# HG changeset patch
# User nipkow
# Date 1528727378 -7200
# Node ID b001bef9aa39981e85655abe790af8c53d534295
# Parent  07f8c09e3f79c36592b76f483461c85b8be581d0# Parent  b56ed5010e6970b77688d0d033457e12a3ca849c
merged

diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/AA_Map.thy
--- a/src/HOL/Data_Structures/AA_Map.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/AA_Map.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -9,21 +9,21 @@
 begin
 
 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) aa_tree \<Rightarrow> ('a*'b) aa_tree" where
-"update x y Leaf = Node 1 Leaf (x,y) Leaf" |
-"update x y (Node lv t1 (a,b) t2) =
+"update x y Leaf = Node Leaf (x,y) 1 Leaf" |
+"update x y (Node t1 (a,b) lv t2) =
   (case cmp x a of
-     LT \<Rightarrow> split (skew (Node lv (update x y t1) (a,b) t2)) |
-     GT \<Rightarrow> split (skew (Node lv t1 (a,b) (update x y t2))) |
-     EQ \<Rightarrow> Node lv t1 (x,y) t2)"
+     LT \<Rightarrow> split (skew (Node (update x y t1) (a,b) lv t2)) |
+     GT \<Rightarrow> split (skew (Node t1 (a,b) lv (update x y t2))) |
+     EQ \<Rightarrow> Node t1 (x,y) lv t2)"
 
 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) aa_tree \<Rightarrow> ('a*'b) aa_tree" where
 "delete _ Leaf = Leaf" |
-"delete x (Node lv l (a,b) r) =
+"delete x (Node l (a,b) lv r) =
   (case cmp x a of
-     LT \<Rightarrow> adjust (Node lv (delete x l) (a,b) r) |
-     GT \<Rightarrow> adjust (Node lv l (a,b) (delete x r)) |
+     LT \<Rightarrow> adjust (Node (delete x l) (a,b) lv r) |
+     GT \<Rightarrow> adjust (Node l (a,b) lv (delete x r)) |
      EQ \<Rightarrow> (if l = Leaf then r
-            else let (l',ab') = split_max l in adjust (Node lv l' ab' r)))"
+            else let (l',ab') = split_max l in adjust (Node l' ab' lv r)))"
 
 
 subsection "Invariance"
@@ -46,7 +46,7 @@
 
 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)
+  case (2 x y t1 a b lv t2)
   consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a" 
     using less_linear by blast 
   thus ?case proof cases
@@ -64,7 +64,7 @@
 qed simp
 
 lemma lvl_update_incr_iff: "(lvl(update a b t) = lvl t + 1) \<longleftrightarrow>
-  (\<exists>l x r. update a b t = Node (lvl t + 1) l x r \<and> lvl l = lvl r)"
+  (\<exists>l x r. update a b t = Node l x (lvl t + 1) 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)
@@ -72,12 +72,12 @@
 
 lemma invar_update: "invar t \<Longrightarrow> invar(update a b t)"
 proof(induction t)
-  case N: (Node n l xy r)
+  case N: (Node l xy 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]
   obtain x y where [simp]: "xy = (x,y)" by fastforce
-  let ?t = "Node n l xy r"
+  let ?t = "Node l xy n r"
   have "a < x \<or> a = x \<or> x < a" by auto
   moreover
   have ?case if "a < x"
@@ -87,13 +87,13 @@
       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"
+    then obtain t1 w t2 where ial[simp]: "update a b 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_update_incr_iff tree.inject)
-    have "update a b ?t = split(skew(Node n (update a b l) xy r))"
+    have "update a b ?t = split(skew(Node (update a b l) xy n 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)"
+    also have "skew(Node (update a b l) xy n r) = Node t1 w n (Node t2 xy n r)"
       by(simp)
     also have "invar(split \<dots>)"
     proof (cases r)
@@ -101,7 +101,7 @@
       hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
       thus ?thesis using Leaf ial by simp
     next
-      case [simp]: (Node m t3 y t4)
+      case [simp]: (Node t3 y m t4)
       show ?thesis (*using N(3) iil l12 by(auto)*)
       proof cases
         assume "m = n" thus ?thesis using N(3) iil by(auto)
@@ -118,14 +118,14 @@
     thus ?case
     proof
       assume 0: "n = lvl r"
-      have "update a b ?t = split(skew(Node n l xy (update a b r)))"
+      have "update a b ?t = split(skew(Node l xy n (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)"
+      also have "skew(Node l xy n (update a b r)) = Node l xy n (update a b r)"
         using N.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"
+        obtain t1 p t2 where iar: "update a b r = Node t1 p 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)
@@ -157,12 +157,12 @@
 
 theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
 proof (induction t)
-  case (Node lv l ab r)
+  case (Node l ab lv 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"
+  let ?t = "Node l ab lv r" let ?t' = "delete x ?t"
 
   from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)
 
