--- a/src/HOL/Library/Comparator.thy Fri Oct 26 14:12:08 2018 +0200
+++ b/src/HOL/Library/Comparator.thy Fri Oct 26 08:20:45 2018 +0000
@@ -184,6 +184,8 @@
end
+text \<open>Fundamental comparator combinators\<close>
+
lift_definition reversed :: "'a comparator \<Rightarrow> 'a comparator"
is "\<lambda>cmp a b. cmp b a"
proof -
--- a/src/HOL/Library/Library.thy Fri Oct 26 14:12:08 2018 +0200
+++ b/src/HOL/Library/Library.thy Fri Oct 26 08:20:45 2018 +0000
@@ -13,7 +13,6 @@
Code_Lazy
Code_Test
Combine_PER
- Comparator
Complete_Partial_Order2
Conditional_Parametricity
Countable
@@ -82,6 +81,7 @@
State_Monad
Stirling
Stream
+ Sorting_Algorithms
Sublist
Sum_of_Squares
Transitive_Closure_Table
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Sorting_Algorithms.thy Fri Oct 26 08:20:45 2018 +0000
@@ -0,0 +1,321 @@
+(* Title: HOL/Library/Sorting_Algorithms.thy
+ Author: Florian Haftmann, TU Muenchen
+*)
+
+theory Sorting_Algorithms
+ imports Main Multiset Comparator
+begin
+
+text \<open>Prelude\<close>
+
+hide_const (open) sorted insort sort
+
+
+section \<open>Stably sorted lists\<close>
+
+abbreviation (input) stable_segment :: "'a comparator \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
+ where "stable_segment cmp x \<equiv> filter (\<lambda>y. compare cmp x y = Equiv)"
+
+fun sorted :: "'a comparator \<Rightarrow> 'a list \<Rightarrow> bool"
+ where sorted_Nil: "sorted cmp [] \<longleftrightarrow> True"
+ | sorted_single: "sorted cmp [x] \<longleftrightarrow> True"
+ | sorted_rec: "sorted cmp (y # x # xs) \<longleftrightarrow> compare cmp x y \<noteq> Less \<and> sorted cmp (x # xs)"
+
+lemma sorted_ConsI:
+ "sorted cmp (x # xs)" if "sorted cmp xs"
+ and "\<And>y ys. xs = y # ys \<Longrightarrow> compare cmp y x \<noteq> Less"
+ using that by (cases xs) simp_all
+
+lemma sorted_induct [consumes 1, case_names Nil Cons, induct pred: sorted]:
+ "P xs" if "sorted cmp xs" and "P []"
+ and *: "\<And>x xs. sorted cmp xs \<Longrightarrow> P xs
+ \<Longrightarrow> (\<And>y. y \<in> set xs \<Longrightarrow> compare cmp y x \<noteq> Less) \<Longrightarrow> P (x # xs)"
+using \<open>sorted cmp xs\<close> proof (induction xs)
+ case Nil
+ show ?case
+ by (rule \<open>P []\<close>)
+next
+ case (Cons x xs)
+ from \<open>sorted cmp (x # xs)\<close> have "sorted cmp xs"
+ by (cases xs) simp_all
+ moreover have "P xs" using \<open>sorted cmp xs\<close>
+ by (rule Cons.IH)
+ moreover have "compare cmp y x \<noteq> Less" if "y \<in> set xs" for y
+ using that \<open>sorted cmp (x # xs)\<close> proof (induction xs)
+ case Nil
+ then show ?case
+ by simp
+ next
+ case (Cons z zs)
+ then show ?case
+ proof (cases zs)
+ case Nil
+ with Cons.prems show ?thesis
+ by simp
+ next
+ case (Cons w ws)
+ with Cons.prems have "compare cmp w z \<noteq> Less" "compare cmp z x \<noteq> Less"
+ by auto
+ then have "compare cmp w x \<noteq> Less"
+ by (auto dest: compare.trans_not_less)
+ with Cons show ?thesis
+ using Cons.prems Cons.IH by auto
+ qed
+ qed
+ ultimately show ?case
+ by (rule *)
+qed
+
+lemma sorted_induct_remove1 [consumes 1, case_names Nil minimum]:
+ "P xs" if "sorted cmp xs" and "P []"
+ and *: "\<And>x xs. sorted cmp xs \<Longrightarrow> P (remove1 x xs)
+ \<Longrightarrow> x \<in> set xs \<Longrightarrow> hd (stable_segment cmp x xs) = x \<Longrightarrow> (\<And>y. y \<in> set xs \<Longrightarrow> compare cmp y x \<noteq> Less)
+ \<Longrightarrow> P xs"
+using \<open>sorted cmp xs\<close> proof (induction xs)
+ case Nil
+ show ?case
+ by (rule \<open>P []\<close>)
