--- a/src/HOL/Data_Structures/AA_Set.thy Tue Sep 24 17:36:14 2019 +0200
+++ b/src/HOL/Data_Structures/AA_Set.thy Wed Sep 25 17:22:57 2019 +0200
@@ -10,67 +10,67 @@
Cmp
begin
-type_synonym 'a aa_tree = "('a,nat) tree"
+type_synonym 'a aa_tree = "('a*nat) tree"
definition empty :: "'a aa_tree" where
"empty = Leaf"
fun lvl :: "'a aa_tree \<Rightarrow> nat" where
"lvl Leaf = 0" |
-"lvl (Node _ _ lv _) = lv"
+"lvl (Node _ (_, lv) _) = lv"
fun invar :: "'a aa_tree \<Rightarrow> bool" where
"invar Leaf = True" |
-"invar (Node l a h r) =
+"invar (Node l (a, h) r) =
(invar l \<and> invar r \<and>
- h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = Node lr b h rr \<and> h = lvl rr + 1)))"
+ h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = Node lr (b,h) rr \<and> h = lvl rr + 1)))"
fun skew :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
-"skew (Node (Node t1 b lvb t2) a lva t3) =
- (if lva = lvb then Node t1 b lvb (Node t2 a lva t3) else Node (Node t1 b lvb t2) a lva t3)" |
+"skew (Node (Node t1 (b, lvb) t2) (a, lva) t3) =
+ (if lva = lvb then Node t1 (b, lvb) (Node t2 (a, lva) t3) else Node (Node t1 (b, lvb) t2) (a, lva) t3)" |
"skew t = t"
fun split :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
-"split (Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4))) =
+"split (Node t1 (a, lva) (Node t2 (b, lvb) (Node t3 (c, lvc) t4))) =
(if lva = lvb \<and> lvb = lvc \<comment> \<open>\<open>lva = lvc\<close> suffices\<close>
- then Node (Node t1 a lva t2) b (lva+1) (Node t3 c lva t4)
- else Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4)))" |
+ then Node (Node t1 (a,lva) t2) (b,lva+1) (Node t3 (c, lva) t4)
+ else Node t1 (a,lva) (Node t2 (b,lvb) (Node t3 (c,lvc) t4)))" |
"split t = t"
hide_const (open) insert
fun insert :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
-"insert x Leaf = Node Leaf x 1 Leaf" |
-"insert x (Node t1 a lv t2) =
+"insert x Leaf = Node Leaf (x, 1) Leaf" |
+"insert x (Node t1 (a,lv) t2) =
(case cmp x a of
- LT \<Rightarrow> split (skew (Node (insert x t1) a lv t2)) |
- GT \<Rightarrow> split (skew (Node t1 a lv (insert x t2))) |
- EQ \<Rightarrow> Node t1 x lv t2)"
+ LT \<Rightarrow> split (skew (Node (insert x t1) (a,lv) t2)) |
+ GT \<Rightarrow> split (skew (Node t1 (a,lv) (insert x t2))) |
+ EQ \<Rightarrow> Node t1 (x, lv) t2)"
fun sngl :: "'a aa_tree \<Rightarrow> bool" where
"sngl Leaf = False" |
-"sngl (Node _ _ _ Leaf) = True" |
-"sngl (Node _ _ lva (Node _ _ lvb _)) = (lva > lvb)"
+"sngl (Node _ _ Leaf) = True" |
+"sngl (Node _ (_, lva) (Node _ (_, lvb) _)) = (lva > lvb)"
definition adjust :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
"adjust t =
(case t of
- Node l x lv r \<Rightarrow>
+ Node l (x,lv) r \<Rightarrow>
(if lvl l >= lv-1 \<and> lvl r >= lv-1 then t else
- if lvl r < lv-1 \<and> sngl l then skew (Node l x (lv-1) r) else
+ if lvl r < lv-1 \<and> sngl l then skew (Node l (x,lv-1) r) else
if lvl r < lv-1