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/AA_Set.thy
--- a/src/HOL/Data_Structures/AA_Set.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/AA_Set.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -14,60 +14,60 @@
 
 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 h l a 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 h lr b 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 lva (Node lvb t1 b t2) a t3) =
-  (if lva = lvb then Node lva t1 b (Node lva t2 a t3) else Node lva (Node lvb t1 b t2) a 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 lva t1 a (Node lvb t2 b (Node lvc t3 c 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 (lva+1) (Node lva t1 a t2) b (Node lva t3 c t4)
-    else Node lva t1 a (Node lvb t2 b (Node lvc t3 c 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 1 Leaf x Leaf" |
-"insert x (Node lv t1 a 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 lv (insert x t1) a t2)) |
-     GT \<Rightarrow> split (skew (Node lv t1 a (insert x t2))) |
-     EQ \<Rightarrow> Node lv t1 x 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 _ _ lva (Node _ _ lvb _)) = (lva > lvb)"
 
 definition adjust :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
 "adjust t =
  (case t of
-  Node lv l x 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 (lv-1) l x 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 lva t1 a (Node lvb t2 b t3)
-             \<Rightarrow> Node (lvb+1) (Node lva t1 a t2) b (Node (lv-1) t3 x 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 (lv-1) l x r)
+    if lvl r < lv then split (Node l x (lv-1) r)
     else
       case r of
-        Node lvb t1 b t4 \<Rightarrow>
+        Node t1 b lvb t4 \<Rightarrow>
           (case t1 of
-             Node lva t2 a t3
-               \<Rightarrow> Node (lva+1) (Node (lv-1) l x t2) a
-                    (split (Node (if sngl t1 then lva else lva+1) t3 b 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 adjust} is expressed with the help of an
 incorrect auxiliary function \texttt{nlvl}.
@@ -78,20 +78,20 @@
 is not restored.\<close>
 
 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))"
+"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 lv l a r) =
+"delete x (Node l a lv r) =
   (case cmp x a of
-     LT \<Rightarrow> adjust (Node lv (delete x l) a r) |
-     GT \<Rightarrow> adjust (Node lv l a (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 lv l' b r)))"
+            else let (l',b) = split_max l in adjust (Node l' b lv r)))"
 
 fun pre_adjust where
-"pre_adjust (Node lv l a 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))))"
 
@@ -100,23 +100,23 @@
 subsection "Auxiliary Proofs"
 
 lemma split_case: "split t = (case t of
-  Node lvx a x (Node lvy b y (Node lvz c z d)) \<Rightarrow>
+  Node t1 x lvx (Node t2 y lvy (Node t3 z lvz t4)) \<Rightarrow>
    (if lvx = lvy \<and> lvy = lvz
-    then Node (lvx+1) (Node lvx a x b) y (Node lvx c z d)
+    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 lvx (Node lvy a y b) x c \<Rightarrow>
-  (if lvx = lvy then Node lvx a y (Node lvx b x c) 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 (Suc n) l a 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"
@@ -125,16 +125,16 @@
 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 lv l x (Node rlv rl rx rr)) =
-     (invar l \<and> invar \<langle>rlv, rl, rx, 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 lv l x 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 n l a 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+
 
 
@@ -156,7 +156,7 @@
 
 lemma lvl_insert_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(insert x t) = lvl t"
 proof (induction t rule: insert.induct)
-  case (2 x lv t1 a t2)
+  case (2 x t1 a lv t2)
   consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a" 
     using less_linear by blast 
   thus ?case proof cases
@@ -180,20 +180,20 @@
 by(cases t rule: split.cases) clarsimp+
 
 lemma invar_NodeL:
-  "\<lbrakk> invar(Node n l x r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node n l' x 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 n l x r); n = lvl r + 1; invar r'; lvl r' = lvl r \<rbrakk> \<Longrightarrow> invar(Node n l x 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 n l x r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n \<rbrakk> \<Longrightarrow> invar(Node n l x 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 (lvl t + 1) l x r \<and> lvl l = lvl r)"
+  (\<exists>l x r. insert a t = Node l x (lvl t + 1) 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)
@@ -201,11 +201,11 @@
 
 lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
 proof(induction t)
-  case N: (Node n l x r)
+  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 n l x 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"
@@ -215,13 +215,13 @@
       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 n t1 w 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 n (insert a l) x 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 n (insert a l) x r) = Node n t1 w (Node n t2 x 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)
@@ -229,7 +229,7 @@
       hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
       thus ?thesis using Leaf ial by simp
     next
-      case [simp]: (Node m t3 y t4)
+      case [simp]: (Node t3 y m t4)
       show ?thesis (*using N(3) iil l12 by(auto)*)
       proof cases
         assume "m = n" thus ?thesis using N(3) iil by(auto)
@@ -246,14 +246,14 @@
     thus ?case
     proof
       assume 0: "n = lvl r"
-      have "insert a ?t = split(skew(Node n l x (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 n l x (insert a r)) = Node n l x (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 n t1 y 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)
@@ -282,21 +282,21 @@
 
 subsubsection "Proofs for delete"
 
