--- a/src/HOL/Data_Structures/List_Ins_Del.thy Wed Nov 11 16:42:30 2015 +0100
+++ b/src/HOL/Data_Structures/List_Ins_Del.thy Wed Nov 11 18:32:26 2015 +0100
@@ -1,149 +1,149 @@
-(* 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)
-
-text{* The above two rules introduce quantifiers. It turns out
-that in practice this is not a problem because of the simplicity of
-the "isin" functions that implement @{const elems}. Nevertheless
-it is possible to avoid the quantifiers with the help of some rewrite rules: *}
-
-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)
-
-lemma sorted_ConsD2: "sorted (y # xs) \<Longrightarrow> x \<le> y \<Longrightarrow> x \<notin> elems xs"
-using leD sorted_ConsD by blast
-
-lemma sorted_snocD2: "sorted (xs @ [y]) \<Longrightarrow> y \<le> x \<Longrightarrow> x \<notin> elems xs"
-using leD sorted_snocD by blast
-
-lemmas elems_simps = sorted_lems elems_app
-lemmas elems_simps1 = elems_simps sorted_Cons_iff sorted_snoc_iff
-lemmas elems_simps2 = elems_simps sorted_ConsD sorted_snocD sorted_ConsD2 sorted_snocD2
-
-
-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 (a#xs) =
- (if x < a then x#a#xs else if x=a then a#xs else a # ins_list x xs)"
-
-lemma set_ins_list: "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_sorted: "sorted (xs @ [a]) \<Longrightarrow>
- ins_list x (xs @ a # ys) =
- (if a \<le> x then xs @ ins_list x (a#ys) else ins_list x xs @ (a#ys))"
-by(induction xs) (auto simp: sorted_lems)
-
-text\<open>In principle, @{thm ins_list_sorted} suffices, but the following two
-corollaries speed up proofs.\<close>
-
-corollary ins_list_sorted1: "sorted (xs @ [a]) \<Longrightarrow> a \<le> x \<Longrightarrow>
- ins_list x (xs @ a # ys) = xs @ ins_list x (a#ys)"
-by(simp add: ins_list_sorted)
-
-corollary ins_list_sorted2: "sorted (xs @ [a]) \<Longrightarrow> x < a \<Longrightarrow>
- ins_list x (xs @ a # ys) = ins_list x xs @ (a#ys)"
-by(auto simp: ins_list_sorted)
-
-lemmas ins_list_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 x [] = []" |
-"del_list x (a#xs) = (if x=a then xs else a # del_list x 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:
- "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_sorted: "sorted (xs @ a # ys) \<Longrightarrow>
- del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
-by(induction xs)
- (fastforce simp: sorted_lems sorted_Cons_iff elems_eq_set intro!: del_list_idem)+
-
-text\<open>In principle, @{thm del_list_sorted} suffices, but the following
-corollaries speed up proofs.\<close>
-
-corollary del_list_sorted1: "sorted (xs @ a # ys) \<Longrightarrow> a \<le> x \<Longrightarrow>
- del_list x (xs @ a # ys) = xs @ del_list x (a # ys)"
-by (auto simp: del_list_sorted)
-
-corollary del_list_sorted2: "sorted (xs @ a # ys) \<Longrightarrow> x < a \<Longrightarrow>
- del_list x (xs @ a # ys) = del_list x xs @ a # ys"
-by (auto simp: del_list_sorted)
-
-corollary del_list_sorted3:
- "sorted (xs @ a # ys @ b # zs) \<Longrightarrow> x < b \<Longrightarrow>
- del_list x (xs @ a # ys @ b # zs) = del_list x (xs @ a # ys) @ b # zs"
-by (auto simp: del_list_sorted sorted_lems)
-
-corollary del_list_sorted4:
- "sorted (xs @ a # ys @ b # zs @ c # us) \<Longrightarrow> x < c \<Longrightarrow>
- del_list x (xs @ a # ys @ b # zs @ c # us) = del_list x (xs @ a # ys @ b # zs) @ c # us"
-by (auto simp: del_list_sorted sorted_lems)
-
-corollary del_list_sorted5:
- "sorted (xs @ a # ys @ b # zs @ c # us @ d # vs) \<Longrightarrow> x < d \<Longrightarrow>
- del_list x (xs @ a # ys @ b # zs @ c # us @ d # vs) =
- del_list x (xs @ a # ys @ b # zs @ c # us) @ d # vs"
-by (auto simp: del_list_sorted sorted_lems)
-
-lemmas del_list_simps = sorted_lems
- del_list_sorted1