then case l of
- Node t1 a lva (Node t2 b lvb t3)
- \<Rightarrow> Node (Node t1 a lva t2) b (lvb+1) (Node t3 x (lv-1) r)
+ Node t1 (a,lva) (Node t2 (b,lvb) t3)
+ \<Rightarrow> Node (Node t1 (a,lva) t2) (b,lvb+1) (Node t3 (x,lv-1) r)
else
- if lvl r < lv then split (Node l x (lv-1) r)
+ if lvl r < lv then split (Node l (x,lv-1) r)
else
case r of
- Node t1 b lvb t4 \<Rightarrow>
+ Node t1 (b,lvb) t4 \<Rightarrow>
(case t1 of
- Node t2 a lva t3
- \<Rightarrow> Node (Node l x (lv-1) t2) a (lva+1)
- (split (Node t3 b (if sngl t1 then lva else lva+1) t4)))))"
+ Node t2 (a,lva) t3
+ \<Rightarrow> Node (Node l (x,lv-1) t2) (a,lva+1)
+ (split (Node t3 (b, if sngl t1 then lva else lva+1) t4)))))"
text\<open>In the paper, the last case of \<^const>\<open>adjust\<close> is expressed with the help of an
incorrect auxiliary function \texttt{nlvl}.
@@ -81,20 +81,20 @@
is not restored.\<close>
fun split_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
-"split_max (Node l a lv Leaf) = (l,a)" |
-"split_max (Node l a lv r) = (let (r',b) = split_max r in (adjust(Node l a lv r'), b))"
+"split_max (Node l (a,lv) Leaf) = (l,a)" |
+"split_max (Node l (a,lv) r) = (let (r',b) = split_max r in (adjust(Node l (a,lv) r'), b))"
fun delete :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
"delete _ Leaf = Leaf" |
-"delete x (Node l a lv r) =
+"delete x (Node l (a,lv) r) =
(case cmp x a of
- LT \<Rightarrow> adjust (Node (delete x l) a lv r) |
- GT \<Rightarrow> adjust (Node l a lv (delete x r)) |
+ LT \<Rightarrow> adjust (Node (delete x l) (a,lv) r) |
+ GT \<Rightarrow> adjust (Node l (a,lv) (delete x r)) |
EQ \<Rightarrow> (if l = Leaf then r
- else let (l',b) = split_max l in adjust (Node l' b lv r)))"
+ else let (l',b) = split_max l in adjust (Node l' (b,lv) r)))"
fun pre_adjust where
-"pre_adjust (Node l a lv r) = (invar l \<and> invar r \<and>
+"pre_adjust (Node l (a,lv) r) = (invar l \<and> invar r \<and>
((lv = lvl l + 1 \<and> (lv = lvl r + 1 \<or> lv = lvl r + 2 \<or> lv = lvl r \<and> sngl r)) \<or>
(lv = lvl l + 2 \<and> (lv = lvl r + 1 \<or> lv = lvl r \<and> sngl r))))"
@@ -103,23 +103,23 @@
subsection "Auxiliary Proofs"
lemma split_case: "split t = (case t of
- Node t1 x lvx (Node t2 y lvy (Node t3 z lvz t4)) \<Rightarrow>
+ Node t1 (x,lvx) (Node t2 (y,lvy) (Node t3 (z,lvz) t4)) \<Rightarrow>
(if lvx = lvy \<and> lvy = lvz
- then Node (Node t1 x lvx t2) y (lvx+1) (Node t3 z lvx t4)
+ then Node (Node t1 (x,lvx) t2) (y,lvx+1) (Node t3 (z,lvx) t4)
else t)
| t \<Rightarrow> t)"
by(auto split: tree.split)
lemma skew_case: "skew t = (case t of
- Node (Node t1 y lvy t2) x lvx t3 \<Rightarrow>
- (if lvx = lvy then Node t1 y lvx (Node t2 x lvx t3) else t)
+ Node (Node t1 (y,lvy) t2) (x,lvx) t3 \<Rightarrow>
+ (if lvx = lvy then Node t1 (y, lvx) (Node t2 (x,lvx) t3) else t)
| t \<Rightarrow> t)"
by(auto split: tree.split)
lemma lvl_0_iff: "invar t \<Longrightarrow> lvl t = 0 \<longleftrightarrow> t = Leaf"