-lemma invarL: "ASSUMPTION(invar \<langle>lv, l, a, 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"
 by(simp add: ASSUMPTION_def)
 
 lemma sngl_NodeI:
-  "sngl (Node lv l a r) \<Longrightarrow> sngl (Node lv l' a' r)"
+  "sngl (Node l a lv r) \<Longrightarrow> sngl (Node l' a' lv r)"
 by(cases r) (simp_all)
 
 
 declare invarL[simp] invarR[simp]
 
 lemma pre_cases:
-assumes "pre_adjust (Node lv l x 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" |
@@ -314,38 +314,38 @@
 declare invar.simps(2)[simp del] invar_2Nodes[simp add]
 
 lemma invar_adjust:
-  assumes pre: "pre_adjust (Node lv l a r)"
-  shows  "invar(adjust (Node lv l a 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 llv ll la 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 rlv rl ra 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 rrlv rrr rra 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)
     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 lv l a r)"
-  shows "lv = lvl (adjust(Node lv l a r)) \<or> lv = lvl (adjust(Node lv l a r)) + 1"
+  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>lvl r, l, a, 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 lv l a r)"
-  "sngl \<langle>lv, l, a, r\<rangle>" "lv = lvl (adjust \<langle>lv, l, a, r\<rangle>)"
-  shows "sngl (adjust \<langle>lv, l, a, 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
@@ -363,13 +363,13 @@
   (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
 
 lemma pre_adj_if_postL:
-  "invar\<langle>lv, l, a, r\<rangle> \<Longrightarrow> post_del l l' \<Longrightarrow> pre_adjust \<langle>lv, l', b, 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>lv, l, a, r\<rangle>; pre_adjust \<langle>lv, l', b, r\<rangle> \<rbrakk>
-  \<Longrightarrow> post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l', b, 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))
 
@@ -412,10 +412,10 @@
 
 theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
 proof (induction t)
-  case (Node lv l a r)
+  case (Node l a lv r)
 
   let ?l' = "delete x l" and ?r' = "delete x r"
-  let ?t = "Node lv l a 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)
 
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/AVL_Map.thy
--- a/src/HOL/Data_Structures/AVL_Map.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/AVL_Map.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -9,16 +9,16 @@
 begin
 
 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
-"update x y Leaf = Node 1 Leaf (x,y) Leaf" |
-"update x y (Node h l (a,b) r) = (case cmp x a of
-   EQ \<Rightarrow> Node h l (x,y) r |
+"update x y Leaf = Node Leaf (x,y) 1 Leaf" |
+"update x y (Node l (a,b) h r) = (case cmp x a of
+   EQ \<Rightarrow> Node l (x,y) h r |
    LT \<Rightarrow> balL (update x y l) (a,b) r |
    GT \<Rightarrow> balR l (a,b) (update x y r))"
 
 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) avl_tree \<Rightarrow> ('a*'b) avl_tree" where
 "delete _ Leaf = Leaf" |
-"delete x (Node h l (a,b) r) = (case cmp x a of
-   EQ \<Rightarrow> del_root (Node h l (a,b) r) |
+"delete x (Node l (a,b) h r) = (case cmp x a of
+   EQ \<Rightarrow> del_root (Node l (a,b) h r) |
    LT \<Rightarrow> balR (delete x l) (a,b) r |
    GT \<Rightarrow> balL l (a,b) (delete x r))"
 
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/AVL_Set.thy
--- a/src/HOL/Data_Structures/AVL_Set.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/AVL_Set.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -18,25 +18,25 @@
 
 fun avl :: "'a avl_tree \<Rightarrow> bool" where
 "avl Leaf = True" |
-"avl (Node h l a r) =
+"avl (Node l a h r) =
  ((height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1) \<and> 
   h = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
 
 fun ht :: "'a avl_tree \<Rightarrow> nat" where
 "ht Leaf = 0" |
-"ht (Node h l a r) = h"
+"ht (Node l a h r) = h"
 
 definition node :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
-"node l a r = Node (max (ht l) (ht r) + 1) l a r"
+"node l a r = Node l a (max (ht l) (ht r) + 1) r"
 
 definition balL :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
 "balL l a r =
   (if ht l = ht r + 2 then
      case l of 
-       Node _ bl b br \<Rightarrow>
+       Node bl b _ br \<Rightarrow>
          if ht bl < ht br then
            case br of
-             Node _ cl c cr \<Rightarrow> node (node bl b cl) c (node cr a r)
+             Node cl c _ cr \<Rightarrow> node (node bl b cl) c (node cr a r)
          else node bl b (node br a r)
    else node l a r)"
 