+next
+ case (Cons x xs)
+ then have "sorted cmp (x # xs)"
+ by (simp add: sorted_ConsI)
+ moreover note Cons.IH
+ moreover have "\<And>y. compare cmp y x = Less \<Longrightarrow> y \<in> set xs \<Longrightarrow> False"
+ using Cons.hyps by simp
+ ultimately show ?case
+ by (auto intro!: * [of "x # xs" x]) blast
+qed
+
+lemma sorted_remove1:
+ "sorted cmp (remove1 x xs)" if "sorted cmp xs"
+proof (cases "x \<in> set xs")
+ case False
+ with that show ?thesis
+ by (simp add: remove1_idem)
+next
+ case True
+ with that show ?thesis proof (induction xs)
+ case Nil
+ then show ?case
+ by simp
+ next
+ case (Cons y ys)
+ show ?case proof (cases "x = y")
+ case True
+ with Cons.hyps show ?thesis
+ by simp
+ next
+ case False
+ then have "sorted cmp (remove1 x ys)"
+ using Cons.IH Cons.prems by auto
+ then have "sorted cmp (y # remove1 x ys)"
+ proof (rule sorted_ConsI)
+ fix z zs
+ assume "remove1 x ys = z # zs"
+ with \<open>x \<noteq> y\<close> have "z \<in> set ys"
+ using notin_set_remove1 [of z ys x] by auto
+ then show "compare cmp z y \<noteq> Less"
+ by (rule Cons.hyps(2))
+ qed
+ with False show ?thesis
+ by simp
+ qed
+ qed
+qed
+
+primrec insort :: "'a comparator \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
+ where "insort cmp y [] = [y]"
+ | "insort cmp y (x # xs) = (if compare cmp y x \<noteq> Greater
+ then y # x # xs
+ else x # insort cmp y xs)"
+
+lemma mset_insort [simp]:
+ "mset (insort cmp x xs) = add_mset x (mset xs)"
+ by (induction xs) simp_all
+
+lemma length_insort [simp]:
+ "length (insort cmp x xs) = Suc (length xs)"
+ by (induction xs) simp_all
+
+lemma sorted_insort:
+ "sorted cmp (insort cmp x xs)" if "sorted cmp xs"
+using that proof (induction xs)
+ case Nil
+ then show ?case
+ by simp
+next
+ case (Cons y ys)
+ then show ?case by (cases ys)
+ (auto, simp_all add: compare.greater_iff_sym_less)
+qed
+
+lemma stable_insort_equiv:
+ "stable_segment cmp y (insort cmp x xs) = x # stable_segment cmp y xs"
+ if "compare cmp y x = Equiv"
+proof (induction xs)
+ case Nil
+ from that show ?case
+ by simp
+next
+ case (Cons z xs)
+ moreover from that have "compare cmp y z = Equiv \<Longrightarrow> compare cmp z x = Equiv"
+ by (auto intro: compare.trans_equiv simp add: compare.sym)
+ ultimately show ?case
+ using that by (auto simp add: compare.greater_iff_sym_less)
+qed
+
+lemma stable_insort_not_equiv:
+ "stable_segment cmp y (insort cmp x xs) = stable_segment cmp y xs"
+ if "compare cmp y x \<noteq> Equiv"
+ using that by (induction xs) simp_all
+
+lemma remove1_insort_same_eq [simp]:
+ "remove1 x (insort cmp x xs) = xs"
+ by (induction xs) simp_all
+
+lemma insort_eq_ConsI:
+ "insort cmp x xs = x # xs"
+ if "sorted cmp xs" "\<And>y. y \<in> set xs \<Longrightarrow> compare cmp y x \<noteq> Less"
+ using that by (induction xs) (simp_all add: compare.greater_iff_sym_less)
+
+lemma remove1_insort_not_same_eq [simp]:
+ "remove1 y (insort cmp x xs) = insort cmp x (remove1 y xs)"
+ if "sorted cmp xs" "x \<noteq> y"
+using that proof (induction xs)
+ case Nil
+ then show ?case
+ by simp
+next
+ case (Cons z zs)
+ show ?case
+ proof (cases "compare cmp z x = Less")
+ case True
+ with Cons show ?thesis
+ by (simp add: compare.greater_iff_sym_less)
+ next
+ case False
+ have "compare cmp y x \<noteq> Less" if "y \<in> set zs" for y
+ using that False Cons.hyps by (auto dest: compare.trans_not_less)
+ with Cons show ?thesis
+ by (simp add: insort_eq_ConsI)
+ qed
+qed
+
+lemma insort_remove1_same_eq:
+ "insort cmp x (remove1 x xs) = xs"