by(cases t) auto
-lemma lvl_Suc_iff: "lvl t = Suc n \<longleftrightarrow> (\<exists> l a r. t = Node l a (Suc n) r)"
+lemma lvl_Suc_iff: "lvl t = Suc n \<longleftrightarrow> (\<exists> l a r. t = Node l (a,Suc n) r)"
by(cases t) auto
lemma lvl_skew: "lvl (skew t) = lvl t"
@@ -128,16 +128,16 @@
lemma lvl_split: "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)"
by(cases t rule: split.cases) auto
-lemma invar_2Nodes:"invar (Node l x lv (Node rl rx rlv rr)) =
- (invar l \<and> invar \<langle>rl, rx, rlv, rr\<rangle> \<and> lv = Suc (lvl l) \<and>
+lemma invar_2Nodes:"invar (Node l (x,lv) (Node rl (rx, rlv) rr)) =
+ (invar l \<and> invar \<langle>rl, (rx, rlv), rr\<rangle> \<and> lv = Suc (lvl l) \<and>
(lv = Suc rlv \<or> rlv = lv \<and> lv = Suc (lvl rr)))"
by simp
lemma invar_NodeLeaf[simp]:
- "invar (Node l x lv Leaf) = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
+ "invar (Node l (x,lv) Leaf) = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
by simp
-lemma sngl_if_invar: "invar (Node l a n r) \<Longrightarrow> n = lvl r \<Longrightarrow> sngl r"
+lemma sngl_if_invar: "invar (Node l (a, n) r) \<Longrightarrow> n = lvl r \<Longrightarrow> sngl r"
by(cases r rule: sngl.cases) clarsimp+
@@ -167,7 +167,8 @@
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)
+ thus ?thesis using 2
+ proof (cases t1 rule: tree2_cases)
case Node
thus ?thesis using 2 GT
apply (auto simp add: skew_case split_case split: tree.splits)
@@ -183,32 +184,32 @@
by(cases t rule: split.cases) clarsimp+
lemma invar_NodeL:
- "\<lbrakk> invar(Node l x n r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node l' x n r)"
+ "\<lbrakk> invar(Node l (x, n) r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node l' (x, n) r)"
by(auto)
lemma invar_NodeR:
- "\<lbrakk> invar(Node l x n r); n = lvl r + 1; invar r'; lvl r' = lvl r \<rbrakk> \<Longrightarrow> invar(Node l x n r')"
+ "\<lbrakk> invar(Node l (x, n) r); n = lvl r + 1; invar r'; lvl r' = lvl r \<rbrakk> \<Longrightarrow> invar(Node l (x, n) r')"
by(auto)
lemma invar_NodeR2:
- "\<lbrakk> invar(Node l x n r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n \<rbrakk> \<Longrightarrow> invar(Node l x n r')"
+ "\<lbrakk> invar(Node l (x, n) r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n \<rbrakk> \<Longrightarrow> invar(Node l (x, n) r')"
by(cases r' rule: sngl.cases) clarsimp+
lemma lvl_insert_incr_iff: "(lvl(insert a t) = lvl t + 1) \<longleftrightarrow>
- (\<exists>l x r. insert a t = Node l x (lvl t + 1) r \<and> lvl l = lvl r)"
-apply(cases t)
+ (\<exists>l x r. insert a t = Node l (x, lvl t + 1) r \<and> lvl l = lvl r)"
+apply(cases t rule: tree2_cases)
apply(auto simp add: skew_case split_case split: if_splits)
apply(auto split: tree.splits if_splits)
done
lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
-proof(induction t)
+proof(induction t rule: tree2_induct)
case N: (Node l x n r)
hence il: "invar l" and ir: "invar r" by auto