@@ -44,38 +44,38 @@
 "balR l a r =
    (if ht r = ht l + 2 then
       case r of
-        Node _ bl b br \<Rightarrow>
+        Node bl b _ br \<Rightarrow>
           if ht bl > ht br then
             case bl of
-              Node _ cl c cr \<Rightarrow> node (node l a cl) c (node cr b br)
+              Node cl c _ cr \<Rightarrow> node (node l a cl) c (node cr b br)
           else node (node l a bl) b br
   else node l a r)"
 
 fun insert :: "'a::linorder \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
-"insert x Leaf = Node 1 Leaf x Leaf" |
-"insert x (Node h l a r) = (case cmp x a of
-   EQ \<Rightarrow> Node h l a r |
+"insert x Leaf = Node Leaf x 1 Leaf" |
+"insert x (Node l a h r) = (case cmp x a of
+   EQ \<Rightarrow> Node l a h r |
    LT \<Rightarrow> balL (insert x l) a r |
    GT \<Rightarrow> balR l a (insert x r))"
 
 fun split_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
-"split_max (Node _ l a r) =
+"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 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') = split_max l in balR l' a' r)"
+"del_root (Node Leaf a h r) = r" |
+"del_root (Node l a h Leaf) = l" |
+"del_root (Node l a h 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]
 
 fun delete :: "'a::linorder \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
 "delete _ Leaf = Leaf" |
-"delete x (Node h l a r) =
+"delete x (Node l a h r) =
   (case cmp x a of
-     EQ \<Rightarrow> del_root (Node h l a r) |
+     EQ \<Rightarrow> del_root (Node l a h r) |
      LT \<Rightarrow> balR (delete x l) a r |
      GT \<Rightarrow> balL l a (delete x r))"
 
@@ -110,8 +110,8 @@
   (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)
+  "inorder (del_root (Node l a h r)) = inorder l @ inorder r"
+by(cases "Node l a h r" rule: del_root.cases)
   (auto simp: inorder_balL inorder_balR inorder_split_maxD split: if_splits prod.splits)
 
 theorem inorder_delete:
@@ -188,7 +188,7 @@
   case Leaf
   with assms show ?thesis by (simp add: node_def balL_def)
 next
-  case (Node ln ll lr lh)
+  case Node
   with assms show ?thesis
   proof(cases "height l = height r + 2")
     case True
@@ -208,7 +208,7 @@
   case Leaf
   with assms show ?thesis by (simp add: node_def balR_def)
 next
-  case (Node rn rl rr rh)
+  case Node
   with assms show ?thesis
   proof(cases "height r = height l + 2")
     case True
@@ -230,7 +230,7 @@
         "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
 using assms
 proof (induction t)
-  case (Node h l a r)
+  case (Node l a h r)
   case 1
   with Node show ?case
   proof(cases "x = a")
@@ -307,14 +307,14 @@
          height x = height(fst (split_max x)) + 1"
 using assms
 proof (induct x rule: split_max_induct)
-  case (Node h l a r)
+  case (Node l a h r)
   case 1
   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 r)
+  case (Node l a h r)
   case 2
   let ?r' = "fst (split_max r)"
   from \<open>avl x\<close> Node 2 have "avl l" and "avl r" by simp_all
@@ -327,9 +327,9 @@
   shows "avl(del_root t)" 
 using assms
 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"
+  case (Node_Node ll ln lh lr n h rl rn rh rr)
+  let ?l = "Node ll ln lh lr"
+  let ?r = "Node rl rn rh rr"
   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
@@ -347,9 +347,9 @@
   shows "height t = height(del_root t) \<or> height t = height(del_root t) + 1"
 using assms
 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"
+  case (Node_Node ll ln lh lr n h rl rn rh rr)
+  let ?l = "Node ll ln lh lr"
+  let ?r = "Node rl rn rh rr"
   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
@@ -382,7 +382,7 @@
   shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
 using assms
 proof (induct t)
-  case (Node h l n r)
+  case (Node l n h r)
   case 1
   with Node show ?case
   proof(cases "x = n")
@@ -403,8 +403,8 @@
   with Node show ?case
   proof(cases "x = n")
     case True
-    with 1 have "height (Node h l n r) = height(del_root (Node h l n r))
-      \<or> height (Node h l n r) = height(del_root (Node h l n r)) + 1"
+    with 1 have "height (Node l n h r) = height(del_root (Node l n h r))
+      \<or> height (Node l n h r) = height(del_root (Node l n h r)) + 1"
       by (subst height_del_root,simp_all)
     with True show ?thesis by simp
   next
@@ -459,7 +459,7 @@
 
 lemma height_invers: 
   "(height t = 0) = (t = Leaf)"
-  "avl t \<Longrightarrow> (height t = Suc h) = (\<exists> l a r . t = Node (Suc h) l a r)"
+  "avl t \<Longrightarrow> (height t = Suc h) = (\<exists> l a r . t = Node l a (Suc h) r)"
 by (induction t) auto
 
