Quelle  BPlusTree.thy

theory BPlusTree                 
  imports Main "HOL-Data_Structures.Sorted_Less" "HOL-Data_Structures.Cmp" "HOL-Library.Multiset"
begin

(* some setup to cover up the redefinition of sorted in Sorted_Less
   but keep the lemmas *)
hide_const (open) Sorted_Less.sorted
abbreviation "sorted_less ≡ Sorted_Less.sorted"

section "Definition of the B-Plus-Tree"

subsection "Datatype definition"

text "B-Plus-Trees are basically B-Trees, that don't have empty Leafs but Leafs that contain
the relevant data. "


datatype 'a bplustree = Leaf (vals: "'a list") | Node (keyvals: "('a bplustree * 'a) list") (lasttree: "'a bplustree")

type_synonym 'a bplustree_list =  "('a bplustree * 'a) list"
type_synonym 'a bplustree_pair =  "('a bplustree * 'a)"

abbreviation subtrees where "subtrees xs ≡ (map fst xs)"
abbreviation separators where "separators xs ≡ (map snd xs)"

subsection "Inorder and Set"

text "The set of B-Plus-tree needs to be manually defined, regarding only the leaves.
This overrides the default instantiation."

fun set_nodes :: "'a bplustree ==> 'a set" where
  "set_nodes (Leaf ks) = {}" |
  "set_nodes (Node ts t) = ∪(set (map set_nodes (subtrees ts))) ∪ (set (separators ts)) ∪ set_nodes t"

fun set_leaves :: "'a bplustree ==> 'a set" where
  "set_leaves (Leaf ks) = set ks" |
  "set_leaves (Node ts t) = ∪(set (map set_leaves (subtrees ts))) ∪ set_leaves t"

text "The inorder is a view of only internal seperators"

fun inorder :: "'a bplustree ==> 'a list" where
  "inorder (Leaf ks) = []" |
  "inorder (Node ts t) = concat (map (λ (sub, sep). inorder sub @ [sep]) ts) @ inorder t"

abbreviation "inorder_list ts ≡ concat (map (λ (sub, sep). inorder sub @ [sep]) ts)"

text "The leaves view considers only its leafs."

fun leaves :: "'a bplustree ==> 'a list" where
  "leaves (Leaf ks) = ks" |
  "leaves (Node ts t) = concat (map leaves (subtrees ts)) @ leaves t"

abbreviation "leaves_list ts ≡ concat (map leaves (subtrees ts))"

fun leaf_nodes where
"leaf_nodes (Leaf xs) = [Leaf xs]" |
"leaf_nodes (Node ts t) = concat (map leaf_nodes (subtrees ts)) @ leaf_nodes t"

abbreviation "leaf_nodes_list ts ≡ concat (map leaf_nodes (subtrees ts))"



text "And the elems view contains all elements of the tree"
(* NOTE this doesn't help *)

fun elems :: "'a bplustree ==> 'a list" where
  "elems (Leaf ks) = ks" |
  "elems (Node ts t) = concat (map (λ (sub, sep). elems sub @ [sep]) ts) @ elems t"

abbreviation "elems_list ts ≡ concat (map (λ (sub, sep). elems sub @ [sep]) ts)"

(* this abbreviation makes handling the list much nicer *)
thm leaves.simps
thm inorder.simps
thm elems.simps

value "leaves (Node [(Leaf [], (0::nat)), (Node [(Leaf [], 1), (Leaf [], 10)] (Leaf []), 12), ((Leaf []), 30), ((Leaf []), 100)] (Leaf []))"

subsection "Height and Balancedness"

class height =
  fixes height :: "'a ==> nat"

instantiation bplustree :: (type) height
begin

fun height_bplustree :: "'a bplustree ==> nat" where
  "height (Leaf ks) = 0" |
  "height (Node ts t) = Suc (Max (height ` (set (subtrees ts@[t]))))"

instance ..

