diff --git a/src/HOL/Data_Structures/AVL_Set.thy b/src/HOL/Data_Structures/AVL_Set.thy
--- a/src/HOL/Data_Structures/AVL_Set.thy
+++ b/src/HOL/Data_Structures/AVL_Set.thy
@@ -1,555 +1,555 @@
 (*
 Author:     Tobias Nipkow, Daniel Stüwe
 Largely derived from AFP entry AVL.
 *)
 
 section "AVL Tree Implementation of Sets"
 
 theory AVL_Set
 imports
   Cmp
   Isin2
   "HOL-Number_Theory.Fib"
 begin
 
 type_synonym 'a avl_tree = "('a*nat) tree"
 
 definition empty :: "'a avl_tree" where
 "empty = Leaf"
 
 text \<open>Invariant:\<close>
 
 fun avl :: "'a avl_tree \<Rightarrow> bool" where
 "avl Leaf = True" |
-"avl (Node l (a,h) r) =
+"avl (Node l (a,n) 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)"
+  n = 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 l (a,h) r) = h"
+"ht (Node l (a,n) r) = n"
 
 definition node :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
 "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>
          if ht bl < ht br then
            case br of
              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)"
 
 definition balR :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
 "balR l a r =
    (if ht r = ht l + 2 then
       case r of
         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)
           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 Leaf (x, 1) Leaf" |
-"insert x (Node l (a, h) r) = (case cmp x a of
-   EQ \<Rightarrow> Node l (a, h) r |
+"insert x (Node l (a, n) r) = (case cmp x a of
+   EQ \<Rightarrow> Node l (a, n) 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) =
   (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 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)"
+"del_root (Node Leaf (a,_) r) = r" |
+"del_root (Node l (a,_) Leaf) = l" |
+"del_root (Node l (a,_) r) = (let (l', a') = split_max l in balR l' a' r)"
 
 lemmas del_root_cases = del_root.cases[split_format(complete), 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 l (a, h) r) =
+"delete x (Node l (a, n) r) =
   (case cmp x a of
-     EQ \<Rightarrow> del_root (Node l (a, h) r) |
+     EQ \<Rightarrow> del_root (Node l (a, n) r) |
      LT \<Rightarrow> balR (delete x l) a r |
      GT \<Rightarrow> balL l a (delete x r))"
 
 
 subsection \<open>Functional Correctness Proofs\<close>
 
 text\<open>Very different from the AFP/AVL proofs\<close>
 
 
 subsubsection "Proofs for insert"
 
 lemma inorder_balL:
   "inorder (balL l a r) = inorder l @ a # inorder r"
 by (auto simp: node_def balL_def split:tree.splits)
 
 lemma inorder_balR:
   "inorder (balR l a r) = inorder l @ a # inorder r"
 by (auto simp: node_def balR_def split:tree.splits)
 
 theorem inorder_insert:
   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
 by (induct t) 
    (auto simp: ins_list_simps inorder_balL inorder_balR)
 
 
 subsubsection "Proofs for delete"
 
 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: split_max.induct)
   (auto simp: inorder_balL split: if_splits prod.splits tree.split)
 
 lemma inorder_del_root:
   "inorder (del_root (Node l ah r)) = inorder l @ inorder r"
 by(cases "Node l ah r" rule: del_root.cases)
   (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_split_maxD split: prod.splits)
 
 
 subsection \<open>AVL invariants\<close>
 
 text\<open>Essentially the AFP/AVL proofs\<close>
 
 
 subsubsection \<open>Insertion maintains AVL balance\<close>
 
 declare Let_def [simp]
 
 lemma ht_height[simp]: "avl t \<Longrightarrow> ht t = height t"
 by (cases t rule: tree2_cases) simp_all
 
 lemma height_balL:
   "\<lbrakk> height l = height r + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
    height (balL l a r) = height r + 2 \<or>
    height (balL l a r) = height r + 3"
 by (cases l) (auto simp:node_def balL_def split:tree.split)
        
 lemma height_balR:
   "\<lbrakk> height r = height l + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
    height (balR l a r) = height l + 2 \<or>
    height (balR l a r) = height l + 3"
 by (cases r) (auto simp add:node_def balR_def split:tree.split)
 
 lemma height_node[simp]: "height(node l a r) = max (height l) (height r) + 1"
 by (simp add: node_def)
 
 lemma avl_node:
   "\<lbrakk> avl l; avl r;
      height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1
    \<rbrakk> \<Longrightarrow> avl(node l a r)"
 by (auto simp add:max_def node_def)
 
 lemma height_balL2:
   "\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
    height (balL l a r) = (1 + max (height l) (height r))"
 by (cases l, cases r) (simp_all add: balL_def)
 
 lemma height_balR2:
   "\<lbrakk> avl l;  avl r;  height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
    height (balR l a r) = (1 + max (height l) (height r))"
 by (cases l, cases r) (simp_all add: balR_def)
 
 lemma avl_balL: 
   assumes "avl l" "avl r" and "height l = height r \<or> height l = height r + 1
     \<or> height r = height l + 1 \<or> height l = height r + 2" 
   shows "avl(balL l a r)"
 proof(cases l rule: tree2_cases)
   case Leaf
   with assms show ?thesis by (simp add: node_def balL_def)
 next
   case Node
   with assms show ?thesis
   proof(cases "height l = height r + 2")
     case True
     from True Node assms show ?thesis
       by (auto simp: balL_def intro!: avl_node split: tree.split) arith+
   next
     case False
     with assms show ?thesis by (simp add: avl_node balL_def)
   qed
 qed
 
 lemma avl_balR: 
   assumes "avl l" and "avl r" and "height l = height r \<or> height l = height r + 1
     \<or> height r = height l + 1 \<or> height r = height l + 2" 
   shows "avl(balR l a r)"
 proof(cases r rule: tree2_cases)
   case Leaf
   with assms show ?thesis by (simp add: node_def balR_def)
 next
   case Node
   with assms show ?thesis
   proof(cases "height r = height l + 2")
     case True
       from True Node assms show ?thesis
         by (auto simp: balR_def intro!: avl_node split: tree.split) arith+
   next
     case False
     with assms show ?thesis by (simp add: balR_def avl_node)
   qed
 qed
 