 text \<open>Any AVL tree of height \<open>h\<close> has at least \<open>fib (h+2)\<close> leaves:\<close>
@@ -472,7 +472,7 @@
 next
   case (3 h)
   from "3.prems" obtain l a r where
-    [simp]: "t = Node (Suc(Suc h)) l a r" "avl l" "avl r"
+    [simp]: "t = Node l a (Suc(Suc h)) r" "avl l" "avl r"
     and C: "
       height r = Suc h \<and> height l = Suc h
     \<or> height r = Suc h \<and> height l = h
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/Isin2.thy
--- a/src/HOL/Data_Structures/Isin2.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/Isin2.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -11,7 +11,7 @@
 
 fun isin :: "('a::linorder,'b) tree \<Rightarrow> 'a \<Rightarrow> bool" where
 "isin Leaf x = False" |
-"isin (Node _ l a r) x =
+"isin (Node l a _ r) x =
   (case cmp x a of
      LT \<Rightarrow> isin l x |
      EQ \<Rightarrow> True |
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/Leftist_Heap.thy
--- a/src/HOL/Data_Structures/Leftist_Heap.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/Leftist_Heap.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -12,7 +12,7 @@
 
 fun mset_tree :: "('a,'b) tree \<Rightarrow> 'a multiset" where
 "mset_tree Leaf = {#}" |
-"mset_tree (Node _ l a r) = {#a#} + mset_tree l + mset_tree r"
+"mset_tree (Node l a _ r) = {#a#} + mset_tree l + mset_tree r"
 
 type_synonym 'a lheap = "('a,nat)tree"
 
@@ -22,27 +22,27 @@
 
 fun rk :: "'a lheap \<Rightarrow> nat" where
 "rk Leaf = 0" |
-"rk (Node n _ _ _) = n"
+"rk (Node _ _ n _) = n"
 
 text\<open>The invariants:\<close>
 
 fun (in linorder) heap :: "('a,'b) tree \<Rightarrow> bool" where
 "heap Leaf = True" |
-"heap (Node _ l m r) =
+"heap (Node l m _ r) =
   (heap l \<and> heap r \<and> (\<forall>x \<in> set_mset(mset_tree l + mset_tree r). m \<le> x))"
 
 fun ltree :: "'a lheap \<Rightarrow> bool" where
 "ltree Leaf = True" |
-"ltree (Node n l a r) =
+"ltree (Node l a n r) =
  (n = rank r + 1 \<and> rank l \<ge> rank r \<and> ltree l & ltree r)"
 
 definition node :: "'a lheap \<Rightarrow> 'a \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
 "node l a r =
  (let rl = rk l; rr = rk r
-  in if rl \<ge> rr then Node (rr+1) l a r else Node (rl+1) r a l)"
+  in if rl \<ge> rr then Node l a (rr+1) r else Node r a (rl+1) l)"
 
 fun get_min :: "'a lheap \<Rightarrow> 'a" where
-"get_min(Node n l a r) = a"
+"get_min(Node l a n r) = a"
 
 text \<open>For function \<open>merge\<close>:\<close>
 unbundle pattern_aliases
@@ -51,25 +51,25 @@
 fun merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
 "merge Leaf t2 = t2" |
 "merge t1 Leaf = t1" |
-"merge (Node n1 l1 a1 r1 =: t1) (Node n2 l2 a2 r2 =: t2) =
+"merge (Node l1 a1 n1 r1 =: t1) (Node l2 a2 n2 r2 =: t2) =
    (if a1 \<le> a2 then node l1 a1 (merge r1 t2)
     else node l2 a2 (merge r2 t1))"
 
 lemma merge_code: "merge t1 t2 = (case (t1,t2) of
   (Leaf, _) \<Rightarrow> t2 |
   (_, Leaf) \<Rightarrow> t1 |
-  (Node n1 l1 a1 r1, Node n2 l2 a2 r2) \<Rightarrow>
+  (Node l1 a1 n1 r1, Node l2 a2 n2 r2) \<Rightarrow>
     if a1 \<le> a2 then node l1 a1 (merge r1 t2) else node l2 a2 (merge r2 t1))"
 by(induction t1 t2 rule: merge.induct) (simp_all split: tree.split)
 
 hide_const (open) insert
 
 definition insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> 'a lheap" where
-"insert x t = merge (Node 1 Leaf x Leaf) t"
+"insert x t = merge (Node Leaf x 1 Leaf) t"
 
 fun del_min :: "'a::ord lheap \<Rightarrow> 'a lheap" where
 "del_min Leaf = Leaf" |
-"del_min (Node n l x r) = merge l r"
+"del_min (Node l x n r) = merge l r"
 