end

text "Balancedness is defined is close accordance to the definition by Ernst"

fun bal:: "'a bplustree ==> bool" where
  "bal (Leaf ks) = True" |
  "bal (Node ts t) = (
    (∀sub ∈ set (subtrees ts). height sub = height t) ∧
    (∀sub ∈ set (subtrees ts). bal sub) ∧ bal t
  )"


value "height (Node [(Leaf [], (0::nat)), (Node [(Leaf [], 1), (Leaf [], 10)] (Leaf []), 12), ((Leaf []), 30), ((Leaf []), 100)] (Leaf []))"
value "bal (Node [(Leaf [], (0::nat)), (Node [(Leaf [], 1), (Leaf [], 10)] (Leaf []), 12), ((Leaf []), 30), ((Leaf []), 100)] (Leaf []))"


subsection "Order"

text "The order of a B-tree is defined just as in the original paper by Bayer."

(* alt1: following knuths definition to allow for any
   natural number as order and resolve ambiguity *)
(* alt2: use range [k,2*k] allowing for valid bplustrees
   from k=1 onwards NOTE this is what I ended up implementing *)

fun order:: "nat ==> 'a bplustree ==> bool" where
  "order k (Leaf ks) = ((length ks ≥ k) ∧ (length ks ≤ 2*k))" |
  "order k (Node ts t) = (
  (length ts ≥ k) ∧
  (length ts ≤ 2*k) ∧
  (∀sub ∈ set (subtrees ts). order k sub) ∧ order k t
)"

text ‹The special condition for the root is called \textit{root\_order}›

(* the invariant for the root of the bplustree *)
fun root_order:: "nat ==> 'a bplustree ==> bool" where
  "root_order k (Leaf ks) = (length ks ≤ 2*k)" |
  "root_order k (Node ts t) = (
  (length ts > 0) ∧
  (length ts ≤ 2*k) ∧
  (∀s ∈ set (subtrees ts). order k s) ∧ order k t
)"


subsection "Auxiliary Lemmas"

(* auxiliary lemmas when handling sets *)
lemma separators_split:
  "set (separators (l@(a,b)#r)) = set (separators l) ∪ set (separators r) ∪ {b}"
  by simp

lemma subtrees_split:
  "set (subtrees (l@(a,b)#r)) = set (subtrees l) ∪ set (subtrees r) ∪ {a}"
  by simp

(* height and set lemmas *)


lemma finite_set_ins_swap:
  assumes "finite A"
  shows "max a (Max (Set.insert b A)) = max b (Max (Set.insert a A))"
  using Max_insert assms max.commute max.left_commute by fastforce

lemma finite_set_in_idem:
  assumes "finite A"
  shows "max a (Max (Set.insert a A)) = Max (Set.insert a A)"
  using Max_insert assms max.commute max.left_commute by fastforce

lemma height_Leaf: "height t = 0 ⟷ (∃ks. t = (Leaf ks))"
  by (induction t) (auto)

lemma height_bplustree_order:
  "height (Node (ls@[a]) t) = height (Node (a#ls) t)"
  by simp

lemma height_bplustree_sub: 
  "height (Node ((sub,x)#ls) t) = max (height (Node ls t)) (Suc (height sub))"
  by simp

lemma height_bplustree_last: 
  "height (Node ((sub,x)#ts) t) = max (height (Node ts sub)) (Suc (height t))"
  by (induction ts) auto


lemma set_leaves_leaves: "set (leaves t) = set_leaves t"
  apply(induction t)
   apply(auto)
  done

lemma set_nodes_nodes: "set (inorder t) = set_nodes t"
  apply(induction t)
   apply(auto simp add: rev_image_eqI)
  done


lemma child_subset_leaves: "p ∈ set t ==> set_leaves (fst p) ⊆ set_leaves (Node t n)"
  apply(induction p arbitrary: t n)
  apply(auto)
  done

lemma child_subset: "p ∈ set t ==> set_nodes (fst p) ⊆ set_nodes (Node t n)"
  apply(induction p arbitrary: t n)
  apply(auto)
  done

lemma some_child_sub: 
  assumes "(sub,sep) ∈ set t"
  shows "sub ∈ set (subtrees t)"
    and "sep ∈ set (separators t)"
  using assms by force+ 