note iil = N.IH(1)[OF il]
note iir = N.IH(2)[OF ir]
- let ?t = "Node l x n r"
+ let ?t = "Node l (x, n) r"
have "a < x \<or> a = x \<or> x < a" by auto
moreover
have ?case if "a < x"
@@ -218,16 +219,16 @@
by (simp add: skew_invar split_invar del: invar.simps)
next
case (Incr)
- then obtain t1 w t2 where ial[simp]: "insert a l = Node t1 w n t2"
+ then obtain t1 w t2 where ial[simp]: "insert a l = Node t1 (w, n) t2"
using N.prems by (auto simp: lvl_Suc_iff)
have l12: "lvl t1 = lvl t2"
by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
- have "insert a ?t = split(skew(Node (insert a l) x n r))"
+ have "insert a ?t = split(skew(Node (insert a l) (x,n) r))"
by(simp add: \<open>a<x\<close>)
- also have "skew(Node (insert a l) x n r) = Node t1 w n (Node t2 x n r)"
+ also have "skew(Node (insert a l) (x,n) r) = Node t1 (w,n) (Node t2 (x,n) r)"
by(simp)
also have "invar(split \<dots>)"
- proof (cases r)
+ proof (cases r rule: tree2_cases)
case Leaf
hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
thus ?thesis using Leaf ial by simp
@@ -249,14 +250,14 @@
thus ?case
proof
assume 0: "n = lvl r"
- have "insert a ?t = split(skew(Node l x n (insert a r)))"
+ have "insert a ?t = split(skew(Node l (x, n) (insert a r)))"
using \<open>a>x\<close> by(auto)
- also have "skew(Node l x n (insert a r)) = Node l x n (insert a r)"
+ also have "skew(Node l (x,n) (insert a r)) = Node l (x,n) (insert a r)"
using N.prems by(simp add: skew_case split: tree.split)
also have "invar(split \<dots>)"
proof -
from lvl_insert_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a]
- obtain t1 y t2 where iar: "insert a r = Node t1 y n t2"
+ obtain t1 y t2 where iar: "insert a r = Node t1 (y,n) t2"
using N.prems 0 by (auto simp: lvl_Suc_iff)
from N.prems iar 0 iir
show ?thesis by (auto simp: split_case split: tree.splits)
@@ -285,21 +286,21 @@
subsubsection "Proofs for delete"
-lemma invarL: "ASSUMPTION(invar \<langle>l, a, lv, r\<rangle>) \<Longrightarrow> invar l"
+lemma invarL: "ASSUMPTION(invar \<langle>l, (a, lv), r\<rangle>) \<Longrightarrow> invar l"
by(simp add: ASSUMPTION_def)
-lemma invarR: "ASSUMPTION(invar \<langle>lv, l, a, r\<rangle>) \<Longrightarrow> invar r"
+lemma invarR: "ASSUMPTION(invar \<langle>l, (a,lv), r\<rangle>) \<Longrightarrow> invar r"
by(simp add: ASSUMPTION_def)
lemma sngl_NodeI:
- "sngl (Node l a lv r) \<Longrightarrow> sngl (Node l' a' lv r)"
-by(cases r) (simp_all)
+ "sngl (Node l (a,lv) r) \<Longrightarrow> sngl (Node l' (a', lv) r)"
+by(cases r rule: tree2_cases) (simp_all)
declare invarL[simp] invarR[simp]
lemma pre_cases:
-assumes "pre_adjust (Node l x lv r)"
+assumes "pre_adjust (Node l (x,lv) r)"
obtains
(tSngl) "invar l \<and> invar r \<and>
lv = Suc (lvl r) \<and> lvl l = lvl r" |
@@ -317,38 +318,39 @@
declare invar.simps(2)[simp del] invar_2Nodes[simp add]
lemma invar_adjust:
- assumes pre: "pre_adjust (Node l a lv r)"
- shows "invar(adjust (Node l a lv r))"