 
 subsection "Lemmas"
@@ -104,7 +104,7 @@
 
 lemma ltree_merge: "\<lbrakk> ltree l; ltree r \<rbrakk> \<Longrightarrow> ltree (merge l r)"
 proof(induction l r rule: merge.induct)
-  case (3 n1 l1 a1 r1 n2 l2 a2 r2)
+  case (3 l1 a1 n1 r1 l2 a2 n2 r2)
   show ?case (is "ltree(merge ?t1 ?t2)")
   proof cases
     assume "a1 \<le> a2"
@@ -173,14 +173,14 @@
 proof(induction t)
   case Leaf show ?case by simp
 next
-  case (Node n l a r)
+  case (Node l a n r)
   hence "rank r \<le> rank l" by simp
   hence *: "(2::nat) ^ rank r \<le> 2 ^ rank l" by simp
-  have "(2::nat) ^ rank \<langle>n, l, a, r\<rangle> = 2 ^ rank r + 2 ^ rank r"
+  have "(2::nat) ^ rank \<langle>l, a, n, r\<rangle> = 2 ^ rank r + 2 ^ rank r"
     by(simp add: mult_2)
   also have "\<dots> \<le> size1 l + size1 r"
     using Node * by (simp del: power_increasing_iff)
-  also have "\<dots> = size1 \<langle>n, l, a, r\<rangle>" by simp
+  also have "\<dots> = size1 \<langle>l, a, n, r\<rangle>" by simp
   finally show ?case .
 qed
 
@@ -189,16 +189,16 @@
 fun t_merge :: "'a::ord lheap \<Rightarrow> 'a lheap \<Rightarrow> nat" where
 "t_merge Leaf t2 = 1" |
 "t_merge t2 Leaf = 1" |
-"t_merge (Node n1 l1 a1 r1 =: t1) (Node n2 l2 a2 r2 =: t2) =
+"t_merge (Node l1 a1 n1 r1 =: t1) (Node l2 a2 n2 r2 =: t2) =
   (if a1 \<le> a2 then 1 + t_merge r1 t2
    else 1 + t_merge r2 t1)"
 
 definition t_insert :: "'a::ord \<Rightarrow> 'a lheap \<Rightarrow> nat" where
-"t_insert x t = t_merge (Node 1 Leaf x Leaf) t"
+"t_insert x t = t_merge (Node Leaf x 1 Leaf) t"
 
 fun t_del_min :: "'a::ord lheap \<Rightarrow> nat" where
 "t_del_min Leaf = 1" |
-"t_del_min (Node n l a r) = t_merge l r"
+"t_del_min (Node l a n r) = t_merge l r"
 
 lemma t_merge_rank: "t_merge l r \<le> rank l + rank r + 1"
 proof(induction l r rule: merge.induct)
@@ -213,7 +213,7 @@
 by linarith
 
 corollary t_insert_log: "ltree t \<Longrightarrow> t_insert x t \<le> log 2 (size1 t) + 2"
-using t_merge_log[of "Node 1 Leaf x Leaf" t]
+using t_merge_log[of "Node Leaf x 1 Leaf" t]
 by(simp add: t_insert_def split: tree.split)
 
 (* FIXME mv ? *)
@@ -234,7 +234,7 @@
 proof(cases t)
   case Leaf thus ?thesis using assms by simp
 next
-  case [simp]: (Node _ t1 _ t2)
+  case [simp]: (Node t1 _ _ t2)
   have "t_del_min t = t_merge t1 t2" by simp
   also have "\<dots> \<le> log 2 (size1 t1) + log 2 (size1 t2) + 1"
     using \<open>ltree t\<close> by (auto simp: t_merge_log simp del: t_merge.simps)
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/Lookup2.thy
--- a/src/HOL/Data_Structures/Lookup2.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/Lookup2.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -11,7 +11,7 @@
 
 fun lookup :: "('a::linorder * 'b, 'c) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
 "lookup Leaf x = None" |
-"lookup (Node _ l (a,b) r) x =
+"lookup (Node l (a,b) _ r) x =
   (case cmp x a of LT \<Rightarrow> lookup l x | GT \<Rightarrow> lookup r x | EQ \<Rightarrow> Some b)"
 
 lemma lookup_map_of:
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/RBT.thy
--- a/src/HOL/Data_Structures/RBT.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/RBT.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -10,8 +10,8 @@
 
 type_synonym 'a rbt = "('a,color)tree"
 