(* balancedness lemmas *)


lemma bal_all_subtrees_equal: "bal (Node ts t) ==> (∀s1 ∈ set (subtrees ts). ∀s2 ∈ set (subtrees ts). height s1 = height s2)"
  by (metis BPlusTree.bal.simps(2))


lemma fold_max_set: "∀x ∈ set t. x = f ==> fold max t f = f"
  apply(induction t)
   apply(auto simp add: max_def_raw)
  done

lemma height_bal_tree: "bal (Node ts t) ==> height (Node ts t) = Suc (height t)"
  by (induction ts) auto



lemma bal_split_last: 
  assumes "bal (Node (ls@(sub,sep)#rs) t)"
  shows "bal (Node (ls@rs) t)"
    and "height (Node (ls@(sub,sep)#rs) t) = height (Node (ls@rs) t)"
  using assms by auto


lemma bal_split_right: 
  assumes "bal (Node (ls@rs) t)"
  shows "bal (Node rs t)"
    and "height (Node rs t) = height (Node (ls@rs) t)"
  using assms by (auto simp add: image_constant_conv)

lemma bal_split_left:
  assumes "bal (Node (ls@(a,b)#rs) t)"
  shows "bal (Node ls a)"
    and "height (Node ls a) = height (Node (ls@(a,b)#rs) t)"
  using assms by (auto simp add: image_constant_conv)


lemma bal_substitute: "[bal (Node (ls@(a,b)#rs) t); height t = height c; bal c] ==> bal (Node (ls@(c,b)#rs) t)"
  unfolding bal.simps
  by auto

lemma bal_substitute_subtree: "[bal (Node (ls@(a,b)#rs) t); height a = height c; bal c] ==> bal (Node (ls@(c,b)#rs) t)"
  using bal_substitute
  by auto

lemma bal_substitute_separator: "bal (Node (ls@(a,b)#rs) t) ==> bal (Node (ls@(a,c)#rs) t)"
  unfolding bal.simps
  by auto

(* order lemmas *)

lemma order_impl_root_order: "[k > 0; order k t] ==> root_order k t"
  apply(cases t)
   apply(auto)
  done


(* sorted leaves implies that some sublists are sorted. This can be followed directly *)

lemma sorted_inorder_list_separators: "sorted_less (inorder_list ts) ==> sorted_less (separators ts)"
  apply(induction ts)
   apply (auto simp add: sorted_lems)
  done

corollary sorted_inorder_separators: "sorted_less (inorder (Node ts t)) ==> sorted_less (separators ts)"
  using sorted_inorder_list_separators sorted_wrt_append
  by auto


lemma sorted_inorder_list_subtrees:
  "sorted_less (inorder_list ts) ==> ∀ sub ∈ set (subtrees ts). sorted_less (inorder sub)"
  apply(induction ts)
   apply (auto simp add: sorted_lems)+
  done

corollary sorted_inorder_subtrees: "sorted_less (inorder (Node ts t)) ==> ∀ sub ∈ set (subtrees ts). sorted_less (inorder sub)"
  using sorted_inorder_list_subtrees sorted_wrt_append by auto

lemma sorted_inorder_list_induct_subtree:
  "sorted_less (inorder_list (ls@(sub,sep)#rs)) ==> sorted_less (inorder sub)"
  by (simp add: sorted_wrt_append)

corollary sorted_inorder_induct_subtree:
  "sorted_less (inorder (Node (ls@(sub,sep)#rs) t)) ==> sorted_less (inorder sub)"
  by (simp add: sorted_wrt_append)

lemma sorted_inorder_induct_last: "sorted_less (inorder (Node ts t)) ==> sorted_less (inorder t)"
  by (simp add: sorted_wrt_append)


lemma sorted_leaves_list_subtrees:
  "sorted_less (leaves_list ts) ==> ∀ sub ∈ set (subtrees ts). sorted_less (leaves sub)"
  apply(induction ts)
   apply (auto simp add: sorted_wrt_append)+
  done

corollary sorted_leaves_subtrees: "sorted_less (leaves (Node ts t)) ==> ∀ sub ∈ set (subtrees ts). sorted_less (leaves sub)"
  using sorted_leaves_list_subtrees sorted_wrt_append by auto

