del_max -> split_max
authornipkow
Mon Apr 23 08:09:50 2018 +0200 (13 months ago)
changeset 6802375130777ece4
parent 68022 c8a506be83bd
child 68024 b5e29bf0aeab
del_max -> split_max
src/HOL/Data_Structures/AA_Map.thy
src/HOL/Data_Structures/AA_Set.thy
src/HOL/Data_Structures/AVL_Map.thy
src/HOL/Data_Structures/AVL_Set.thy
     1.1 --- a/src/HOL/Data_Structures/AA_Map.thy	Sun Apr 22 21:05:14 2018 +0100
     1.2 +++ b/src/HOL/Data_Structures/AA_Map.thy	Mon Apr 23 08:09:50 2018 +0200
     1.3 @@ -23,7 +23,7 @@
     1.4       LT \<Rightarrow> adjust (Node lv (delete x l) (a,b) r) |
     1.5       GT \<Rightarrow> adjust (Node lv l (a,b) (delete x r)) |
     1.6       EQ \<Rightarrow> (if l = Leaf then r
     1.7 -            else let (l',ab') = del_max l in adjust (Node lv l' ab' r)))"
     1.8 +            else let (l',ab') = split_max l in adjust (Node lv l' ab' r)))"
     1.9  
    1.10  
    1.11  subsection "Invariance"
    1.12 @@ -187,7 +187,7 @@
    1.13          by(auto simp: post_del_def invar.simps(2))
    1.14      next
    1.15        assume "l \<noteq> Leaf" thus ?thesis using equal Node.prems
    1.16 -        by simp (metis inv_l post_del_adjL post_del_max pre_adj_if_postL)
    1.17 +        by simp (metis inv_l post_del_adjL post_split_max pre_adj_if_postL)
    1.18      qed
    1.19    qed
    1.20  qed (simp add: post_del_def)
    1.21 @@ -204,7 +204,7 @@
    1.22    inorder (delete x t) = del_list x (inorder t)"
    1.23  by(induction t)
    1.24    (auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR 
    1.25 -              post_del_max post_delete del_maxD split: prod.splits)
    1.26 +              post_split_max post_delete split_maxD split: prod.splits)
    1.27  
    1.28  interpretation I: Map_by_Ordered
    1.29  where empty = Leaf and lookup = lookup and update = update and delete = delete
     2.1 --- a/src/HOL/Data_Structures/AA_Set.thy	Sun Apr 22 21:05:14 2018 +0100
     2.2 +++ b/src/HOL/Data_Structures/AA_Set.thy	Mon Apr 23 08:09:50 2018 +0200
     2.3 @@ -72,14 +72,14 @@
     2.4  text\<open>In the paper, the last case of @{const adjust} is expressed with the help of an
     2.5  incorrect auxiliary function \texttt{nlvl}.
     2.6  
     2.7 -Function @{text del_max} below is called \texttt{dellrg} in the paper.
     2.8 +Function @{text split_max} below is called \texttt{dellrg} in the paper.