 (* It appears that these two properties need to be proved simultaneously: *)
 
 text\<open>Insertion maintains the AVL property:\<close>
 
 theorem avl_insert:
   assumes "avl t"
   shows "avl(insert x t)"
         "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
 using assms
 proof (induction t rule: tree2_induct)
-  case (Node l a h r)
+  case (Node l a _ r)
   case 1
   show ?case
   proof(cases "x = a")
     case True with Node 1 show ?thesis by simp
   next
     case False
     show ?thesis 
     proof(cases "x<a")
       case True with Node 1 show ?thesis by (auto simp add:avl_balL)
     next
       case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balR)
     qed
   qed
   case 2
   show ?case
   proof(cases "x = a")
     case True with Node 1 show ?thesis by simp
   next
     case False
     show ?thesis 
     proof(cases "x<a")
       case True
       show ?thesis
       proof(cases "height (insert x l) = height r + 2")
         case False with Node 2 \<open>x < a\<close> show ?thesis by (auto simp: height_balL2)
       next
         case True 
         hence "(height (balL (insert x l) a r) = height r + 2) \<or>
           (height (balL (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
           using Node 2 by (intro height_balL) simp_all
         thus ?thesis
         proof
           assume ?A with 2 \<open>x < a\<close> show ?thesis by (auto)
         next
           assume ?B with True 1 Node(2) \<open>x < a\<close> show ?thesis by (simp) arith
         qed
       qed
     next
       case False
       show ?thesis 
       proof(cases "height (insert x r) = height l + 2")
         case False with Node 2 \<open>\<not>x < a\<close> show ?thesis by (auto simp: height_balR2)
       next
         case True 
         hence "(height (balR l a (insert x r)) = height l + 2) \<or>
           (height (balR l a (insert x r)) = height l + 3)"  (is "?A \<or> ?B")
           using Node 2 by (intro height_balR) simp_all
         thus ?thesis 
         proof
           assume ?A with 2 \<open>\<not>x < a\<close> show ?thesis by (auto)
         next
           assume ?B with True 1 Node(4) \<open>\<not>x < a\<close> show ?thesis by (simp) arith
         qed
       qed
     qed
   qed
 qed simp_all
 
 
 subsubsection \<open>Deletion maintains AVL balance\<close>
 
 lemma avl_split_max:
   assumes "avl x" and "x \<noteq> Leaf"
   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: split_max_induct)
-  case (Node l a h r)
+  case Node
   case 1
   thus ?case using Node
     by (auto simp: height_balL height_balL2 avl_balL split:prod.split)
 next
-  case (Node l a h r)
+  case (Node l a _ 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
   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_del_root:
   assumes "avl t" and "t \<noteq> Leaf"
   shows "avl(del_root t)" 
 using assms
 proof (cases t rule:del_root_cases)
   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
   hence "avl(?l')" "height ?l = height(?l') \<or>
          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(split_max ?l)) ?r)"
     by (rule avl_balR)
   with Node_Node show ?thesis by (auto split:prod.splits)
 qed simp_all
 
 lemma height_del_root:
   assumes "avl t" and "t \<noteq> Leaf" 
   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 ll ln lh lr n h rl rn rh rr)
-  let ?l = "Node ll (ln, lh) lr"
-  let ?r = "Node rl (rn, rh) rr"
+  case (Node_Node ll lx lh lr x h rl rx rh rr)
+  let ?l = "Node ll (lx, lh) lr"
+  let ?r = "Node rl (rx, 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
   from \<open>avl t\<close> and Node_Node have "avl ?l" by simp
   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")
     case False
     show ?thesis using l'_height t_height False
       by (subst height_balR2[OF \<open>avl ?l'\<close> \<open>avl ?r\<close> False])+ arith
   next
     case True
     show ?thesis
     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
       case 2 thus ?thesis using l'_height t_height True by arith
     qed
   qed
   thus ?thesis using Node_Node by (auto split:prod.splits)
 qed simp_all
 
 text\<open>Deletion maintains the AVL property:\<close>
 
 theorem avl_delete:
   assumes "avl t" 
   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 rule: tree2_induct)
-  case (Node l n h r)
+  case (Node l a n r)
   case 1
   show ?case
-  proof(cases "x = n")
+  proof(cases "x = a")
     case True with Node 1 show ?thesis by (auto simp:avl_del_root)
   next
     case False
     show ?thesis 
-    proof(cases "x<n")
+    proof(cases "x<a")
       case True with Node 1 show ?thesis by (auto simp add:avl_balR)
     next
-      case False with Node 1 \<open>x\<noteq>n\<close> show ?thesis by (auto simp add:avl_balL)
+      case False with Node 1 \<open>x\<noteq>a\<close> show ?thesis by (auto simp add:avl_balL)
     qed
   qed
   case 2
   show ?case
-  proof(cases "x = n")
+  proof(cases "x = a")
     case True
-    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"
+    with 1 have "height (Node l (a,n) r) = height(del_root (Node l (a,n) r))
+      \<or> height (Node l (a,n) r) = height(del_root (Node l (a,n) r)) + 1"
       by (subst height_del_root,simp_all)
     with True show ?thesis by simp
   next
     case False
     show ?thesis 
-    proof(cases "x<n")
+    proof(cases "x<a")
       case True
       show ?thesis
       proof(cases "height r = height (delete x l) + 2")
-        case False with Node 1 \<open>x < n\<close> show ?thesis by(auto simp: balR_def)
+        case False with Node 1 \<open>x < a\<close> show ?thesis by(auto simp: balR_def)
       next
         case True 
-        hence "(height (balR (delete x l) n r) = height (delete x l) + 2) \<or>
-          height (balR (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
+        hence "(height (balR (delete x l) a r) = height (delete x l) + 2) \<or>
+          height (balR (delete x l) a r) = height (delete x l) + 3" (is "?A \<or> ?B")
           using Node 2 by (intro height_balR) auto
         thus ?thesis 
         proof
-          assume ?A with \<open>x < n\<close> Node 2 show ?thesis by(auto simp: balR_def)
+          assume ?A with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
         next
-          assume ?B with \<open>x < n\<close> Node 2 show ?thesis by(auto simp: balR_def)
+          assume ?B with \<open>x < a\<close> Node 2 show ?thesis by(auto simp: balR_def)
         qed
       qed
     next
       case False
       show ?thesis
       proof(cases "height l = height (delete x r) + 2")
-        case False with Node 1 \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> show ?thesis by(auto simp: balL_def)
+        case False with Node 1 \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> show ?thesis by(auto simp: balL_def)
       next
         case True 
-        hence "(height (balL l n (delete x r)) = height (delete x r) + 2) \<or>
-          height (balL l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
+        hence "(height (balL l a (delete x r)) = height (delete x r) + 2) \<or>
+          height (balL l a (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
           using Node 2 by (intro height_balL) auto
         thus ?thesis 
         proof
-          assume ?A with \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> Node 2 show ?thesis by(auto simp: balL_def)
+          assume ?A with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
         next
-          assume ?B with \<open>\<not>x < n\<close> \<open>x \<noteq> n\<close> Node 2 show ?thesis by(auto simp: balL_def)
+          assume ?B with \<open>\<not>x < a\<close> \<open>x \<noteq> a\<close> Node 2 show ?thesis by(auto simp: balL_def)
         qed
       qed
     qed
   qed
 qed simp_all
 