lemma sorted_leaves_list_induct_subtree:
  "sorted_less (leaves_list (ls@(sub,sep)#rs)) ==> sorted_less (leaves sub)"
  by (simp add: sorted_wrt_append)

corollary sorted_leaves_induct_subtree:
  "sorted_less (leaves (Node (ls@(sub,sep)#rs) t)) ==> sorted_less (leaves sub)"
  by (simp add: sorted_wrt_append)

lemma sorted_leaves_induct_last: "sorted_less (leaves (Node ts t)) ==> sorted_less (leaves t)"
  by (simp add: sorted_wrt_append)

text "Additional lemmas on the sortedness of the whole tree, which is
correct alignment of navigation structure and leave data"

fun inbetween where
"inbetween f l Nil t u = f l t u" |
"inbetween f l ((sub,sep)#xs) t u = (f l sub sep ∧ inbetween f sep xs t u)"

thm fold_cong

lemma cong_inbetween[fundef_cong]: "
\<lbrakk>a = b; xs = ys; ∧l' u' sub sep. (sub,sep) ∈ set ys ==> f l' sub u' = g l' sub u'; ∧l' u'. f l' a u' = g l' b u']
  ==> inbetween f l xs a u = inbetween g l ys b u"
  apply(induction ys arbitrary: l a b u xs)
  apply auto
  apply fastforce+
  done

(* adding l < u makes sorted_less inorder not necessary anymore *)
fun aligned :: "'a ::linorder ==> _" where 
"aligned l (Leaf ks) u = (l < u ∧ (∀x ∈ set ks. l < x ∧ x ≤ u))" |
"aligned l (Node ts t) u = (inbetween aligned l ts t u)"

lemma sorted_less_merge: "sorted_less (as@[a]) ==> sorted_less (a#bs) ==> sorted_less (as@a#bs)"
  using sorted_mid_iff by blast


thm aligned.simps

lemma leaves_cases: "x ∈ set (leaves (Node ts t)) ==> (∃(sub,sep) ∈ set ts. x ∈ set (leaves sub)) ∨ x ∈ set (leaves t)"
  apply (induction ts)
  apply auto
  done

lemma align_sub: "aligned l (Node ts t) u ==> (sub,sep) ∈ set ts ==> ∃l' ∈ set (separators ts) ∪ {l}. aligned l' sub sep" 
  apply(induction ts arbitrary: l)
  apply auto
  done

lemma align_last: "aligned l (Node (ts@[(sub,sep)]) t) u ==> aligned sep t u" 
  apply(induction ts arbitrary: l)
  apply auto
  done

lemma align_last': "aligned l (Node ts t) u ==> ∃l' ∈ set (separators ts) ∪ {l}. aligned l' t u" 
  apply(induction ts arbitrary: l)
  apply auto
  done

lemma aligned_sorted_inorder: "aligned l t u ==> sorted_less (l#(inorder t)@[u])" 
proof(induction l t u rule: aligned.induct)
  case (2 l ts t u)
  then show ?case 
  proof(cases ts)
    case Nil
    then show ?thesis
      using 2 by auto 
  next
    case Cons
    then obtain ts' sub sep where ts_split: "ts = ts'@[(sub,sep)]"
      by (metis list.distinct(1) rev_exhaust surj_pair)
    moreover from 2 have "sorted_less (l#(inorder_list ts))"
    proof (induction ts arbitrary: l)
      case (Cons a ts')
      then show ?case 
      proof (cases a)
        case (Pair sub sep)
        then have "aligned l sub sep" "inbetween aligned sep ts' t u" 
          using "Cons.prems" by simp+
        then have "aligned sep (Node ts' t) u"
          by simp
        then have "sorted_less (sep#inorder_list ts')"
          using Cons 
          by (metis insert_iff list.set(2))
        moreover have "sorted_less (l#inorder sub@[sep])"
          using Cons 
          by (metis Pair ‹aligned l sub sep› list.set_intros(1))
        ultimately show ?thesis
          using Pair sorted_less_merge[of "l#inorder sub" sep "inorder_list ts'"] 
          by simp
      qed
    qed simp
    moreover have "sorted_less (sep#inorder t@[u])"
    proof -
      from 2 have "aligned sep t u"
        using align_last ts_split by blast
      then show ?thesis
        using "2.IH" by blast
    qed
    ultimately show ?thesis
      using sorted_less_merge[of "l#inorder_list ts'@inorder sub" sep "inorder t@[u]"]
      by simp
  qed
qed simp

