--- a/src/HOL/Data_Structures/AA_Map.thy Sun Apr 22 21:05:14 2018 +0100
+++ b/src/HOL/Data_Structures/AA_Map.thy Mon Apr 23 08:09:50 2018 +0200
@@ -23,7 +23,7 @@
LT \<Rightarrow> adjust (Node lv (delete x l) (a,b) r) |
GT \<Rightarrow> adjust (Node lv l (a,b) (delete x r)) |
EQ \<Rightarrow> (if l = Leaf then r
- else let (l',ab') = del_max l in adjust (Node lv l' ab' r)))"
+ else let (l',ab') = split_max l in adjust (Node lv l' ab' r)))"
subsection "Invariance"
@@ -187,7 +187,7 @@
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)
+ by simp (metis inv_l post_del_adjL post_split_max pre_adj_if_postL)
qed
qed
qed (simp add: post_del_def)
@@ -204,7 +204,7 @@
inorder (delete x t) = del_list x (inorder t)"
by(induction t)
(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)
+ post_split_max post_delete split_maxD split: prod.splits)
interpretation I: Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
--- a/src/HOL/Data_Structures/AA_Set.thy Sun Apr 22 21:05:14 2018 +0100
+++ b/src/HOL/Data_Structures/AA_Set.thy Mon Apr 23 08:09:50 2018 +0200
@@ -72,14 +72,14 @@
text\<open>In the paper, the last case of @{const adjust} is expressed with the help of an
incorrect auxiliary function \texttt{nlvl}.
-Function @{text del_max} below is called \texttt{dellrg} in the paper.
+Function @{text split_max} below is called \texttt{dellrg} in the paper.
The latter is incorrect for two reasons: \texttt{dellrg} is meant to delete the largest
element but recurses on the left instead of the right subtree; the invariant
is not restored.\<close>
-fun del_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
-"del_max (Node lv l a Leaf) = (l,a)" |
-"del_max (Node lv l a r) = (let (r',b) = del_max r in (adjust(Node lv l a r'), b))"
+fun split_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
+"split_max (Node lv l a Leaf) = (l,a)" |
+"split_max (Node lv l a r) = (let (r',b) = split_max r in (adjust(Node lv l a r'), b))"
fun delete :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
"delete _ Leaf = Leaf" |
@@ -88,7 +88,7 @@
LT \<Rightarrow> adjust (Node lv (delete x l) a r) |
GT \<Rightarrow> adjust (Node lv l a (delete x r)) |
EQ \<Rightarrow> (if l = Leaf then r
- else let (l',b) = del_max l in adjust (Node lv l' b r)))"
+ else let (l',b) = split_max l in adjust (Node lv l' b r)))"
fun pre_adjust where
"pre_adjust (Node lv l a r) = (invar l \<and> invar r \<and>
@@ -397,13 +397,13 @@
declare prod.splits[split]
-theorem post_del_max:
- "\<lbrakk> invar t; (t', x) = del_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
-proof (induction t arbitrary: t' rule: del_max.induct)
+theorem post_split_max:
+ "\<lbrakk> invar t; (t', x) = split_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
+proof (induction t arbitrary: t' rule: split_max.induct)
case (2 lv l a lvr rl ra rr)
let ?r = "\<langle>lvr, rl, ra, rr\<rangle>"
let ?t = "\<langle>lv, l, a, ?r\<rangle>"
- from "2.prems"(2) obtain r' where r': "(r', x) = del_max ?r"
+ from "2.prems"(2) obtain r' where r': "(r', x) = split_max ?r"
and [simp]: "t' = adjust \<langle>lv, l, a, r'\<rangle>" by auto
from "2.IH"[OF _ r'] \<open>invar ?t\<close> have post: "post_del ?r r'" by simp
note preR = pre_adj_if_postR[OF \<open>invar ?t\<close> post]
@@ -440,7 +440,7 @@
by(auto simp: post_del_def invar.simps(2))
next
assume "l \<noteq> Leaf" thus ?thesis using equal
- by simp (metis Node.prems inv_l post_del_adjL post_del_max pre_adj_if_postL)
+ by simp (metis Node.prems inv_l post_del_adjL post_split_max pre_adj_if_postL)
qed
qed
qed (simp add: post_del_def)
@@ -471,16 +471,16 @@
(auto simp: adjust_def inorder_skew inorder_split invar.simps(2) pre_adjust.simps
split: tree.splits)
-lemma del_maxD:
- "\<lbrakk> del_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
-by(induction t arbitrary: t' rule: del_max.induct)
- (auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_del_max split: prod.splits)
+lemma split_maxD:
+ "\<lbrakk> split_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
+by(induction t arbitrary: t' rule: split_max.induct)
+ (auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_split_max split: prod.splits)
lemma inorder_delete:
"invar t \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
by(induction t)
(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)
+ post_split_max post_delete split_maxD split: prod.splits)
interpretation I: Set_by_Ordered
where empty = Leaf and isin = isin and insert = insert and delete = delete
--- a/src/HOL/Data_Structures/AVL_Map.thy Sun Apr 22 21:05:14 2018 +0100
+++ b/src/HOL/Data_Structures/AVL_Map.thy Mon Apr 23 08:09:50 2018 +0200
@@ -34,7 +34,7 @@
"sorted1(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
by(induction t)
(auto simp: del_list_simps inorder_balL inorder_balR
- inorder_del_root inorder_del_maxD split: prod.splits)
+ inorder_del_root inorder_split_maxD split: prod.splits)