+ assumes pre: "pre_adjust (Node l (a,lv) r)"
+ shows "invar(adjust (Node l (a,lv) r))"
using pre proof (cases rule: pre_cases)
case (tDouble) thus ?thesis unfolding adjust_def by (cases r) (auto simp: invar.simps(2))
next
case (rDown)
- from rDown obtain llv ll la lr where l: "l = Node ll la llv lr" by (cases l) auto
+ from rDown obtain llv ll la lr where l: "l = Node ll (la, llv) lr" by (cases l) auto
from rDown show ?thesis unfolding adjust_def by (auto simp: l invar.simps(2) split: tree.splits)
next
case (lDown_tDouble)
- from lDown_tDouble obtain rlv rr ra rl where r: "r = Node rl ra rlv rr" by (cases r) auto
+ from lDown_tDouble obtain rlv rr ra rl where r: "r = Node rl (ra, rlv) rr" by (cases r) auto
from lDown_tDouble and r obtain rrlv rrr rra rrl where
- rr :"rr = Node rrr rra rrlv rrl" by (cases rr) auto
+ rr :"rr = Node rrr (rra, rrlv) rrl" by (cases rr) auto
from lDown_tDouble show ?thesis unfolding adjust_def r rr
- apply (cases rl) apply (auto simp add: invar.simps(2) split!: if_split)
+ apply (cases rl rule: tree2_cases) apply (auto simp add: invar.simps(2) split!: if_split)
using lDown_tDouble by (auto simp: split_case lvl_0_iff elim:lvl.elims split: tree.split)
qed (auto simp: split_case invar.simps(2) adjust_def split: tree.splits)
lemma lvl_adjust:
- assumes "pre_adjust (Node l a lv r)"
- shows "lv = lvl (adjust(Node l a lv r)) \<or> lv = lvl (adjust(Node l a lv r)) + 1"
-using assms(1) proof(cases rule: pre_cases)
+ assumes "pre_adjust (Node l (a,lv) r)"
+ shows "lv = lvl (adjust(Node l (a,lv) r)) \<or> lv = lvl (adjust(Node l (a,lv) r)) + 1"
+using assms(1)
+proof(cases rule: pre_cases)
case lDown_tSngl thus ?thesis
- using lvl_split[of "\<langle>l, a, lvl r, r\<rangle>"] by (auto simp: adjust_def)
+ using lvl_split[of "\<langle>l, (a, lvl r), r\<rangle>"] by (auto simp: adjust_def)
next
case lDown_tDouble thus ?thesis
by (auto simp: adjust_def invar.simps(2) split: tree.split)
qed (auto simp: adjust_def split: tree.splits)
-lemma sngl_adjust: assumes "pre_adjust (Node l a lv r)"
- "sngl \<langle>l, a, lv, r\<rangle>" "lv = lvl (adjust \<langle>l, a, lv, r\<rangle>)"
- shows "sngl (adjust \<langle>l, a, lv, r\<rangle>)"
+lemma sngl_adjust: assumes "pre_adjust (Node l (a,lv) r)"
+ "sngl \<langle>l, (a, lv), r\<rangle>" "lv = lvl (adjust \<langle>l, (a, lv), r\<rangle>)"
+ shows "sngl (adjust \<langle>l, (a, lv), r\<rangle>)"
using assms proof (cases rule: pre_cases)
case rDown
thus ?thesis using assms(2,3) unfolding adjust_def
@@ -361,38 +363,38 @@
(lvl t' = lvl t \<and> sngl t \<longrightarrow> sngl t')"
lemma pre_adj_if_postR:
- "invar\<langle>lv, l, a, r\<rangle> \<Longrightarrow> post_del r r' \<Longrightarrow> pre_adjust \<langle>lv, l, a, r'\<rangle>"
+ "invar\<langle>lv, (l, a), r\<rangle> \<Longrightarrow> post_del r r' \<Longrightarrow> pre_adjust \<langle>lv, (l, a), r'\<rangle>"
by(cases "sngl r")
(auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
