf" - shows "avl(delete_root t)" + shows "avl(del_root t)" using assms -proof (cases t rule:delete_root_cases) +proof (cases t rule:del_root_cases) case (Node_Node h lh ll ln lr n rh rl rn rr) let ?l = "Node lh ll ln lr" let ?r = "Node rh rl rn rr" - let ?l' = "fst (delete_max ?l)" + let ?l' = "fst (del_max ?l)" from `avl t` and Node_Node have "avl ?r" by simp from `avl t` and Node_Node have "avl ?l" by simp hence "avl(?l')" "height ?l = height(?l') \ - height ?l = height(?l') + 1" by (rule avl_delete_max,simp)+ + height ?l = height(?l') + 1" by (rule avl_del_max,simp)+ with `avl t` Node_Node have "height ?l' = height ?r \ height ?l' = height ?r + 1 \ height ?r = height ?l' + 1 \ height ?r = height ?l' + 2" by fastforce - with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(delete_max ?l)) ?r)" + with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(del_max ?l)) ?r)" by (rule avl_balR) with Node_Node show ?thesis by (auto split:prod.splits) qed simp_all -lemma height_delete_root: +lemma height_del_root: assumes "avl t" and "t \ Leaf" - shows "height t = height(delete_root t) \ height t = height(delete_root t) + 1" + shows "height t = height(del_root t) \ height t = height(del_root t) + 1" using assms -proof (cases t rule: delete_root_cases) +proof (cases t rule: del_root_cases) case (Node_Node h lh ll ln lr n rh rl rn rr) let ?l = "Node lh ll ln lr" let ?r = "Node rh rl rn rr" - let ?l' = "fst (delete_max ?l)" - let ?t' = "balR ?l' (snd(delete_max ?l)) ?r" + let ?l' = "fst (del_max ?l)" + let ?t' = "balR ?l' (snd(del_max ?l)) ?r" from `avl t` and Node_Node have "avl ?r" by simp from `avl t` and Node_Node have "avl ?l" by simp - hence "avl(?l')" by (rule avl_delete_max,simp) - have l'_height: "height ?l = height ?l' \ height ?l = height ?l' + 1" using `avl ?l` by (intro avl_delete_max) auto + hence "avl(?l')" by (rule avl_del_max,simp) + have l'_height: "height ?l = height ?l' \ height ?l = height ?l' + 1" using `avl ?l` by (intro avl_del_max) auto have t_height: "height t = 1 + max (height ?l) (height ?r)" using `avl t` Node_Node by simp have "height t = height ?t' \ height t = height ?t' + 1" using `avl t` Node_Node proof(cases "height ?r = height ?l' + 2") @@ -362,7 +356,7 @@ next case True show ?thesis - proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]]) + proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (del_max ?l)"]]) case 1 thus ?thesis using l'_height t_height True by arith next @@ -385,7 +379,7 @@ with Node show ?case proof(cases "x = n") case True - with Node 1 show ?thesis by (auto simp:avl_delete_root) + with Node 1 show ?thesis by (auto simp:avl_del_root) next case False with Node 1 show ?thesis @@ -401,9 +395,9 @@ with Node show ?case proof(cases "x = n") case True - with 1 have "height (Node h l n r) = height(delete_root (Node h l n r)) - \ height (Node h l n r) = height(delete_root (Node h l n r)) + 1" - by (subst height_delete_root,simp_all) + with 1 have "height (Node h l n r) = height(del_root (Node h l n r)) + \ height (Node h l n r) = height(del_root (Node h l n r)) + 1" + by (subst height_del_root,simp_all) with True show ?thesis by simp next case False diff -r 61995131cf28 -r 5121b9a57cce src/HOL/Data_Structures/Tree_Map.thy --- a/src/HOL/Data_Structures/Tree_Map.thy Thu Nov 12 21:12:09 2015 +0100 +++ b/src/HOL/Data_Structures/Tree_Map.thy Fri Nov 13 12:06:50 2015 +0100 @@ -44,7 +44,7 @@ "del_min t = (x,t') \ t \ Leaf \ sorted1(inorder t) \ x # inorder t' = inorder t" by(induction t arbitrary: t' rule: del_min.induct) - (auto simp: del_list_simps split: prod.splits) + (auto simp: del_list_simps split: prod.splits if_splits) lemma inorder_delete: "sorted1(inorder t) \ inorder(delete x t) = del_list x (inorder t)" diff -r 61995131cf28 -r 5121b9a57cce src/HOL/Data_Structures/Tree_Set.thy --- a/src/HOL/Data_Structures/Tree_Set.thy Thu Nov 12 21:12:09 2015 +0100 +++ b/src/HOL/Data_Structures/Tree_Set.thy Fri Nov 13 12:06:50 2015 +0100 @@ -24,8 +24,8 @@ GT \ Node l a (insert x r))" fun del_min :: "'a tree \ 'a * 'a tree" where -"del_min (Node Leaf a r) = (a, r)" | -"del_min (Node l a r) = (let (x,l') = del_min l in (x, Node l' a r))" +"del_min (Node l a r) = (if l = Leaf then (a,r) + else let (x,l') = del_min l in (x, Node l' a r))" fun delete :: "'a::cmp \ 'a tree \ 'a tree" where "delete x Leaf = Leaf" | @@ -53,7 +53,7 @@ "del_min t = (x,t') \ t \ Leaf \ sorted(inorder t) \ x # inorder t' = inorder t" by(induction t arbitrary: t' rule: del_min.induct) - (auto simp: sorted_lems split: prod.splits) + (auto simp: sorted_lems split: prod.splits if_splits) lemma inorder_delete: "sorted(inorder t) \ inorder(delete x t) = del_list x (inorder t)"