interpretation Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
--- a/src/HOL/Data_Structures/AVL_Set.thy Sun Apr 22 21:05:14 2018 +0100
+++ b/src/HOL/Data_Structures/AVL_Set.thy Mon Apr 23 08:09:50 2018 +0200
@@ -58,16 +58,16 @@
LT \<Rightarrow> balL (insert x l) a r |
GT \<Rightarrow> balR l a (insert x r))"
-fun del_max :: "'a avl_tree \<Rightarrow> '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'))"
+fun split_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
+"split_max (Node _ l a r) =
+ (if r = Leaf then (l,a) else let (r',a') = split_max r in (balL l a r', a'))"
-lemmas del_max_induct = del_max.induct[case_names Node Leaf]
+lemmas split_max_induct = split_max.induct[case_names Node Leaf]
fun del_root :: "'a avl_tree \<Rightarrow> '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)"
+"del_root (Node h l a r) = (let (l', a') = split_max l in balR l' a' r)"
lemmas del_root_cases = del_root.cases[case_names Leaf_t Node_Leaf Node_Node]
@@ -103,22 +103,22 @@
subsubsection "Proofs for delete"
-lemma inorder_del_maxD:
- "\<lbrakk> del_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
+lemma inorder_split_maxD:
+ "\<lbrakk> split_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
inorder t' @ [a] = inorder t"
-by(induction t arbitrary: t' rule: del_max.induct)
+by(induction t arbitrary: t' rule: split_max.induct)
(auto simp: inorder_balL split: if_splits prod.splits tree.split)
lemma inorder_del_root:
"inorder (del_root (Node h l a r)) = inorder l @ inorder r"
by(cases "Node h l a r" rule: del_root.cases)
- (auto simp: inorder_balL inorder_balR inorder_del_maxD split: if_splits prod.splits)
+ (auto simp: inorder_balL inorder_balR inorder_split_maxD split: if_splits prod.splits)
theorem inorder_delete:
"sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
by(induction t)
(auto simp: del_list_simps inorder_balL inorder_balR
- inorder_del_root inorder_del_maxD split: prod.splits)
+ inorder_del_root inorder_split_maxD split: prod.splits)
subsubsection "Overall functional correctness"
@@ -301,12 +301,12 @@
subsubsection \<open>Deletion maintains AVL balance\<close>
-lemma avl_del_max:
+lemma avl_split_max:
assumes "avl x" and "x \<noteq> Leaf"
- shows "avl (fst (del_max x))" "height x = height(fst (del_max x)) \<or>
- height x = height(fst (del_max x)) + 1"
+ shows "avl (fst (split_max x))" "height x = height(fst (split_max x)) \<or>
+ height x = height(fst (split_max x)) + 1"
using assms
-proof (induct x rule: del_max_induct)
+proof (induct x rule: split_max_induct)
case (Node h l a r)
case 1
thus ?case using Node
@@ -316,7 +316,7 @@
next
case (Node h l a r)
case 2
- let ?r' = "fst (del_max r)"
+ let ?r' = "fst (split_max r)"
from \<open>avl x\<close> 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+
@@ -330,14 +330,14 @@
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 (del_max ?l)"
+ let ?l' = "fst (split_max ?l)"
from \<open>avl t\<close> and Node_Node have "avl ?r" by simp
from \<open>avl t\<close> and Node_Node have "avl ?l" by simp
hence "avl(?l')" "height ?l = height(?l') \<or>
- height ?l = height(?l') + 1" by (rule avl_del_max,simp)+
+ height ?l = height(?l') + 1" by (rule avl_split_max,simp)+
with \<open>avl t\<close> Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
\<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
- with \<open>avl ?l'\<close> \<open>avl ?r\<close> have "avl(balR ?l' (snd(del_max ?l)) ?r)"
+ with \<open>avl ?l'\<close> \<open>avl ?r\<close> have "avl(balR ?l' (snd(split_max ?l)) ?r)"
by (rule avl_balR)
with Node_Node show ?thesis by (auto split:prod.splits)
qed simp_all
@@ -350,12 +350,12 @@
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 (del_max ?l)"
- let ?t' = "balR ?l' (snd(del_max ?l)) ?r"
+ let ?l' = "fst (split_max ?l)"
+ let ?t' = "balR ?l' (snd(split_max ?l)) ?r"
from \<open>avl t\<close> and Node_Node have "avl ?r" by simp
from \<open>avl t\<close> and Node_Node have "avl ?l" by simp
- hence "avl(?l')" by (rule avl_del_max,simp)
- have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using \<open>avl ?l\<close> by (intro avl_del_max) auto
+ hence "avl(?l')" by (rule avl_split_max,simp)
+ have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using \<open>avl ?l\<close> by (intro avl_split_max) auto
have t_height: "height t = 1 + max (height ?l) (height ?r)" using \<open>avl t\<close> Node_Node by simp
have "height t = height ?t' \<or> height t = height ?t' + 1" using \<open>avl t\<close> Node_Node
proof(cases "height ?r = height ?l' + 2")
@@ -364,7 +364,7 @@
next
case True
show ?thesis
- proof(cases rule: disjE[OF height_balR[OF True \<open>avl ?l'\<close> \<open>avl ?r\<close>, of "snd (del_max ?l)"]])
+ proof(cases rule: disjE[OF height_balR[OF True \<open>avl ?l'\<close> \<open>avl ?r\<close>, of "snd (split_max ?l)"]])
case 1
thus ?thesis using l'_height t_height True by arith
next