- del_list_sorted2
- del_list_sorted3
- del_list_sorted4
- del_list_sorted5
-
-end
+(* 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)
+
+text{* The above two rules introduce quantifiers. It turns out
+that in practice this is not a problem because of the simplicity of
+the "isin" functions that implement @{const elems}. Nevertheless
+it is possible to avoid the quantifiers with the help of some rewrite rules: *}
+
+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)
+
+lemma sorted_ConsD2: "sorted (y # xs) \<Longrightarrow> x \<le> y \<Longrightarrow> x \<notin> elems xs"
+using leD sorted_ConsD by blast
+
+lemma sorted_snocD2: "sorted (xs @ [y]) \<Longrightarrow> y \<le> x \<Longrightarrow> x \<notin> elems xs"
+using leD sorted_snocD by blast
+
+lemmas elems_simps = sorted_lems elems_app
+lemmas elems_simps1 = elems_simps sorted_Cons_iff sorted_snoc_iff
+lemmas elems_simps2 = elems_simps sorted_ConsD sorted_snocD sorted_ConsD2 sorted_snocD2
+
+
+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 (a#xs) =
+ (if x < a then x#a#xs else if x=a then a#xs else a # ins_list x xs)"
+
+lemma set_ins_list: "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_sorted: "sorted (xs @ [a]) \<Longrightarrow>
+ ins_list x (xs @ a # ys) =
+ (if a \<le> x then xs @ ins_list x (a#ys) else ins_list x xs @ (a#ys))"
+by(induction xs) (auto simp: sorted_lems)
+
+text\<open>In principle, @{thm ins_list_sorted} suffices, but the following two
+corollaries speed up proofs.\<close>
+
+corollary ins_list_sorted1: "sorted (xs @ [a]) \<Longrightarrow> a \<le> x \<Longrightarrow>
+ ins_list x (xs @ a # ys) = xs @ ins_list x (a#ys)"
+by(simp add: ins_list_sorted)
+
+corollary ins_list_sorted2: "sorted (xs @ [a]) \<Longrightarrow> x < a \<Longrightarrow>
+ ins_list x (xs @ a # ys) = ins_list x xs @ (a#ys)"
+by(auto simp: ins_list_sorted)
+
+lemmas ins_list_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 x [] = []" |
+"del_list x (a#xs) = (if x=a then xs else a # del_list x 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:
+ "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_sorted: "sorted (xs @ a # ys) \<Longrightarrow>
+ del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
+by(induction xs)
+ (fastforce simp: sorted_lems sorted_Cons_iff elems_eq_set intro!: del_list_idem)+
+
+text\<open>In principle, @{thm del_list_sorted} suffices, but the following
+corollaries speed up proofs.\<close>
+
+corollary del_list_sorted1: "sorted (xs @ a # ys) \<Longrightarrow> a \<le> x \<Longrightarrow>
+ del_list x (xs @ a # ys) = xs @ del_list x (a # ys)"
+by (auto simp: del_list_sorted)
+
+corollary del_list_sorted2: "sorted (xs @ a # ys) \<Longrightarrow> x < a \<Longrightarrow>
+ del_list x (xs @ a # ys) = del_list x xs @ a # ys"
+by (auto simp: del_list_sorted)
+
+corollary del_list_sorted3:
+ "sorted (xs @ a # ys @ b # zs) \<Longrightarrow> x < b \<Longrightarrow>
+ del_list x (xs @ a # ys @ b # zs) = del_list x (xs @ a # ys) @ b # zs"
+by (auto simp: del_list_sorted sorted_lems)
+
+corollary del_list_sorted4:
+ "sorted (xs @ a # ys @ b # zs @ c # us) \<Longrightarrow> x < c \<Longrightarrow>
+ del_list x (xs @ a # ys @ b # zs @ c # us) = del_list x (xs @ a # ys @ b # zs) @ c # us"
+by (auto simp: del_list_sorted sorted_lems)
+
+corollary del_list_sorted5:
+ "sorted (xs @ a # ys @ b # zs @ c # us @ d # vs) \<Longrightarrow> x < d \<Longrightarrow>
+ del_list x (xs @ a # ys @ b # zs @ c # us @ d # vs) =
+ del_list x (xs @ a # ys @ b # zs @ c # us) @ d # vs"
+by (auto simp: del_list_sorted sorted_lems)
+
+lemmas del_list_simps = sorted_lems
+ del_list_sorted1
+ del_list_sorted2
+ del_list_sorted3
+ del_list_sorted4
+ del_list_sorted5
+
+end