-abbreviation R where "R l a r \<equiv> Node Red l a r"
-abbreviation B where "B l a r \<equiv> Node Black l a r"
+abbreviation R where "R l a r \<equiv> Node l a Red r"
+abbreviation B where "B l a r \<equiv> Node l a Black r"
 
 fun baliL :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "baliL (R (R t1 a1 t2) a2 t3) a3 t4 = R (B t1 a1 t2) a2 (B t3 a3 t4)" |
@@ -25,7 +25,7 @@
 
 fun paint :: "color \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "paint c Leaf = Leaf" |
-"paint c (Node _ l a r) = Node c l a r"
+"paint c (Node l a _ r) = Node l a c r"
 
 fun baldL :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "baldL (R t1 x t2) y t3 = R (B t1 x t2) y t3" |
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/RBT_Map.thy
--- a/src/HOL/Data_Structures/RBT_Map.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/RBT_Map.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -27,7 +27,7 @@
 and delR :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
 where
 "del x Leaf = Leaf" |
-"del x (Node c t1 (a,b) t2) = (case cmp x a of
+"del x (Node t1 (a,b) c t2) = (case cmp x a of
   LT \<Rightarrow> delL x t1 (a,b) t2 |
   GT \<Rightarrow> delR x t1 (a,b) t2 |
   EQ \<Rightarrow> combine t1 t2)" |
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/RBT_Set.thy
--- a/src/HOL/Data_Structures/RBT_Set.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/RBT_Set.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -28,11 +28,11 @@
 
 fun color :: "'a rbt \<Rightarrow> color" where
 "color Leaf = Black" |
-"color (Node c _ _ _) = c"
+"color (Node _ _ c _) = c"
 
 fun del :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "del x Leaf = Leaf" |
-"del x (Node _ l a r) =
+"del x (Node l a _ r) =
   (case cmp x a of
      LT \<Rightarrow> if l \<noteq> Leaf \<and> color l = Black
            then baldL (del x l) a r else R (del x l) a r |
@@ -97,20 +97,20 @@
 
 fun bheight :: "'a rbt \<Rightarrow> nat" where
 "bheight Leaf = 0" |
-"bheight (Node c l x r) = (if c = Black then bheight l + 1 else bheight l)"
+"bheight (Node l x c r) = (if c = Black then bheight l + 1 else bheight l)"
 
 fun invc :: "'a rbt \<Rightarrow> bool" where
 "invc Leaf = True" |
-"invc (Node c l a r) =
+"invc (Node l a c r) =
   (invc l \<and> invc r \<and> (c = Red \<longrightarrow> color l = Black \<and> color r = Black))"
 
 fun invc2 :: "'a rbt \<Rightarrow> bool" \<comment> \<open>Weaker version\<close> where
 "invc2 Leaf = True" |
-"invc2 (Node c l a r) = (invc l \<and> invc r)"
+"invc2 (Node l a c r) = (invc l \<and> invc r)"
 
 fun invh :: "'a rbt \<Rightarrow> bool" where
 "invh Leaf = True" |
-"invh (Node c l x r) = (invh l \<and> invh r \<and> bheight l = bheight r)"
+"invh (Node l x c r) = (invh l \<and> invh r \<and> bheight l = bheight r)"
 
 lemma invc2I: "invc t \<Longrightarrow> invc2 t"
 by (cases t) simp+
@@ -232,7 +232,7 @@
    (color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or>
     color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
 proof (induct x t rule: del.induct)
-case (2 x c _ y)
+case (2 x _ y c)
   have "x = y \<or> x < y \<or> x > y" by auto
   thus ?case proof (elim disjE)
     assume "x = y"
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/Set2_Join.thy
--- a/src/HOL/Data_Structures/Set2_Join.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/Set2_Join.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -28,7 +28,7 @@
   \<Longrightarrow> bst (join l a r)"
 assumes inv_Leaf: "inv \<langle>\<rangle>"
 assumes inv_join: "\<lbrakk> inv l; inv r \<rbrakk> \<Longrightarrow> inv (join l k r)"
-assumes inv_Node: "\<lbrakk> inv (Node h l x r) \<rbrakk> \<Longrightarrow> inv l \<and> inv r"
+assumes inv_Node: "\<lbrakk> inv (Node l x h r) \<rbrakk> \<Longrightarrow> inv l \<and> inv r"
 begin
 
 declare set_join [simp]
@@ -36,7 +36,7 @@
 subsection "\<open>split_min\<close>"
 
 fun split_min :: "('a,'b) tree \<Rightarrow> 'a \<times> ('a,'b) tree" where
-"split_min (Node _ l x r) =
+"split_min (Node l x _ r) =
   (if l = Leaf then (x,r) else let (m,l') = split_min l in (m, join l' x r))"
 
