src/HOL/Data_Structures/List_Ins_Del.thy

+(* Author: Tobias Nipkow *)
+
+section {* List Insertion and Deletion *}
+
+theory List_Ins_Del
+imports Sorted_Less
+begin
+
+subsection \<open>Elements in a list\<close>
+
+fun elems :: "'a list \<Rightarrow> 'a set" where
+"elems [] = {}" |
+"elems (x#xs) = Set.insert x (elems xs)"
+
+lemma elems_app: "elems (xs @ ys) = (elems xs \<union> elems ys)"
+by (induction xs) auto
+
+lemma elems_eq_set: "elems xs = set xs"
+by (induction xs) auto
+
+lemma sorted_Cons_iff:
+  "sorted(x # xs) = (sorted xs \<and> (\<forall>y \<in> elems xs. x < y))"
+by(simp add: elems_eq_set Sorted_Less.sorted_Cons_iff)
+
+lemma sorted_snoc_iff:
+  "sorted(xs @ [x]) = (sorted xs \<and> (\<forall>y \<in> elems xs. y < x))"
+by(simp add: elems_eq_set Sorted_Less.sorted_snoc_iff)
+
+lemma sorted_ConsD: "sorted (y # xs) \<Longrightarrow> x \<in> elems xs \<Longrightarrow> y < x"
+by (simp add: sorted_Cons_iff)
+
+lemma sorted_snocD: "sorted (xs @ [y]) \<Longrightarrow> x \<in> elems xs \<Longrightarrow> x < y"
+by (simp add: sorted_snoc_iff)
+
+lemmas elems_simps0 = sorted_lems elems_app
+lemmas elems_simps = elems_simps0 sorted_Cons_iff sorted_snoc_iff
+lemmas sortedD = sorted_ConsD sorted_snocD
+
+
+subsection \<open>Inserting into an ordered list without duplicates:\<close>
+
+fun ins_list :: "'a::linorder \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+"ins_list x [] = [x]" |
+"ins_list x (y#zs) =
+  (if x < y then x#y#zs else if x=y then x#zs else y # ins_list x zs)"
+
+lemma set_ins_list[simp]: "elems (ins_list x xs) = insert x (elems xs)"
+by(induction xs) auto
+
+lemma distinct_if_sorted: "sorted xs \<Longrightarrow> distinct xs"
+apply(induction xs rule: sorted.induct)
+apply auto
+by (metis in_set_conv_decomp_first less_imp_not_less sorted_mid_iff2)
+
+lemma sorted_ins_list: "sorted xs \<Longrightarrow> sorted(ins_list x xs)"
+by(induction xs rule: sorted.induct) auto
+
+lemma ins_list_sorted1: "sorted (xs @ [y]) \<Longrightarrow> y \<le> x \<Longrightarrow>
+  ins_list x (xs @ y # ys) = xs @ ins_list x (y#ys)"
+by(induction xs) (auto simp: sorted_lems)
+
+lemma ins_list_sorted2: "sorted (xs @ [y]) \<Longrightarrow> x < y \<Longrightarrow>
+  ins_list x (xs @ y # ys) = ins_list x xs @ (y#ys)"
+by(induction xs) (auto simp: sorted_lems)
+
+lemmas ins_simps = sorted_lems ins_list_sorted1 ins_list_sorted2
+
+
+subsection \<open>Delete one occurrence of an element from a list:\<close>
+
+fun del_list :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+"del_list a [] = []" |
+"del_list a (x#xs) = (if a=x then xs else x # del_list a xs)"
+
+lemma del_list_idem: "x \<notin> elems xs \<Longrightarrow> del_list x xs = xs"
+by (induct xs) simp_all
+
+lemma elems_del_list_eq [simp]:
+  "distinct xs \<Longrightarrow> elems (del_list x xs) = elems xs - {x}"
+apply(induct xs)
+ apply simp
+apply (simp add: elems_eq_set)
+apply blast
+done
+
+lemma sorted_del_list: "sorted xs \<Longrightarrow> sorted(del_list x xs)"
+apply(induction xs rule: sorted.induct)
+apply auto
+by (meson order.strict_trans sorted_Cons_iff)
+
+lemma del_list_sorted1: "sorted (xs @ [x]) \<Longrightarrow> x \<le> y \<Longrightarrow>
+  del_list y (xs @ x # ys) = xs @ del_list y (x # ys)"
+by (induction xs) (auto simp: sorted_mid_iff2)
+
+lemma del_list_sorted2: "sorted (xs @ x # ys) \<Longrightarrow> y < x \<Longrightarrow>
+  del_list y (xs @ x # ys) = del_list y xs @ x # ys"
+by (induction xs) (auto simp: sorted_Cons_iff intro!: del_list_idem)
+
+lemma del_list_sorted3:
+  "sorted (xs @ x # ys @ y # zs) \<Longrightarrow> a < y \<Longrightarrow>
+  del_list a (xs @ x # ys @ y # zs) = del_list a (xs @ x # ys) @ y # zs"
+by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted2)
+
+lemma del_list_sorted4:
+  "sorted (xs @ x # ys @ y # zs @ z # us) \<Longrightarrow> a < z \<Longrightarrow>
+  del_list a (xs @ x # ys @ y # zs @ z # us) = del_list a (xs @ x # ys @ y # zs) @ z # us"
+by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted3)
+
+lemma del_list_sorted5:
+  "sorted (xs @ x # ys @ y # zs @ z # us @ u # vs) \<Longrightarrow> a < u \<Longrightarrow>
+   del_list a (xs @ x # ys @ y # zs @ z # us @ u # vs) =
+   del_list a (xs @ x # ys @ y # zs @ z # us) @ u # vs" 
+by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted4)
+
+lemmas del_simps = sorted_lems
+  del_list_sorted1
+  del_list_sorted2
+  del_list_sorted3
+  del_list_sorted4
+  del_list_sorted5
+
+end