 
 subsection "Overall correctness"
 
 interpretation S: Set_by_Ordered
 where empty = empty and isin = isin and insert = insert and delete = delete
 and inorder = inorder and inv = avl
 proof (standard, goal_cases)
   case 1 show ?case by (simp add: empty_def)
 next
   case 2 thus ?case by(simp add: isin_set_inorder)
 next
   case 3 thus ?case by(simp add: inorder_insert)
 next
   case 4 thus ?case by(simp add: inorder_delete)
 next
   case 5 thus ?case by (simp add: empty_def)
 next
   case 6 thus ?case by (simp add: avl_insert(1))
 next
   case 7 thus ?case by (simp add: avl_delete(1))
 qed
 
 
 subsection \<open>Height-Size Relation\<close>
 
 text \<open>Based on theorems by Daniel St\"uwe, Manuel Eberl and Peter Lammich.\<close>
 
 lemma height_invers: 
   "(height t = 0) = (t = Leaf)"
-  "avl t \<Longrightarrow> (height t = Suc h) = (\<exists> l a r . t = Node l (a,Suc h) r)"
+  "avl t \<Longrightarrow> (height t = Suc n) = (\<exists> l a r . t = Node l (a,Suc n) 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>
+text \<open>Any AVL tree of height \<open>n\<close> has at least \<open>fib (n+2)\<close> leaves:\<close>
 
-lemma avl_fib_bound: "avl t \<Longrightarrow> height t = h \<Longrightarrow> fib (h+2) \<le> size1 t"
-proof (induction h arbitrary: t rule: fib.induct)
+lemma avl_fib_bound: "avl t \<Longrightarrow> height t = n \<Longrightarrow> fib (n+2) \<le> size1 t"
+proof (induction n arbitrary: t rule: fib.induct)
   case 1 thus ?case by (simp add: height_invers)
 next
   case 2 thus ?case by (cases t) (auto simp: height_invers)
 next
   case (3 h)
   from "3.prems" obtain l a r where
     [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
     \<or> height r = h \<and> height l = Suc h" (is "?C1 \<or> ?C2 \<or> ?C3")
     by (cases t) (simp, fastforce)
   {
     assume ?C1
     with "3.IH"(1)
     have "fib (h + 3) \<le> size1 l" "fib (h + 3) \<le> size1 r"
       by (simp_all add: eval_nat_numeral)
     hence ?case by (auto simp: eval_nat_numeral)
   } moreover {
     assume ?C2
     hence ?case using "3.IH"(1)[of r] "3.IH"(2)[of l] by auto
   } moreover {
     assume ?C3
     hence ?case using "3.IH"(1)[of l] "3.IH"(2)[of r] by auto
   } ultimately show ?case using C by blast
 qed
 
 lemma fib_alt_induct [consumes 1, case_names 1 2 rec]:
   assumes "n > 0" "P 1" "P 2" "\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc n) \<Longrightarrow> P (Suc (Suc n))"
   shows   "P n"
   using assms(1)
 proof (induction n rule: fib.induct)
   case (3 n)
   thus ?case using assms by (cases n) (auto simp: eval_nat_numeral)
 qed (insert assms, auto)
 
 text \<open>An exponential lower bound for \<^const>\<open>fib\<close>:\<close>
 
 lemma fib_lowerbound:
   defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
   defines "c \<equiv> 1 / \<phi> ^ 2"
   assumes "n > 0"
   shows   "real (fib n) \<ge> c * \<phi> ^ n"
 proof -
   have "\<phi> > 1" by (simp add: \<phi>_def)
   hence "c > 0" by (simp add: c_def)
   from \<open>n > 0\<close> show ?thesis
   proof (induction n rule: fib_alt_induct)
     case (rec n)
     have "c * \<phi> ^ Suc (Suc n) = \<phi> ^ 2 * (c * \<phi> ^ n)"
       by (simp add: field_simps power2_eq_square)
     also have "\<dots> \<le> (\<phi> + 1) * (c * \<phi> ^ n)"
       by (rule mult_right_mono) (insert \<open>c > 0\<close>, simp_all add: \<phi>_def power2_eq_square field_simps)
     also have "\<dots> = c * \<phi> ^ Suc n + c * \<phi> ^ n"
       by (simp add: field_simps)
     also have "\<dots> \<le> real (fib (Suc n)) + real (fib n)"
       by (intro add_mono rec.IH)
     finally show ?case by simp
   qed (insert \<open>\<phi> > 1\<close>, simp_all add: c_def power2_eq_square eval_nat_numeral)
 qed
 
 text \<open>The size of an AVL tree is (at least) exponential in its height:\<close>
 
 lemma avl_size_lowerbound:
   defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
   assumes "avl t"
   shows   "\<phi> ^ (height t) \<le> size1 t"
 proof -
   have "\<phi> > 0" by(simp add: \<phi>_def add_pos_nonneg)
   hence "\<phi> ^ height t = (1 / \<phi> ^ 2) * \<phi> ^ (height t + 2)"
     by(simp add: field_simps power2_eq_square)
   also have "\<dots> \<le> fib (height t + 2)"
     using fib_lowerbound[of "height t + 2"] by(simp add: \<phi>_def)
   also have "\<dots> \<le> size1 t"
     using avl_fib_bound[of t "height t"] assms by simp
   finally show ?thesis .
 qed
 