     2.9  The latter is incorrect for two reasons: \texttt{dellrg} is meant to delete the largest
    2.10  element but recurses on the left instead of the right subtree; the invariant
    2.11  is not restored.\<close>
    2.12  
    2.13 -fun del_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
    2.14 -"del_max (Node lv l a Leaf) = (l,a)" |
    2.15 -"del_max (Node lv l a r) = (let (r',b) = del_max r in (adjust(Node lv l a r'), b))"
    2.16 +fun split_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
    2.17 +"split_max (Node lv l a Leaf) = (l,a)" |
    2.18 +"split_max (Node lv l a r) = (let (r',b) = split_max r in (adjust(Node lv l a r'), b))"
    2.19  
    2.20  fun delete :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
    2.21  "delete _ Leaf = Leaf" |
    2.22 @@ -88,7 +88,7 @@
    2.23       LT \<Rightarrow> adjust (Node lv (delete x l) a r) |
    2.24       GT \<Rightarrow> adjust (Node lv l a (delete x r)) |
    2.25       EQ \<Rightarrow> (if l = Leaf then r
    2.26 -            else let (l',b) = del_max l in adjust (Node lv l' b r)))"
    2.27 +            else let (l',b) = split_max l in adjust (Node lv l' b r)))"
    2.28  
    2.29  fun pre_adjust where
    2.30  "pre_adjust (Node lv l a r) = (invar l \<and> invar r \<and>
    2.31 @@ -397,13 +397,13 @@
    2.32  
    2.33  declare prod.splits[split]
    2.34  
    2.35 -theorem post_del_max:
    2.36 - "\<lbrakk> invar t; (t', x) = del_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
    2.37 -proof (induction t arbitrary: t' rule: del_max.induct)
    2.38 +theorem post_split_max:
    2.39 + "\<lbrakk> invar t; (t', x) = split_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
    2.40 +proof (induction t arbitrary: t' rule: split_max.induct)
    2.41    case (2 lv l a lvr rl ra rr)
    2.42    let ?r =  "\<langle>lvr, rl, ra, rr\<rangle>"
    2.43    let ?t = "\<langle>lv, l, a, ?r\<rangle>"
    2.44 -  from "2.prems"(2) obtain r' where r': "(r', x) = del_max ?r"
    2.45 +  from "2.prems"(2) obtain r' where r': "(r', x) = split_max ?r"
    2.46      and [simp]: "t' = adjust \<langle>lv, l, a, r'\<rangle>" by auto
    2.47    from  "2.IH"[OF _ r'] \<open>invar ?t\<close> have post: "post_del ?r r'" by simp
    2.48    note preR = pre_adj_if_postR[OF \<open>invar ?t\<close> post]
    2.49 @@ -440,7 +440,7 @@
    2.50          by(auto simp: post_del_def invar.simps(2))
    2.51      next
    2.52        assume "l \<noteq> Leaf" thus ?thesis using equal
    2.53 -        by simp (metis Node.prems inv_l post_del_adjL post_del_max pre_adj_if_postL)
    2.54 +        by simp (metis Node.prems inv_l post_del_adjL post_split_max pre_adj_if_postL)
    2.55      qed
    2.56    qed
    2.57  qed (simp add: post_del_def)
    2.58 @@ -471,16 +471,16 @@
    2.59    (auto simp: adjust_def inorder_skew inorder_split invar.simps(2) pre_adjust.simps
    2.60       split: tree.splits)
    2.61  
    2.62 -lemma del_maxD:
    2.63 -  "\<lbrakk> del_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
    2.64 -by(induction t arbitrary: t' rule: del_max.induct)
    2.65 -  (auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_del_max split: prod.splits)
    2.66 +lemma split_maxD:
    2.67 +  "\<lbrakk> split_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
    2.68 +by(induction t arbitrary: t' rule: split_max.induct)
    2.69 +  (auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_split_max split: prod.splits)
    2.70  
    2.71  lemma inorder_delete:
    2.72    "invar t \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
    2.73  by(induction t)
    2.74    (auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR 
    2.75 -              post_del_max post_delete del_maxD split: prod.splits)