 lemma split_min_set:
@@ -84,7 +84,7 @@
 
 fun split :: "('a,'b)tree \<Rightarrow> 'a \<Rightarrow> ('a,'b)tree \<times> bool \<times> ('a,'b)tree" where
 "split Leaf k = (Leaf, False, Leaf)" |
-"split (Node _ l a r) k =
+"split (Node l a _ r) k =
   (if k < a then let (l1,b,l2) = split l k in (l1, b, join l2 a r) else
    if a < k then let (r1,b,r2) = split r k in (join l a r1, b, r2)
    else (l, True, r))"
@@ -145,7 +145,7 @@
 "union t1 t2 =
   (if t1 = Leaf then t2 else
    if t2 = Leaf then t1 else
-   case t1 of Node _ l1 k r1 \<Rightarrow>
+   case t1 of Node l1 k _ r1 \<Rightarrow>
    let (l2,_ ,r2) = split t2 k;
        l' = union l1 l2; r' = union r1 r2
    in join l' k r')"
@@ -181,7 +181,7 @@
 "inter t1 t2 =
   (if t1 = Leaf then Leaf else
    if t2 = Leaf then Leaf else
-   case t1 of Node _ l1 k r1 \<Rightarrow>
+   case t1 of Node l1 k _ r1 \<Rightarrow>
    let (l2,kin,r2) = split t2 k;
        l' = inter l1 l2; r' = inter r1 r2
    in if kin then join l' k r' else join2 l' r')"
@@ -196,7 +196,7 @@
   proof (cases t1)
     case Leaf thus ?thesis by (simp add: inter.simps)
   next
-    case [simp]: (Node _ l1 k r1)
+    case [simp]: (Node l1 k _ r1)
     show ?thesis
     proof (cases "t2 = Leaf")
       case True thus ?thesis by (simp add: inter.simps)
@@ -246,7 +246,7 @@
 "diff t1 t2 =
   (if t1 = Leaf then Leaf else
    if t2 = Leaf then t1 else
-   case t2 of Node _ l2 k r2 \<Rightarrow>
+   case t2 of Node l2 k _ r2 \<Rightarrow>
    let (l1,_,r1) = split t1 k;
        l' = diff l1 l2; r' = diff r1 r2
    in join2 l' r')"
@@ -261,7 +261,7 @@
   proof (cases t2)
     case Leaf thus ?thesis by (simp add: diff.simps)
   next
-    case [simp]: (Node _ l2 k r2)
+    case [simp]: (Node l2 k _ r2)
     show ?thesis
     proof (cases "t1 = Leaf")
       case True thus ?thesis by (simp add: diff.simps)
@@ -341,7 +341,7 @@
 end
 
 interpretation unbal: Set2_Join
-where join = "\<lambda>l x r. Node () l x r" and inv = "\<lambda>t. True"
+where join = "\<lambda>l x r. Node l x () r" and inv = "\<lambda>t. True"
 proof (standard, goal_cases)
   case 1 show ?case by simp
 next
diff -r 07f8c09e3f79 -r b001bef9aa39 src/HOL/Data_Structures/Tree2.thy
--- a/src/HOL/Data_Structures/Tree2.thy	Mon Jun 11 14:49:34 2018 +0200
+++ b/src/HOL/Data_Structures/Tree2.thy	Mon Jun 11 16:29:38 2018 +0200
@@ -4,30 +4,30 @@
 
 datatype ('a,'b) tree =
   Leaf ("\<langle>\<rangle>") |
-  Node 'b "('a,'b)tree" 'a "('a,'b) tree" ("(1\<langle>_,/ _,/ _,/ _\<rangle>)")
+  Node "('a,'b)tree" 'a 'b "('a,'b) tree" ("(1\<langle>_,/ _,/ _,/ _\<rangle>)")
 
 fun inorder :: "('a,'b)tree \<Rightarrow> 'a list" where
 "inorder Leaf = []" |
-"inorder (Node _ l a r) = inorder l @ a # inorder r"
+"inorder (Node l a _ r) = inorder l @ a # inorder r"
 
 fun height :: "('a,'b) tree \<Rightarrow> nat" where
 "height Leaf = 0" |
-"height (Node _ l a r) = max (height l) (height r) + 1"
+"height (Node l a _ r) = max (height l) (height r) + 1"
 
 fun set_tree :: "('a,'b) tree \<Rightarrow> 'a set" where
 "set_tree Leaf = {}" |
-"set_tree (Node _ l x r) = Set.insert x (set_tree l \<union> set_tree r)"
+"set_tree (Node l a _ r) = Set.insert a (set_tree l \<union> set_tree r)"
 
 fun bst :: "('a::linorder,'b) tree \<Rightarrow> bool" where
 "bst Leaf = True" |
-"bst (Node _ l a r) = (bst l \<and> bst r \<and> (\<forall>x \<in> set_tree l. x < a) \<and> (\<forall>x \<in> set_tree r. a < x))"
+"bst (Node l a _ r) = (bst l \<and> bst r \<and> (\<forall>x \<in> set_tree l. x < a) \<and> (\<forall>x \<in> set_tree r. a < x))"
 
 definition size1 :: "('a,'b) tree \<Rightarrow> nat" where
 "size1 t = size t + 1"
 
 lemma size1_simps[simp]:
   "size1 \<langle>\<rangle> = 1"
-  "size1 \<langle>u, l, x, r\<rangle> = size1 l + size1 r"
+  "size1 \<langle>l, x, u, r\<rangle> = size1 l + size1 r"
 by (simp_all add: size1_def)
 
 lemma size1_ge0[simp]: "0 < size1 t"