 text \<open>The height of an AVL tree is most \<^term>\<open>(1/log 2 \<phi>)\<close> \<open>\<approx> 1.44\<close> times worse
 than \<^term>\<open>log 2 (size1 t)\<close>:\<close>
 
 lemma  avl_height_upperbound:
   defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
   assumes "avl t"
   shows   "height t \<le> (1/log 2 \<phi>) * log 2 (size1 t)"
 proof -
   have "\<phi> > 0" "\<phi> > 1" by(auto simp: \<phi>_def pos_add_strict)
   hence "height t = log \<phi> (\<phi> ^ height t)" by(simp add: log_nat_power)
   also have "\<dots> \<le> log \<phi> (size1 t)"
     using avl_size_lowerbound[OF assms(2), folded \<phi>_def] \<open>1 < \<phi>\<close>
     by (simp add: le_log_of_power) 
   also have "\<dots> = (1/log 2 \<phi>) * log 2 (size1 t)"
     by(simp add: log_base_change[of 2 \<phi>])
   finally show ?thesis .
 qed
 
 end
diff --git a/src/Pure/Syntax/parser.ML b/src/Pure/Syntax/parser.ML
--- a/src/Pure/Syntax/parser.ML
+++ b/src/Pure/Syntax/parser.ML
@@ -1,719 +1,724 @@
 (*  Title:      Pure/Syntax/parser.ML
     Author:     Carsten Clasohm, Sonia Mahjoub
     Author:     Makarius
 
 General context-free parser for the inner syntax of terms and types.
 *)
 
 signature PARSER =
 sig
   type gram
   val empty_gram: gram
   val extend_gram: Syntax_Ext.xprod list -> gram -> gram
   val make_gram: Syntax_Ext.xprod list -> gram
   val pretty_gram: gram -> Pretty.T list
   datatype parsetree =
     Node of string * parsetree list |
     Tip of Lexicon.token
   exception PARSETREE of parsetree
   val pretty_parsetree: parsetree -> Pretty.T
   val parse: gram -> string -> Lexicon.token list -> parsetree list
   val branching_level: int Config.T
 end;
 
 structure Parser: PARSER =
 struct
 
 (** datatype gram **)
 
 (* nonterminals *)
 
 (*production for the NTs are stored in a vector, indexed by the NT tag*)
 type nt = int;
 
 type tags = nt Symtab.table;
 val tags_empty: tags = Symtab.empty;
 fun tags_content (tags: tags) = sort_by #1 (Symtab.dest tags);
 fun tags_lookup (tags: tags) = Symtab.lookup tags;
 fun tags_insert tag (tags: tags) = Symtab.update_new tag tags;
 fun tags_name (tags: tags) = the o Inttab.lookup (Inttab.make (map swap (Symtab.dest tags)));
 
 type nts = Inttab.set;
 val nts_empty: nts = Inttab.empty;
 val nts_merge: nts * nts -> nts = Inttab.merge (K true);
 fun nts_insert nt : nts -> nts = Inttab.insert_set nt;
 fun nts_member (nts: nts) = Inttab.defined nts;
 fun nts_fold f (nts: nts) = Inttab.fold (f o #1) nts;
 fun nts_subset (nts1: nts, nts2: nts) = Inttab.forall (nts_member nts2 o #1) nts1;
 fun nts_is_empty (nts: nts) = Inttab.is_empty nts;
 fun nts_is_unique (nts: nts) = nts_is_empty nts orelse Inttab.is_single nts;
 
 
 (* tokens *)
 
 structure Tokens = Table(type key = Lexicon.token val ord = Lexicon.tokens_match_ord);
 
 type tokens = Tokens.set;
 val tokens_empty: tokens = Tokens.empty;
 val tokens_merge: tokens * tokens -> tokens = Tokens.merge (K true);
 fun tokens_insert tk : tokens -> tokens = Tokens.insert_set tk;
 fun tokens_remove tk : tokens -> tokens = Tokens.remove_set tk;
 fun tokens_member (tokens: tokens) = Tokens.defined tokens;
 fun tokens_is_empty (tokens: tokens) = Tokens.is_empty tokens;
 fun tokens_fold f (tokens: tokens) = Tokens.fold (f o #1) tokens;
 val tokens_union = tokens_fold tokens_insert;
 val tokens_subtract = tokens_fold tokens_remove;
 fun tokens_find P (tokens: tokens) =
   Tokens.get_first (fn (tok, ()) => if P tok then SOME tok else NONE) tokens;
 fun tokens_add (nt: nt, tokens) = if tokens_is_empty tokens then I else cons (nt, tokens);
 
 
 (* productions *)
 
 datatype symb =
   Terminal of Lexicon.token |
   Nonterminal of nt * int;  (*(tag, prio)*)
 
 type prods = (symb list * string * int) list Tokens.table;  (*start_token ~> [(rhs, name, prio)]*)
 
 val prods_empty: prods = Tokens.empty;
 fun prods_lookup (prods: prods) = Tokens.lookup_list prods;
 fun prods_update entry : prods -> prods = Tokens.update entry;
 fun prods_content (prods: prods) = distinct (op =) (maps #2 (Tokens.dest prods));
 
 type nt_gram = (nts * tokens) * prods; (*dependent_nts, start_tokens, prods*)
   (*depent_nts is a set of all NTs whose lookahead depends on this NT's lookahead*)
 
 val nt_gram_empty: nt_gram = ((nts_empty, tokens_empty), prods_empty);
 
 type chains = unit Int_Graph.T;
 fun chains_preds (chains: chains) = Int_Graph.immediate_preds chains;
 fun chains_all_preds (chains: chains) = Int_Graph.all_preds chains;
 fun chains_all_succs (chains: chains) = Int_Graph.all_succs chains;
 val chains_empty: chains = Int_Graph.empty;
 fun chains_member (chains: chains) = Int_Graph.is_edge chains;
 fun chains_declare nt : chains -> chains = Int_Graph.default_node (nt, ());
 fun chains_insert (from, to) =
   chains_declare from #> chains_declare to #> Int_Graph.add_edge (from, to);
 
 datatype gram =
   Gram of
    {nt_count: int,
     prod_count: int,
     tags: tags,
     chains: chains,
     lambdas: nts,
     prods: nt_gram Vector.vector};
     (*"tags" is used to map NT names (i.e. strings) to tags;
      chain productions are not stored as normal productions
      but instead as an entry in "chains": from -> to;
      lambda productions are stored as normal productions
      and also as an entry in "lambdas"*)
 