    2.76 +              post_split_max post_delete split_maxD split: prod.splits)
    2.77  
    2.78  interpretation I: Set_by_Ordered
    2.79  where empty = Leaf and isin = isin and insert = insert and delete = delete
     3.1 --- a/src/HOL/Data_Structures/AVL_Map.thy	Sun Apr 22 21:05:14 2018 +0100
     3.2 +++ b/src/HOL/Data_Structures/AVL_Map.thy	Mon Apr 23 08:09:50 2018 +0200
     3.3 @@ -34,7 +34,7 @@
     3.4    "sorted1(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
     3.5  by(induction t)
     3.6    (auto simp: del_list_simps inorder_balL inorder_balR
     3.7 -     inorder_del_root inorder_del_maxD split: prod.splits)
     3.8 +     inorder_del_root inorder_split_maxD split: prod.splits)
     3.9  
    3.10  interpretation Map_by_Ordered
    3.11  where empty = Leaf and lookup = lookup and update = update and delete = delete
     4.1 --- a/src/HOL/Data_Structures/AVL_Set.thy	Sun Apr 22 21:05:14 2018 +0100
     4.2 +++ b/src/HOL/Data_Structures/AVL_Set.thy	Mon Apr 23 08:09:50 2018 +0200
     4.3 @@ -58,16 +58,16 @@
     4.4     LT \<Rightarrow> balL (insert x l) a r |
     4.5     GT \<Rightarrow> balR l a (insert x r))"
     4.6  
     4.7 -fun del_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
     4.8 -"del_max (Node _ l a r) =
     4.9 -  (if r = Leaf then (l,a) else let (r',a') = del_max r in (balL l a r', a'))"
    4.10 +fun split_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
    4.11 +"split_max (Node _ l a r) =
    4.12 +  (if r = Leaf then (l,a) else let (r',a') = split_max r in (balL l a r', a'))"
    4.13  
    4.14 -lemmas del_max_induct = del_max.induct[case_names Node Leaf]
    4.15 +lemmas split_max_induct = split_max.induct[case_names Node Leaf]
    4.16  
    4.17  fun del_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
    4.18  "del_root (Node h Leaf a r) = r" |
    4.19  "del_root (Node h l a Leaf) = l" |
    4.20 -"del_root (Node h l a r) = (let (l', a') = del_max l in balR l' a' r)"
    4.21 +"del_root (Node h l a r) = (let (l', a') = split_max l in balR l' a' r)"
    4.22  
    4.23  lemmas del_root_cases = del_root.cases[case_names Leaf_t Node_Leaf Node_Node]
    4.24  
    4.25 @@ -103,22 +103,22 @@
    4.26  
    4.27  subsubsection "Proofs for delete"
    4.28  
    4.29 -lemma inorder_del_maxD:
    4.30 -  "\<lbrakk> del_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
    4.31 +lemma inorder_split_maxD:
    4.32 +  "\<lbrakk> split_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
    4.33     inorder t' @ [a] = inorder t"
    4.34 -by(induction t arbitrary: t' rule: del_max.induct)
    4.35 +by(induction t arbitrary: t' rule: split_max.induct)
    4.36    (auto simp: inorder_balL split: if_splits prod.splits tree.split)
    4.37  
    4.38  lemma inorder_del_root:
    4.39    "inorder (del_root (Node h l a r)) = inorder l @ inorder r"
    4.40  by(cases "Node h l a r" rule: del_root.cases)
    4.41 -  (auto simp: inorder_balL inorder_balR inorder_del_maxD split: if_splits prod.splits)
    4.42 +  (auto simp: inorder_balL inorder_balR inorder_split_maxD split: if_splits prod.splits)
    4.43  
    4.44  theorem inorder_delete:
    4.45    "sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
    4.46  by(induction t)
    4.47    (auto simp: del_list_simps inorder_balL inorder_balR
    4.48 -    inorder_del_root inorder_del_maxD split: prod.splits)
    4.49 +    inorder_del_root inorder_split_maxD split: prod.splits)
    4.50  
    4.51  
    4.52  subsubsection "Overall functional correctness"
    4.53 @@ -301,12 +301,12 @@
    4.54  
    4.55  subsubsection \<open>Deletion maintains AVL balance\<close>
    4.56  
    4.57 -lemma avl_del_max:
    4.58 +lemma avl_split_max:
    4.59    assumes "avl x" and "x \<noteq> Leaf"
    4.60 -  shows "avl (fst (del_max x))" "height x = height(fst (del_max x)) \<or>
    4.61 -         height x = height(fst (del_max x)) + 1"
    4.62 +  shows "avl (fst (split_max x))" "height x = height(fst (split_max x)) \<or>
    4.63 +         height x = height(fst (split_max x)) + 1"
    4.64  using assms
    4.65 -proof (induct x rule: del_max_induct)
    4.66 +proof (induct x rule: split_max_induct)
    4.67    case (Node h l a r)
    4.68    case 1
    4.69    thus ?case using Node
    4.70 @@ -316,7 +316,7 @@
    4.71  next
    4.72    case (Node h l a r)
    4.73    case 2
    4.74 -  let ?r' = "fst (del_max r)"
    4.75 +  let ?r' = "fst (split_max r)"
    4.76    from \<open>avl x\<close> Node 2 have "avl l" and "avl r" by simp_all
    4.77    thus ?case using Node 2 height_balL[of l ?r' a] height_balL2[of l ?r' a]
    4.78      apply (auto split:prod.splits simp del:avl.simps) by arith+
    4.79 @@ -330,14 +330,14 @@
    4.80    case (Node_Node h lh ll ln lr n rh rl rn rr)
    4.81    let ?l = "Node lh ll ln lr"
    4.82    let ?r = "Node rh rl rn rr"
    4.83 -  let ?l' = "fst (del_max ?l)"
    4.84 +  let ?l' = "fst (split_max ?l)"
    4.85    from \<open>avl t\<close> and Node_Node have "avl ?r" by simp
    4.86    from \<open>avl t\<close> and Node_Node have "avl ?l" by simp
    4.87    hence "avl(?l')" "height ?l = height(?l') \<or>
    4.88 -         height ?l = height(?l') + 1" by (rule avl_del_max,simp)+
    4.89 +         height ?l = height(?l') + 1" by (rule avl_split_max,simp)+
    4.90    with \<open>avl t\<close> Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
    4.91              \<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
    4.92 -  with \<open>avl ?l'\<close> \<open>avl ?r\<close> have "avl(balR ?l' (snd(del_max ?l)) ?r)"
    4.93 +  with \<open>avl ?l'\<close> \<open>avl ?r\<close> have "avl(balR ?l' (snd(split_max ?l)) ?r)"
    4.94      by (rule avl_balR)
    4.95    with Node_Node show ?thesis by (auto split:prod.splits)
    4.96  qed simp_all
    4.97 @@ -350,12 +350,12 @@
    4.98    case (Node_Node h lh ll ln lr n rh rl rn rr)
    4.99    let ?l = "Node lh ll ln lr"
   4.100    let ?r = "Node rh rl rn rr"
   4.101 -  let ?l' = "fst (del_max ?l)"
   4.102 -  let ?t' = "balR ?l' (snd(del_max ?l)) ?r"
   4.103 +  let ?l' = "fst (split_max ?l)"
   4.104 +  let ?t' = "balR ?l' (snd(split_max ?l)) ?r"
   4.105    from \<open>avl t\<close> and Node_Node have "avl ?r" by simp
   4.106    from \<open>avl t\<close> and Node_Node have "avl ?l" by simp
   4.107 -  hence "avl(?l')"  by (rule avl_del_max,simp)
   4.108 -  have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using \<open>avl ?l\<close> by (intro avl_del_max) auto
   4.109 +  hence "avl(?l')"  by (rule avl_split_max,simp)
   4.110 +  have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using \<open>avl ?l\<close> by (intro avl_split_max) auto
   4.111    have t_height: "height t = 1 + max (height ?l) (height ?r)" using \<open>avl t\<close> Node_Node by simp
   4.112    have "height t = height ?t' \<or> height t = height ?t' + 1" using  \<open>avl t\<close> Node_Node
   4.113    proof(cases "height ?r = height ?l' + 2")
   4.114 @@ -364,7 +364,7 @@
   4.115    next
   4.116      case True
   4.117      show ?thesis
   4.118 -    proof(cases rule: disjE[OF height_balR[OF True \<open>avl ?l'\<close> \<open>avl ?r\<close>, of "snd (del_max ?l)"]])
   4.119 +    proof(cases rule: disjE[OF height_balR[OF True \<open>avl ?l'\<close> \<open>avl ?r\<close>, of "snd (split_max ?l)"]])
   4.120        case 1
   4.121        thus ?thesis using l'_height t_height True by arith
   4.122      next