lemma separators_in_inorder_list: "set (separators ts) ⊆ set (inorder_list ts)" 
  apply (induction ts)
  apply auto
  done

lemma separators_in_inorder: "set (separators ts) ⊆ set (inorder (Node ts t))" 
  by fastforce

lemma aligned_sorted_separators: "aligned l (Node ts t) u ==> sorted_less (l#(separators ts)@[u])" 
  by (smt (verit, ccfv_threshold) aligned_sorted_inorder separators_in_inorder sorted_inorder_separators sorted_lems(2) sorted_wrt.simps(2) sorted_wrt_append subset_eq)

lemma aligned_leaves_inbetween: "aligned l t u ==> ∀x ∈ set (leaves t). l < x ∧ x ≤ u"
proof (induction l t u rule: aligned.induct)
  case (1 l ks u)
  then show ?case by auto
next
  case (2 l ts t u)
  have *: "sorted_less (l#inorder (Node ts t)@[u])"
    using "2.prems" aligned_sorted_inorder by blast
  show ?case
  proof
    fix x assume "x ∈ set (leaves (Node ts t))"
    then consider (sub) "∃(sub,sep) ∈ set ts. x ∈ set (leaves sub)" | (last) "x ∈ set (leaves t)"
      by fastforce
    then show "l < x ∧ x ≤ u"
    proof (cases)
      case sub
      then obtain sub sep where "(sub,sep) ∈ set ts" "x ∈ set (leaves sub)" by auto 
      then obtain l' where "aligned l' sub sep" "l' ∈ set (separators ts) ∪ {l}"
        using "2.prems"(1) ‹(sub, sep) ∈ set ts› align_sub by blast
      then have "∀x ∈ set (leaves sub). l' < x ∧ x ≤ sep"
        using "2.IH"(1) ‹(sub,sep) ∈ set ts› by auto
      moreover from * have "l ≤ l'"
          by (metis Un_insert_right ‹l' ∈ set (separators ts) ∪ {l}› append_Cons boolean_algebra_cancel.sup0 dual_order.eq_iff insert_iff less_imp_le separators_in_inorder sorted_snoc sorted_wrt.simps(2) subset_eq)
      moreover from * have  "sep ≤ u"
        by (metis ‹(sub, sep) ∈ set ts› less_imp_le list.set_intros(1) separators_in_inorder some_child_sub(2) sorted_mid_iff2 sorted_wrt_append subset_eq)
      ultimately show ?thesis
        by (meson ‹x ∈ set (leaves sub)› order.strict_trans1 order.trans)
    next
      case last
      then obtain l' where "aligned l' t u" "l' ∈ set (separators ts) ∪ {l}"
        using align_last' "2.prems" by blast
      then have "∀x ∈ set (leaves t). l' < x ∧ x ≤ u"
        using "2.IH"(2) by auto
      moreover from * have "l ≤ l'"
        by (metis Un_insert_right ‹l' ∈ set (separators ts) ∪ {l}› append_Cons boolean_algebra_cancel.sup0 dual_order.eq_iff insert_iff less_imp_le separators_in_inorder sorted_snoc sorted_wrt.simps(2) subset_eq)
      ultimately show ?thesis
        by (meson ‹x ∈ set (leaves t)› order.strict_trans1 order.trans)
    qed
  qed
qed
 
lemma aligned_leaves_list_inbetween: "aligned l (Node ts t) u ==> ∀x ∈ set (leaves_list ts). l < x ∧ x ≤ u"
  by (metis Un_iff aligned_leaves_inbetween leaves.simps(2) set_append)

lemma aligned_split_left: "aligned l (Node (ls@(sub,sep)#rs) t) u ==> aligned l (Node ls sub) sep"
  apply(induction ls arbitrary: l)
  apply auto
  done


lemma aligned_split_right: "aligned l (Node (ls@(sub,sep)#rs) t) u ==> aligned sep (Node rs t) u"
  apply(induction ls arbitrary: l)
  apply auto
  done