 (*productions for which no starting token is
   known yet are associated with this token*)
 val unknown_start = Lexicon.eof;
 
 fun get_start tks =
   (case Tokens.min tks of
     SOME (tk, _) => tk
   | NONE => unknown_start);
 
 fun add_production prods (lhs, new_prod as (rhs, _, pri)) (chains, lambdas) =
   let
     (*store chain if it does not already exist*)
     val (chain, new_chain, chains') =
       (case (pri, rhs) of
         (~1, [Nonterminal (from, ~1)]) =>
           if chains_member chains (from, lhs)
           then (SOME from, false, chains)
           else (SOME from, true, chains_insert (from, lhs) chains)
-      | _ => (NONE, false, chains_declare lhs chains));
+      | _ =>
+        let
+          val chains' = chains
+            |> chains_declare lhs
+            |> fold (fn Nonterminal (nt, _) => chains_declare nt | _ => I) rhs;
+        in (NONE, false, chains') end);
 
     (*propagate new chain in lookahead and lambdas;
       added_starts is used later to associate existing
       productions with new starting tokens*)
     val (added_starts, lambdas') =
       if not new_chain then ([], lambdas)
       else
         let (*lookahead of chain's source*)
           val ((_, from_tks), _) = Array.sub (prods, the chain);
 
           (*copy from's lookahead to chain's destinations*)
           fun copy_lookahead to =
             let
               val ((to_nts, to_tks), ps) = Array.sub (prods, to);
 
               val new_tks = tokens_subtract to_tks from_tks;  (*added lookahead tokens*)
               val to_tks' = tokens_merge (to_tks, new_tks);
               val _ = Array.update (prods, to, ((to_nts, to_tks'), ps));
             in tokens_add (to, new_tks) end;
 
           val tos = chains_all_succs chains' [lhs];
         in
           (fold copy_lookahead tos [],
             lambdas |> nts_member lambdas lhs ? fold nts_insert tos)
         end;
 
     (*test if new production can produce lambda
       (rhs must either be empty or only consist of lambda NTs)*)
     val new_lambdas =
       if forall
         (fn Nonterminal (id, _) => nts_member lambdas' id
           | Terminal _ => false) rhs
       then SOME (filter_out (nts_member lambdas') (chains_all_succs chains' [lhs]))
       else NONE;
     val lambdas'' = fold nts_insert (these new_lambdas) lambdas';
 
     (*list optional terminal and all nonterminals on which the lookahead
       of a production depends*)
     fun lookahead_dependency _ [] nts = (NONE, nts)
       | lookahead_dependency _ (Terminal tk :: _) nts = (SOME tk, nts)
       | lookahead_dependency lambdas (Nonterminal (nt, _) :: symbs) nts =
           if nts_member lambdas nt then
             lookahead_dependency lambdas symbs (nts_insert nt nts)
           else (NONE, nts_insert nt nts);
 
     (*get all known starting tokens for a nonterminal*)
     fun starts_for_nt nt = snd (fst (Array.sub (prods, nt)));
 
     (*update prods, lookaheads, and lambdas according to new lambda NTs*)
     val (added_starts', lambdas') =
       let
         (*propagate added lambda NT*)
         fun propagate_lambda [] added_starts lambdas = (added_starts, lambdas)
           | propagate_lambda (l :: ls) added_starts lambdas =
               let
                 (*get lookahead for lambda NT*)
                 val ((dependent, l_starts), _) = Array.sub (prods, l);
 
                 (*check productions whose lookahead may depend on lambda NT*)
                 fun examine_prods [] add_lambda nt_dependencies added_tks nt_prods =
                       (add_lambda, nt_dependencies, added_tks, nt_prods)
                   | examine_prods ((p as (rhs, _, _)) :: ps) add_lambda
                         nt_dependencies added_tks nt_prods =
                       let val (tk, nts) = lookahead_dependency lambdas rhs nts_empty in
                         if nts_member nts l then       (*update production's lookahead*)
                           let
                             val new_lambda =
                               is_none tk andalso nts_subset (nts, lambdas);
 
                             val new_tks =
                               tokens_empty
                               |> fold tokens_insert (the_list tk)
                               |> nts_fold (tokens_union o starts_for_nt) nts
                               |> tokens_subtract l_starts;
 
                             val added_tks' = tokens_merge (added_tks, new_tks);
 
                             val nt_dependencies' = nts_merge (nt_dependencies, nts);
 
                             (*associate production with new starting tokens*)
                             fun copy tk nt_prods =
                               prods_update (tk, p :: prods_lookup nt_prods tk) nt_prods;
 
                             val nt_prods' = nt_prods
                               |> tokens_fold copy new_tks
                               |> new_lambda ? copy Lexicon.dummy;
                           in
                             examine_prods ps (add_lambda orelse new_lambda)
                               nt_dependencies' added_tks' nt_prods'
                           end
                         else (*skip production*)
                           examine_prods ps add_lambda nt_dependencies added_tks nt_prods
                       end;
 
                 (*check each NT whose lookahead depends on new lambda NT*)
                 fun process_nts nt (added_lambdas, added_starts) =
                   let
                     val ((old_nts, old_tks), nt_prods) = Array.sub (prods, nt);
 
                     (*existing productions whose lookahead may depend on l*)
                     val tk_prods = prods_lookup nt_prods (get_start l_starts);
 
                     (*add_lambda is true if an existing production of the nt
                       produces lambda due to the new lambda NT l*)
                     val (add_lambda, nt_dependencies, added_tks, nt_prods') =
                       examine_prods tk_prods false nts_empty tokens_empty nt_prods;
 
                     val new_nts = nts_merge (old_nts, nt_dependencies);
                     val new_tks = tokens_merge (old_tks, added_tks);
 
                     val added_lambdas' = added_lambdas |> add_lambda ? cons nt;
                     val _ = Array.update (prods, nt, ((new_nts, new_tks), nt_prods'));
                       (*N.B. that because the tks component
                         is used to access existing
                         productions we have to add new
                         tokens at the _end_ of the list*)
                     val added_starts' = tokens_add (nt, added_tks) added_starts;
                   in (added_lambdas', added_starts') end;
 
                 val (added_lambdas, added_starts') =
                   nts_fold process_nts dependent ([], added_starts);
                 val added_lambdas' = filter_out (nts_member lambdas) added_lambdas;
               in
                 propagate_lambda (ls @ added_lambdas') added_starts'
                   (fold nts_insert added_lambdas' lambdas)
               end;
       in
         propagate_lambda
           (nts_fold (fn l => not (nts_member lambdas l) ? cons l) lambdas'' [])
           added_starts lambdas''
       end;
 