lemma pre_adj_if_postL:
- "invar\<langle>l, a, lv, r\<rangle> \<Longrightarrow> post_del l l' \<Longrightarrow> pre_adjust \<langle>l', b, lv, r\<rangle>"
+ "invar\<langle>l, (a, lv), r\<rangle> \<Longrightarrow> post_del l l' \<Longrightarrow> pre_adjust \<langle>l', (b, lv), r\<rangle>"
by(cases "sngl r")
(auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
lemma post_del_adjL:
- "\<lbrakk> invar\<langle>l, a, lv, r\<rangle>; pre_adjust \<langle>l', b, lv, r\<rangle> \<rbrakk>
- \<Longrightarrow> post_del \<langle>l, a, lv, r\<rangle> (adjust \<langle>l', b, lv, r\<rangle>)"
+ "\<lbrakk> invar\<langle>l, (a, lv), r\<rangle>; pre_adjust \<langle>l', (b, lv), r\<rangle> \<rbrakk>
+ \<Longrightarrow> post_del \<langle>l, (a, lv), r\<rangle> (adjust \<langle>l', (b, lv), r\<rangle>)"
unfolding post_del_def
by (metis invar_adjust lvl_adjust sngl_NodeI sngl_adjust lvl.simps(2))
lemma post_del_adjR:
-assumes "invar\<langle>lv, l, a, r\<rangle>" "pre_adjust \<langle>lv, l, a, r'\<rangle>" "post_del r r'"
-shows "post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l, a, r'\<rangle>)"
+assumes "invar\<langle>l, (a,lv), r\<rangle>" "pre_adjust \<langle>l, (a,lv), r'\<rangle>" "post_del r r'"
+shows "post_del \<langle>l, (a,lv), r\<rangle> (adjust \<langle>l, (a,lv), r'\<rangle>)"
proof(unfold post_del_def, safe del: disjCI)
- let ?t = "\<langle>lv, l, a, r\<rangle>"
- let ?t' = "adjust \<langle>lv, l, a, r'\<rangle>"
+ let ?t = "\<langle>l, (a,lv), r\<rangle>"
+ let ?t' = "adjust \<langle>l, (a,lv), r'\<rangle>"
show "invar ?t'" by(rule invar_adjust[OF assms(2)])
show "lvl ?t' = lvl ?t \<or> lvl ?t' + 1 = lvl ?t"
using lvl_adjust[OF assms(2)] by auto
show "sngl ?t'" if as: "lvl ?t' = lvl ?t" "sngl ?t"
proof -
- have s: "sngl \<langle>lv, l, a, r'\<rangle>"
- proof(cases r')
+ have s: "sngl \<langle>l, (a,lv), r'\<rangle>"
+ proof(cases r' rule: tree2_cases)
case Leaf thus ?thesis by simp
next
case Node thus ?thesis using as(2) assms(1,3)
- by (cases r) (auto simp: post_del_def)
+ by (cases r rule: tree2_cases) (auto simp: post_del_def)
qed
show ?thesis using as(1) sngl_adjust[OF assms(2) s] by simp
qed
@@ -403,22 +405,22 @@
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>"
+ case (2 l a lv rl bl rr)
+ let ?r = "\<langle>rl, bl, rr\<rangle>"
+ let ?t = "\<langle>l, (a, lv), ?r\<rangle>"
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
+ and [simp]: "t' = adjust \<langle>l, (a, lv), 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]
show ?case by (simp add: post_del_adjR[OF "2.prems"(1) preR post])
qed (auto simp: post_del_def)
theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
-proof (induction t)
+proof (induction t rule: tree2_induct)
case (Node l a lv r)
let ?l' = "delete x l" and ?r' = "delete x r"
- let ?t = "Node l a lv r" let ?t' = "delete x ?t"
+ let ?t = "Node l (a,lv) r" let ?t' = "delete x ?t"
from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)