lemma aligned_subst: "aligned l (Node (ls@(sub', subl)#(sub,subsep)#rs) t) u ==> aligned subl subsub subsep ==>
aligned l (Node (ls@(sub',subl)#(subsub,subsep)#rs) t) u" 
  apply (induction ls arbitrary: l)
  apply auto
  done

lemma aligned_subst_emptyls: "aligned l (Node ((sub,subsep)#rs) t) u ==> aligned l subsub subsep ==>
aligned l (Node ((subsub,subsep)#rs) t) u" 
  by auto

lemma aligned_subst_last: "aligned l (Node (ts'@[(sub', sep')]) t) u ==> aligned sep' t' u ==>
  aligned l (Node (ts'@[(sub', sep')]) t') u" 
  apply (induction ts' arbitrary: l)
  apply auto
  done

fun Laligned :: "'a ::linorder bplustree ==> _" where 
"Laligned (Leaf ks) u = (∀x ∈ set ks. x ≤ u)" |
"Laligned (Node ts t) u = (case ts of [] ==> (Laligned t u) |
 (sub,sep)#ts' ==> ((Laligned sub sep) ∧ inbetween aligned sep ts' t u))"

lemma Laligned_nonempty_Node: "Laligned (Node ((sub,sep)#ts') t) u =
  ((Laligned sub sep) ∧ inbetween aligned sep ts' t u)"
  by simp

lemma aligned_imp_Laligned: "aligned l t u ==> Laligned t u"
  apply (induction l t u rule: aligned.induct)
  apply simp
  subgoal for l ts t u
    apply(cases ts)
     apply auto
    apply blast
    done
  done

lemma Laligned_split_left: "Laligned (Node (ls@(sub,sep)#rs) t) u ==> Laligned (Node ls sub) sep"
  apply(cases ls)
  apply (auto dest!: aligned_imp_Laligned)
  apply (meson aligned.simps(2) aligned_split_left)
  done

lemma Laligned_split_right: "Laligned (Node (ls@(sub,sep)#rs) t) u ==> aligned sep (Node rs t) u"
  apply(cases ls)
  apply (auto split!: list.splits dest!: aligned_imp_Laligned)
  apply (meson aligned.simps(2) aligned_split_right)
  done

lemma Lalign_sub: "Laligned (Node ((a,b)#ts) t) u ==> (sub,sep) ∈ set ts ==> ∃l' ∈ set (separators ts) ∪ {b}. aligned l' sub sep" 
  apply(induction ts arbitrary: a b)
    apply (auto dest!: aligned_imp_Laligned)
  done

lemma Lalign_last: "Laligned (Node (ts@[(sub,sep)]) t) u ==> aligned sep t u" 
  by (cases ts) (auto simp add: align_last)

lemma Lalign_last': "Laligned (Node ((a,b)#ts) t) u ==> ∃l' ∈ set (separators ts) ∪ {b}. aligned l' t u" 
  apply(induction ts arbitrary: a b)
  apply (auto dest!: aligned_imp_Laligned)
  done

lemma Lalign_Llast: "Laligned (Node ts t) u ==> Laligned t u" 
  apply(cases ts)
  apply auto
  using aligned_imp_Laligned Lalign_last' Laligned_nonempty_Node
  by metis


lemma Laligned_sorted_inorder: "Laligned t u ==> sorted_less ((inorder t)@[u])" 
proof(induction t u rule: Laligned.induct)
  case (1 ks u)
  then show ?case by auto
next
  case (2 ts t u)
  then show ?case 
    apply (cases ts)
    apply auto
    by (metis aligned.simps(2) aligned_sorted_inorder append_assoc inorder.simps(2) sorted_less_merge)
qed


lemma Laligned_sorted_separators: "Laligned (Node ts t) u ==> sorted_less ((separators ts)@[u])" 
  by (smt (verit, del_insts) Laligned_sorted_inorder separators_in_inorder sorted_inorder_separators sorted_wrt_append subset_eq)