     (*insert production into grammar*)
     val added_starts' =
       if is_some chain then added_starts'  (*don't store chain production*)
       else
         let
           (*lookahead tokens of new production and on which
             NTs lookahead depends*)
           val (start_tk, start_nts) = lookahead_dependency lambdas' rhs nts_empty;
 
           val start_tks =
             tokens_empty
             |> fold tokens_insert (the_list start_tk)
             |> nts_fold (tokens_union o starts_for_nt) start_nts;
 
           val start_tks' =
             start_tks
             |> (if is_some new_lambdas then tokens_insert Lexicon.dummy
                 else if tokens_is_empty start_tks then tokens_insert unknown_start
                 else I);
 
           (*add lhs NT to list of dependent NTs in lookahead*)
           fun add_nts nt initial =
             (if initial then
               let val ((old_nts, old_tks), ps) = Array.sub (prods, nt) in
                 if nts_member old_nts lhs then ()
                 else Array.update (prods, nt, ((nts_insert lhs old_nts, old_tks), ps))
               end
              else (); false);
 
           (*add new start tokens to chained NTs' lookahead list;
             also store new production for lhs NT*)
           fun add_tks [] added = added
             | add_tks (nt :: nts) added =
                 let
                   val ((old_nts, old_tks), nt_prods) = Array.sub (prods, nt);
 
                   val new_tks = tokens_subtract old_tks start_tks;
 
                   (*store new production*)
                   fun store tk (prods, _) =
                     let
                       val tk_prods = prods_lookup prods tk;
                       val tk_prods' = new_prod :: tk_prods;
                       val prods' = prods_update (tk, tk_prods') prods;
                     in (prods', true) end;
 
                   val (nt_prods', changed) = (nt_prods, false)
                     |> nt = lhs ? tokens_fold store start_tks';
                   val _ =
                     if not changed andalso tokens_is_empty new_tks then ()
                     else
                       Array.update
                         (prods, nt, ((old_nts, tokens_merge (old_tks, new_tks)), nt_prods'));
                 in add_tks nts (tokens_add (nt, new_tks) added) end;
           val _ = nts_fold add_nts start_nts true;
         in add_tks (chains_all_succs chains' [lhs]) [] end;
 
     (*associate productions with new lookaheads*)
     val _ =
       let
         (*propagate added start tokens*)
         fun add_starts [] = ()
           | add_starts ((changed_nt, new_tks) :: starts) =
               let
                 (*token under which old productions which
                   depend on changed_nt could be stored*)
                 val key =
                   tokens_find (not o tokens_member new_tks) (starts_for_nt changed_nt)
                   |> the_default unknown_start;
 
                 (*copy productions whose lookahead depends on changed_nt;
                   if key = SOME unknown_start then tk_prods is used to hold
                   the productions not copied*)
                 fun update_prods [] result = result
                   | update_prods ((p as (rhs, _: string, _: nt)) :: ps)
                         (tk_prods, nt_prods) =
                       let
                         (*lookahead dependency for production*)
                         val (tk, depends) = lookahead_dependency lambdas' rhs nts_empty;
 
                         (*test if this production has to be copied*)
                         val update = nts_member depends changed_nt;
 
                         (*test if production could already be associated with
                           a member of new_tks*)
                         val lambda =
                           not (nts_is_unique depends) orelse
                           not (nts_is_empty depends) andalso is_some tk
                           andalso tokens_member new_tks (the tk);
 
                         (*associate production with new starting tokens*)
                         fun copy tk nt_prods =
                           let
                             val tk_prods = prods_lookup nt_prods tk;
                             val tk_prods' =
                               if not lambda then p :: tk_prods
                               else insert (op =) p tk_prods;
                                (*if production depends on lambda NT we
                                  have to look for duplicates*)
                           in prods_update (tk, tk_prods') nt_prods end;
                         val result =
                           if update then (tk_prods, tokens_fold copy new_tks nt_prods)
                           else if key = unknown_start then (p :: tk_prods, nt_prods)
                           else (tk_prods, nt_prods);
                       in update_prods ps result end;
 
                 (*copy existing productions for new starting tokens*)
                 fun process_nts nt =
                   let
                     val ((nts, tks), nt_prods) = Array.sub (prods, nt);
 
                     val tk_prods = prods_lookup nt_prods key;
 
                     (*associate productions with new lookahead tokens*)
                     val (tk_prods', nt_prods') = update_prods tk_prods ([], nt_prods);
 
                     val nt_prods'' =
                       if key = unknown_start then
                         prods_update (key, tk_prods') nt_prods'
                       else nt_prods';
 
                     val added_tks = tokens_subtract tks new_tks;
                     val tks' = tokens_merge (tks, added_tks);
                     val _ = Array.update (prods, nt, ((nts, tks'), nt_prods''));
                   in tokens_add (nt, added_tks) end;
 
                 val ((dependent, _), _) = Array.sub (prods, changed_nt);
               in add_starts (starts @ nts_fold process_nts dependent []) end;
       in add_starts added_starts' end;
   in (chains', lambdas') end;
 
 
 (* pretty_gram *)
 
 fun pretty_gram (Gram {tags, prods, chains, ...}) =
   let
     val print_nt = tags_name tags;
     fun print_pri p = if p < 0 then "" else Symbol.make_sup ("(" ^ string_of_int p ^ ")");
 
     fun pretty_symb (Terminal tok) =
           if Lexicon.is_literal tok
           then Pretty.quote (Pretty.keyword1 (Lexicon.str_of_token tok))
           else Pretty.str (Lexicon.str_of_token tok)
       | pretty_symb (Nonterminal (tag, p)) = Pretty.str (print_nt tag ^ print_pri p);
 
     fun pretty_const "" = []
       | pretty_const c = [Pretty.str ("\<^bold>\<Rightarrow> " ^ quote c)];
 
     fun prod_of_chain from = ([Nonterminal (from, ~1)], "", ~1);
 
     fun pretty_prod (name, tag) =
       (prods_content (#2 (Vector.sub (prods, tag))) @ map prod_of_chain (chains_preds chains tag))
       |> map (fn (symbs, const, p) =>
           Pretty.block (Pretty.breaks
             (Pretty.str (name ^ print_pri p ^ " =") :: map pretty_symb symbs @ pretty_const const)));
   in maps pretty_prod (tags_content tags) end;
 
