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(* Author: Tobias Nipkow *)
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section {* List Insertion and Deletion *}
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theory List_Ins_Del
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imports Sorted_Less
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begin
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subsection \<open>Elements in a list\<close>
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fun elems :: "'a list \<Rightarrow> 'a set" where
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"elems [] = {}" |
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"elems (x#xs) = Set.insert x (elems xs)"
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lemma elems_app: "elems (xs @ ys) = (elems xs \<union> elems ys)"
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by (induction xs) auto
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lemma elems_eq_set: "elems xs = set xs"
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by (induction xs) auto
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lemma sorted_Cons_iff:
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"sorted(x # xs) = (sorted xs \<and> (\<forall>y \<in> elems xs. x < y))"
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by(simp add: elems_eq_set Sorted_Less.sorted_Cons_iff)
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lemma sorted_snoc_iff:
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"sorted(xs @ [x]) = (sorted xs \<and> (\<forall>y \<in> elems xs. y < x))"
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by(simp add: elems_eq_set Sorted_Less.sorted_snoc_iff)
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text{* The above two rules introduce quantifiers. It turns out
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that in practice this is not a problem because of the simplicity of
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the "isin" functions that implement @{const elems}. Nevertheless
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it is possible to avoid the quantifiers with the help of some rewrite rules: *}
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lemma sorted_ConsD: "sorted (y # xs) \<Longrightarrow> x \<in> elems xs \<Longrightarrow> y < x"
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by (simp add: sorted_Cons_iff)
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lemma sorted_snocD: "sorted (xs @ [y]) \<Longrightarrow> x \<in> elems xs \<Longrightarrow> x < y"
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by (simp add: sorted_snoc_iff)
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lemma sorted_ConsD2: "sorted (y # xs) \<Longrightarrow> x \<le> y \<Longrightarrow> x \<notin> elems xs"
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using leD sorted_ConsD by blast
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lemma sorted_snocD2: "sorted (xs @ [y]) \<Longrightarrow> y \<le> x \<Longrightarrow> x \<notin> elems xs"
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using leD sorted_snocD by blast
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lemmas elems_simps = sorted_lems elems_app
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lemmas elems_simps1 = elems_simps sorted_Cons_iff sorted_snoc_iff
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lemmas elems_simps2 = elems_simps sorted_ConsD sorted_snocD sorted_ConsD2 sorted_snocD2
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subsection \<open>Inserting into an ordered list without duplicates:\<close>
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fun ins_list :: "'a::linorder \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"ins_list x [] = [x]" |
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"ins_list x (a#xs) =
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(if x < a then x#a#xs else if x=a then a#xs else a # ins_list x xs)"
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lemma set_ins_list: "elems (ins_list x xs) = insert x (elems xs)"
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by(induction xs) auto
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lemma distinct_if_sorted: "sorted xs \<Longrightarrow> distinct xs"
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apply(induction xs rule: sorted.induct)
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apply auto
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by (metis in_set_conv_decomp_first less_imp_not_less sorted_mid_iff2)
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lemma sorted_ins_list: "sorted xs \<Longrightarrow> sorted(ins_list x xs)"
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by(induction xs rule: sorted.induct) auto
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lemma ins_list_sorted: "sorted (xs @ [a]) \<Longrightarrow>
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ins_list x (xs @ a # ys) =
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(if a \<le> x then xs @ ins_list x (a#ys) else ins_list x xs @ (a#ys))"
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by(induction xs) (auto simp: sorted_lems)
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text\<open>In principle, @{thm ins_list_sorted} suffices, but the following two
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corollaries speed up proofs.\<close>
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corollary ins_list_sorted1: "sorted (xs @ [a]) \<Longrightarrow> a \<le> x \<Longrightarrow>
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ins_list x (xs @ a # ys) = xs @ ins_list x (a#ys)"
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by(simp add: ins_list_sorted)
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corollary ins_list_sorted2: "sorted (xs @ [a]) \<Longrightarrow> x < a \<Longrightarrow>
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ins_list x (xs @ a # ys) = ins_list x xs @ (a#ys)"
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by(auto simp: ins_list_sorted)
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lemmas ins_list_simps = sorted_lems ins_list_sorted1 ins_list_sorted2
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subsection \<open>Delete one occurrence of an element from a list:\<close>
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fun del_list :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"del_list x [] = []" |
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"del_list x (a#xs) = (if x=a then xs else a # del_list x xs)"
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lemma del_list_idem: "x \<notin> elems xs \<Longrightarrow> del_list x xs = xs"
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by (induct xs) simp_all
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lemma elems_del_list_eq:
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"distinct xs \<Longrightarrow> elems (del_list x xs) = elems xs - {x}"
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apply(induct xs)
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apply simp
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apply (simp add: elems_eq_set)
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apply blast
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done
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lemma sorted_del_list: "sorted xs \<Longrightarrow> sorted(del_list x xs)"
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apply(induction xs rule: sorted.induct)
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apply auto
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by (meson order.strict_trans sorted_Cons_iff)
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lemma del_list_sorted: "sorted (xs @ a # ys) \<Longrightarrow>
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del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
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by(induction xs)
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(fastforce simp: sorted_lems sorted_Cons_iff elems_eq_set intro!: del_list_idem)+
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text\<open>In principle, @{thm del_list_sorted} suffices, but the following
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corollaries speed up proofs.\<close>
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corollary del_list_sorted1: "sorted (xs @ a # ys) \<Longrightarrow> a \<le> x \<Longrightarrow>
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del_list x (xs @ a # ys) = xs @ del_list x (a # ys)"
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by (auto simp: del_list_sorted)
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corollary del_list_sorted2: "sorted (xs @ a # ys) \<Longrightarrow> x < a \<Longrightarrow>
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del_list x (xs @ a # ys) = del_list x xs @ a # ys"
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by (auto simp: del_list_sorted)
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corollary del_list_sorted3:
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"sorted (xs @ a # ys @ b # zs) \<Longrightarrow> x < b \<Longrightarrow>
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del_list x (xs @ a # ys @ b # zs) = del_list x (xs @ a # ys) @ b # zs"
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by (auto simp: del_list_sorted sorted_lems)
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corollary del_list_sorted4:
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"sorted (xs @ a # ys @ b # zs @ c # us) \<Longrightarrow> x < c \<Longrightarrow>
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del_list x (xs @ a # ys @ b # zs @ c # us) = del_list x (xs @ a # ys @ b # zs) @ c # us"
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by (auto simp: del_list_sorted sorted_lems)
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corollary del_list_sorted5:
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"sorted (xs @ a # ys @ b # zs @ c # us @ d # vs) \<Longrightarrow> x < d \<Longrightarrow>
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del_list x (xs @ a # ys @ b # zs @ c # us @ d # vs) =
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del_list x (xs @ a # ys @ b # zs @ c # us) @ d # vs"
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by (auto simp: del_list_sorted sorted_lems)
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lemmas del_list_simps = sorted_lems
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del_list_sorted1
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del_list_sorted2
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del_list_sorted3
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del_list_sorted4
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del_list_sorted5
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end
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