lemma Laligned_leaves_inbetween: "Laligned t u ==> ∀x ∈ set (leaves t). x ≤ u"
proof (induction t u rule: Laligned.induct)
  case (1 ks u)
  then show ?case by auto
next
  case (2 ts t u)
  have *: "sorted_less (inorder (Node ts t)@[u])"
    using "2.prems" Laligned_sorted_inorder by blast
  show ?case
  proof (cases ts)
    case Nil
    show ?thesis
    proof 
      fix x assume "x ∈ set (leaves (Node ts t))"
      then have "x ∈ set (leaves t)"
        using Nil by auto
      moreover have "Laligned t u"
        using "2.prems" Nil by auto
      ultimately show "x ≤ u"
        using "2.IH"(1) Nil
        by simp
    qed
  next
    case (Cons h ts')
    then obtain a b where h_split: "h = (a,b)"
      by (cases h)
    show ?thesis 
    proof
    fix x assume "x ∈ set (leaves (Node ts t))"
    then consider (first) "x ∈ set (leaves a)" | (sub) "∃(sub,sep) ∈ set ts'. x ∈ set (leaves sub)" | (last) "x ∈ set (leaves t)"
      using Cons h_split by fastforce
    then show "x ≤ u"
      proof (cases)
        case first
        moreover have "Laligned a b"
          using "2.prems" Cons h_split by auto
        moreover have "b ≤ u"
          by (metis "*" h_split less_imp_le list.set_intros(1) local.Cons separators_in_inorder some_child_sub(2) sorted_wrt_append subsetD)
        ultimately show ?thesis
          using "2.IH"(2)[OF Cons sym[OF h_split]]
          by auto
      next
        case sub
        then obtain sub sep where "(sub,sep) ∈ set ts'" "x ∈ set (leaves sub)" by auto 
        then obtain l' where "aligned l' sub sep" "l' ∈ set (separators ts') ∪ {b}"
          using "2.prems" Lalign_sub h_split local.Cons by blast
        then have "∀x ∈ set (leaves sub). l' < x ∧ x ≤ sep"
          by (meson aligned_leaves_inbetween)
        moreover from * have  "sep ≤ u"
          by (metis "2.prems" Laligned_sorted_separators ‹(sub, sep) ∈ set ts'› insert_iff less_imp_le list.set(2) local.Cons some_child_sub(2) sorted_wrt_append)
        ultimately show ?thesis
          by (meson ‹x ∈ set (leaves sub)› order.strict_trans1 order.trans)
      next
        case last
        then obtain l' where "aligned l' t u" "l' ∈ set (separators ts') ∪ {b}"
          using "2.prems" Lalign_last' h_split local.Cons by blast
        then have "∀x ∈ set (leaves t). l' < x ∧ x ≤ u"
          by (meson aligned_leaves_inbetween)
        then show ?thesis
          by (meson ‹x ∈ set (leaves t)› order.strict_trans1 order.trans)
      qed
    qed
  qed
qed
 
lemma Laligned_leaves_list_inbetween: "Laligned (Node ts t) u ==> ∀x ∈ set (leaves_list ts). x ≤ u"
  by (metis Un_iff Laligned_leaves_inbetween leaves.simps(2) set_append)

lemma Laligned_subst_last: "Laligned (Node (ts'@[(sub', sep')]) t) u ==> aligned sep' t' u ==>
  Laligned (Node (ts'@[(sub', sep')]) t') u" 
  apply (cases ts')
  apply (auto)
  by (meson aligned.simps(2) aligned_subst_last)

lemma Laligned_subst: "Laligned (Node (ls@(sub', subl)#(sub,subsep)#rs) t) u ==> aligned subl subsub subsep ==>
Laligned (Node (ls@(sub',subl)#(subsub,subsep)#rs) t) u" 
  apply (induction ls)
  apply auto
  apply (meson aligned.simps(2) aligned_subst)
  done

lemma concat_leaf_nodes_leaves: "(concat (map leaves (leaf_nodes t))) = leaves t"
  apply(induction t rule: leaf_nodes.induct)
  subgoal by auto
  subgoal for ts t
    apply(induction ts)
    apply simp
    apply auto
    done
  done

lemma leaf_nodes_not_empty: "leaf_nodes t ≠ []"
  by (induction t) auto

end