 
 
 (** operations on grammars **)
 
 val empty_gram =
   Gram
    {nt_count = 0,
     prod_count = 0,
     tags = tags_empty,
     chains = chains_empty,
     lambdas = nts_empty,
     prods = Vector.fromList [nt_gram_empty]};
 
 
 (*Add productions to a grammar*)
 fun extend_gram [] gram = gram
   | extend_gram xprods (Gram {nt_count, prod_count, tags, chains, lambdas, prods}) =
       let
         (*Get tag for existing nonterminal or create a new one*)
         fun get_tag nt_count tags nt =
           (case tags_lookup tags nt of
             SOME tag => (nt_count, tags, tag)
           | NONE => (nt_count + 1, tags_insert (nt, nt_count) tags, nt_count));
 
         (*Convert symbols to the form used by the parser;
           delimiters and predefined terms are stored as terminals,
           nonterminals are converted to integer tags*)
         fun symb_of [] nt_count tags result = (nt_count, tags, rev result)
           | symb_of (Syntax_Ext.Delim s :: ss) nt_count tags result =
               symb_of ss nt_count tags (Terminal (Lexicon.literal s) :: result)
           | symb_of (Syntax_Ext.Argument (s, p) :: ss) nt_count tags result =
               let
                 val (nt_count', tags', new_symb) =
                   (case Lexicon.get_terminal s of
                     NONE =>
                       let val (nt_count', tags', s_tag) = get_tag nt_count tags s;
                       in (nt_count', tags', Nonterminal (s_tag, p)) end
                   | SOME tk => (nt_count, tags, Terminal tk));
               in symb_of ss nt_count' tags' (new_symb :: result) end
           | symb_of (_ :: ss) nt_count tags result = symb_of ss nt_count tags result;
 
         (*Convert list of productions by invoking symb_of for each of them*)
         fun prod_of [] nt_count prod_count tags result =
               (nt_count, prod_count, tags, result)
           | prod_of (Syntax_Ext.XProd (lhs, xsymbs, const, pri) :: ps)
                 nt_count prod_count tags result =
               let
                 val (nt_count', tags', lhs_tag) = get_tag nt_count tags lhs;
                 val (nt_count'', tags'', prods) = symb_of xsymbs nt_count' tags' [];
               in
                 prod_of ps nt_count'' (prod_count + 1) tags''
                   ((lhs_tag, (prods, const, pri)) :: result)
               end;
 
         val (nt_count', prod_count', tags', xprods') =
           prod_of xprods nt_count prod_count tags [];
 
         (*Copy array containing productions of old grammar;
           this has to be done to preserve the old grammar while being able
           to change the array's content*)
         val prods' =
           let
             fun get_prod i =
               if i < nt_count then Vector.sub (prods, i)
               else nt_gram_empty;
           in Array.tabulate (nt_count', get_prod) end;
 
         val (chains', lambdas') =
           (chains, lambdas) |> fold (add_production prods') xprods';
       in
         Gram
          {nt_count = nt_count',
           prod_count = prod_count',
           tags = tags',
           chains = chains',
           lambdas = lambdas',
           prods = Array.vector prods'}
       end;
 
 fun make_gram xprods = extend_gram xprods empty_gram;
 
 
 
 (** parser **)
 
 (* parsetree *)
 
 datatype parsetree =
   Node of string * parsetree list |
   Tip of Lexicon.token;
 
 exception PARSETREE of parsetree;
 
 fun pretty_parsetree parsetree =
   let
     fun pretty (Node (c, pts)) =
           [Pretty.enclose "(" ")" (Pretty.breaks (Pretty.quote (Pretty.str c) :: maps pretty pts))]
       | pretty (Tip tok) =
           if Lexicon.valued_token tok then [Pretty.str (Lexicon.str_of_token tok)] else [];
   in (case pretty parsetree of [prt] => prt | _ => raise PARSETREE parsetree) end;
 
 
 (* parser state *)
 
 type state =
   nt * int *    (*identification and production precedence*)
   parsetree list *  (*already parsed nonterminals on rhs*)
   symb list *       (*rest of rhs*)
   string *          (*name of production*)
   int;              (*index for previous state list*)
 
 
 (*Get all rhss with precedence >= min_prec*)
 fun get_RHS min_prec = filter (fn (_, _, prec: int) => prec >= min_prec);
 
 (*Get all rhss with precedence >= min_prec and < max_prec*)
 fun get_RHS' min_prec max_prec =
   filter (fn (_, _, prec: int) => prec >= min_prec andalso prec < max_prec);
 
 (*Make states using a list of rhss*)
 fun mk_states i lhs_ID rhss =
   let fun mk_state (rhs, id, prod_prec) = (lhs_ID, prod_prec, [], rhs, id, i);
   in map mk_state rhss end;
 
 (*Add parse tree to list and eliminate duplicates
   saving the maximum precedence*)
 fun conc (t: parsetree list, prec: int) [] = (NONE, [(t, prec)])
   | conc (t, prec) ((t', prec') :: ts) =
       if t = t' then
         (SOME prec',
           if prec' >= prec then (t', prec') :: ts
           else (t, prec) :: ts)
       else
         let val (n, ts') = conc (t, prec) ts
         in (n, (t', prec') :: ts') end;
 
 (*Update entry in used*)
 fun update_trees (A, t) used =
   let
     val (i, ts) = the (Inttab.lookup used A);
     val (n, ts') = conc t ts;
   in (n, Inttab.update (A, (i, ts')) used) end;
 