+ if "sorted cmp xs" and "x \<in> set xs" and "hd (stable_segment cmp x xs) = x"
+using that proof (induction xs)
+ case Nil
+ then show ?case
+ by simp
+next
+ case (Cons y ys)
+ then have "compare cmp x y \<noteq> Less"
+ by auto
+ then consider "compare cmp x y = Greater" | "compare cmp x y = Equiv"
+ by (cases "compare cmp x y") auto
+ then show ?case proof cases
+ case 1
+ with Cons.prems Cons.IH show ?thesis
+ by auto
+ next
+ case 2
+ with Cons.prems have "x = y"
+ by simp
+ with Cons.hyps show ?thesis
+ by (simp add: insort_eq_ConsI)
+ qed
+qed
+
+definition sort :: "'a comparator \<Rightarrow> 'a list \<Rightarrow> 'a list"
+ where "sort cmp xs = foldr (insort cmp) xs []"
+
+lemma sort_simps [simp]:
+ "sort cmp [] = []"
+ "sort cmp (x # xs) = insort cmp x (sort cmp xs)"
+ by (simp_all add: sort_def)
+
+lemma mset_sort [simp]:
+ "mset (sort cmp xs) = mset xs"
+ by (induction xs) simp_all
+
+lemma length_sort [simp]:
+ "length (sort cmp xs) = length xs"
+ by (induction xs) simp_all
+
+lemma sorted_sort [simp]:
+ "sorted cmp (sort cmp xs)"
+ by (induction xs) (simp_all add: sorted_insort)
+
+lemma stable_sort:
+ "stable_segment cmp x (sort cmp xs) = stable_segment cmp x xs"
+ by (induction xs) (simp_all add: stable_insort_equiv stable_insort_not_equiv)
+
+lemma sort_remove1_eq [simp]:
+ "sort cmp (remove1 x xs) = remove1 x (sort cmp xs)"
+ by (induction xs) simp_all
+
+lemma set_insort [simp]:
+ "set (insort cmp x xs) = insert x (set xs)"
+ by (induction xs) auto
+
+lemma set_sort [simp]:
+ "set (sort cmp xs) = set xs"
+ by (induction xs) auto
+
+lemma sort_eqI:
+ "sort cmp ys = xs"
+ if permutation: "mset ys = mset xs"
+ and sorted: "sorted cmp xs"
+ and stable: "\<And>y. y \<in> set ys \<Longrightarrow>
+ stable_segment cmp y ys = stable_segment cmp y xs"
+proof -
+ have stable': "stable_segment cmp y ys =
+ stable_segment cmp y xs" for y
+ proof (cases "\<exists>x\<in>set ys. compare cmp y x = Equiv")
+ case True
+ then obtain z where "z \<in> set ys" and "compare cmp y z = Equiv"
+ by auto
+ then have "compare cmp y x = Equiv \<longleftrightarrow> compare cmp z x = Equiv" for x
+ by (meson compare.sym compare.trans_equiv)
+ moreover have "stable_segment cmp z ys =
+ stable_segment cmp z xs"
+ using \<open>z \<in> set ys\<close> by (rule stable)
+ ultimately show ?thesis
+ by simp
+ next
+ case False
+ moreover from permutation have "set ys = set xs"
+ by (rule mset_eq_setD)
+ ultimately show ?thesis
+ by simp
+ qed
+ show ?thesis
+ using sorted permutation stable' proof (induction xs arbitrary: ys rule: sorted_induct_remove1)
+ case Nil
+ then show ?case
+ by simp
+ next
+ case (minimum x xs)
+ from \<open>mset ys = mset xs\<close> have ys: "set ys = set xs"
+ by (rule mset_eq_setD)
+ then have "compare cmp y x \<noteq> Less" if "y \<in> set ys" for y
+ using that minimum.hyps by simp
+ from minimum.prems have stable: "stable_segment cmp x ys = stable_segment cmp x xs"
+ by simp
+ have "sort cmp (remove1 x ys) = remove1 x xs"
+ by (rule minimum.IH) (simp_all add: minimum.prems filter_remove1)
+ then have "remove1 x (sort cmp ys) = remove1 x xs"
+ by simp
+ then have "insort cmp x (remove1 x (sort cmp ys)) =
+ insort cmp x (remove1 x xs)"
+ by simp
+ also from minimum.hyps ys stable have "insort cmp x (remove1 x (sort cmp ys)) = sort cmp ys"
+ by (simp add: stable_sort insort_remove1_same_eq)
+ also from minimum.hyps have "insort cmp x (remove1 x xs) = xs"
+ by (simp add: insort_remove1_same_eq)
+ finally show ?case .
+ qed
+qed
+
+end