 (*Replace entry in used*)
 fun update_prec (A, prec) =
   Inttab.map_entry A (fn (_, ts) => (prec, ts));
 
 fun getS A max_prec NONE Si =
       filter
         (fn (_, _, _, Nonterminal (B, prec) :: _, _, _) => A = B andalso prec <= max_prec
           | _ => false) Si
   | getS A max_prec (SOME min_prec) Si =
       filter
         (fn (_, _, _, Nonterminal (B, prec) :: _, _, _) =>
             A = B andalso prec > min_prec andalso prec <= max_prec
           | _ => false) Si;
 
 fun get_states Estate j A max_prec =
   filter
     (fn (_, _, _, Nonterminal (B, prec) :: _, _, _) => A = B andalso prec <= max_prec
       | _ => false)
     (Array.sub (Estate, j));
 
 
 fun movedot_term c (A, j, ts, Terminal a :: sa, id, i) =
   if Lexicon.valued_token c orelse id <> ""
   then (A, j, Tip c :: ts, sa, id, i)
   else (A, j, ts, sa, id, i);
 
 fun movedot_nonterm tt (A, j, ts, Nonterminal _ :: sa, id, i) =
   (A, j, tt @ ts, sa, id, i);
 
 fun movedot_lambda [] _ = []
   | movedot_lambda ((t, ki) :: ts) (state as (B, j, tss, Nonterminal (A, k) :: sa, id, i)) =
       if k <= ki then (B, j, t @ tss, sa, id, i) :: movedot_lambda ts state
       else movedot_lambda ts state;
 
 
 (*trigger value for warnings*)
 val branching_level = Config.declare_int ("syntax_branching_level", \<^here>) (K 600);
 
 (*get all productions of a NT and NTs chained to it which can
   be started by specified token*)
 fun prods_for (Gram {prods, chains, ...}) tk nt =
   let
     fun token_prods prods =
       fold cons (prods_lookup prods tk) #>
       fold cons (prods_lookup prods Lexicon.dummy);
     fun nt_prods nt = #2 (Vector.sub (prods, nt));
   in fold (token_prods o nt_prods) (chains_all_preds chains [nt]) [] end;
 
 
 fun PROCESSS gram Estate i c states =
   let
     fun processS _ [] (Si, Sii) = (Si, Sii)
       | processS used (S :: States) (Si, Sii) =
           (case S of
             (_, _, _, Nonterminal (nt, min_prec) :: _, _, _) =>
               let (*predictor operation*)
                 val (used', new_states) =
                   (case Inttab.lookup used nt of
                     SOME (used_prec, l) => (*nonterminal has been processed*)
                       if used_prec <= min_prec then
                         (*wanted precedence has been processed*)
                         (used, movedot_lambda l S)
                       else (*wanted precedence hasn't been parsed yet*)
                         let
                           val tk_prods = prods_for gram c nt;
                           val States' =
                             mk_states i nt (get_RHS' min_prec used_prec tk_prods);
                         in (update_prec (nt, min_prec) used, movedot_lambda l S @ States') end
                   | NONE => (*nonterminal is parsed for the first time*)
                       let
                         val tk_prods = prods_for gram c nt;
                         val States' = mk_states i nt (get_RHS min_prec tk_prods);
                       in (Inttab.update (nt, (min_prec, [])) used, States') end);
               in
                 processS used' (new_states @ States) (S :: Si, Sii)
               end
           | (_, _, _, Terminal a :: _, _, _) => (*scanner operation*)
               processS used States
                 (S :: Si,
                   if Lexicon.tokens_match_ord (a, c) = EQUAL then movedot_term c S :: Sii else Sii)
           | (A, prec, ts, [], id, j) => (*completer operation*)
               let val tt = if id = "" then ts else [Node (id, rev ts)] in
                 if j = i then (*lambda production?*)
                   let
                     val (prec', used') = update_trees (A, (tt, prec)) used;
                     val Slist = getS A prec prec' Si;
                     val States' = map (movedot_nonterm tt) Slist;
                   in processS used' (States' @ States) (S :: Si, Sii) end
                 else
                   let val Slist = get_states Estate j A prec
                   in processS used (map (movedot_nonterm tt) Slist @ States) (S :: Si, Sii) end
               end)
   in processS Inttab.empty states ([], []) end;
 
 
 fun produce gram stateset i indata prev_token =
   (case Array.sub (stateset, i) of
     [] =>
       let
         val toks = if Lexicon.is_eof prev_token then indata else prev_token :: indata;
         val pos = Position.here (Lexicon.pos_of_token prev_token);
       in
         if null toks then
           error ("Inner syntax error: unexpected end of input" ^ pos)
         else
           error ("Inner syntax error" ^ pos ^
             Markup.markup Markup.no_report
               ("\n" ^ Pretty.string_of
                 (Pretty.block [
                   Pretty.str "at", Pretty.brk 1,
                   Pretty.block
                    (Pretty.str "\"" ::
                     Pretty.breaks (map (Pretty.str o Lexicon.str_of_token) (#1 (split_last toks))) @
                     [Pretty.str "\""])])))
       end
   | s =>
       (case indata of
         [] => s
       | c :: cs =>
           let
             val (si, sii) = PROCESSS gram stateset i c s;
             val _ = Array.update (stateset, i, si);
             val _ = Array.update (stateset, i + 1, sii);
           in produce gram stateset (i + 1) cs c end));
 
 
 fun get_trees states = map_filter (fn (_, _, [pt], _, _, _) => SOME pt | _ => NONE) states;
 
 fun earley (gram as Gram {tags, ...}) startsymbol indata =
   let
     val start_tag =
       (case tags_lookup tags startsymbol of
         SOME tag => tag
       | NONE => error ("Inner syntax: bad grammar root symbol " ^ quote startsymbol));
     val S0 = [(~1, 0, [], [Nonterminal (start_tag, 0), Terminal Lexicon.eof], "", 0)];
     val s = length indata + 1;
     val Estate = Array.array (s, []);
     val _ = Array.update (Estate, 0, S0);
   in
     get_trees (produce gram Estate 0 indata Lexicon.eof)
   end;
 
 
 fun parse gram start toks =
   let
     val end_pos =
       (case try List.last toks of
         NONE => Position.none
       | SOME tok => Lexicon.end_pos_of_token tok);
     val r =
       (case earley gram start (toks @ [Lexicon.mk_eof end_pos]) of
         [] => raise Fail "Inner syntax: no parse trees"
       | pts => pts);
   in r end;
 
 end;