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author | eberlm <eberlm@in.tum.de> |

Thu, 29 Sep 2016 16:49:42 +0200 | |

changeset 63965 | d510b816ea41 |

parent 63954 | fb03766658f4 |

child 63966 | 957ba35d1338 |

Set_Permutations replaced by more general Multiset_Permutations

--- a/src/HOL/Codegenerator_Test/Generate_Efficient_Datastructures.thy Thu Sep 29 11:24:36 2016 +0100 +++ b/src/HOL/Codegenerator_Test/Generate_Efficient_Datastructures.thy Thu Sep 29 16:49:42 2016 +0200 @@ -76,6 +76,9 @@ lemma [code, code del]: "permutations_of_set = permutations_of_set" .. +lemma [code, code del]: + "permutations_of_multiset = permutations_of_multiset" .. + (* If the code generation ends with an exception with the following message: '"List.set" is not a constructor, on left hand side of equation: ...',

--- a/src/HOL/Library/Library.thy Thu Sep 29 11:24:36 2016 +0100 +++ b/src/HOL/Library/Library.thy Thu Sep 29 16:49:42 2016 +0200 @@ -51,6 +51,7 @@ Monad_Syntax More_List Multiset_Order + Multiset_Permutations Numeral_Type Omega_Words_Fun OptionalSugar @@ -76,7 +77,6 @@ Reflection Saturated Set_Algebras - Set_Permutations State_Monad Stirling Stream

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Library/Multiset_Permutations.thy Thu Sep 29 16:49:42 2016 +0200 @@ -0,0 +1,510 @@ +(* + File: Multiset_Permutations.thy + Author: Manuel Eberl (TU München) + + Defines the set of permutations of a given multiset (or set), i.e. the set of all lists whose + entries correspond to the multiset (resp. set). +*) +section \<open>Permutations of a Multiset\<close> +theory Multiset_Permutations +imports + Complex_Main + "~~/src/HOL/Library/Multiset" + "~~/src/HOL/Library/Permutations" +begin + +(* TODO Move *) +lemma mset_tl: "xs \<noteq> [] \<Longrightarrow> mset (tl xs) = mset xs - {#hd xs#}" + by (cases xs) simp_all + +lemma mset_set_image_inj: + assumes "inj_on f A" + shows "mset_set (f ` A) = image_mset f (mset_set A)" +proof (cases "finite A") + case True + from this and assms show ?thesis by (induction A) auto +qed (insert assms, simp add: finite_image_iff) + +lemma multiset_remove_induct [case_names empty remove]: + assumes "P {#}" "\<And>A. A \<noteq> {#} \<Longrightarrow> (\<And>x. x \<in># A \<Longrightarrow> P (A - {#x#})) \<Longrightarrow> P A" + shows "P A" +proof (induction A rule: full_multiset_induct) + case (less A) + hence IH: "P B" if "B \<subset># A" for B using that by blast + show ?case + proof (cases "A = {#}") + case True + thus ?thesis by (simp add: assms) + next + case False + hence "P (A - {#x#})" if "x \<in># A" for x + using that by (intro IH) (simp add: mset_subset_diff_self) + from False and this show "P A" by (rule assms) + qed +qed + +lemma map_list_bind: "map g (List.bind xs f) = List.bind xs (map g \<circ> f)" + by (simp add: List.bind_def map_concat) + +lemma mset_eq_mset_set_imp_distinct: + "finite A \<Longrightarrow> mset_set A = mset xs \<Longrightarrow> distinct xs" +proof (induction xs arbitrary: A) + case (Cons x xs A) + from Cons.prems(2) have "x \<in># mset_set A" by simp + with Cons.prems(1) have [simp]: "x \<in> A" by simp + from Cons.prems have "x \<notin># mset_set (A - {x})" by simp + also from Cons.prems have "mset_set (A - {x}) = mset_set A - {#x#}" + by (subst mset_set_Diff) simp_all + also have "mset_set A = mset (x#xs)" by (simp add: Cons.prems) + also have "\<dots> - {#x#} = mset xs" by simp + finally have [simp]: "x \<notin> set xs" by (simp add: in_multiset_in_set) + from Cons.prems show ?case by (auto intro!: Cons.IH[of "A - {x}"] simp: mset_set_Diff) +qed simp_all +(* END TODO *) + + +subsection \<open>Permutations of a multiset\<close> + +definition permutations_of_multiset :: "'a multiset \<Rightarrow> 'a list set" where + "permutations_of_multiset A = {xs. mset xs = A}" + +lemma permutations_of_multisetI: "mset xs = A \<Longrightarrow> xs \<in> permutations_of_multiset A" + by (simp add: permutations_of_multiset_def) + +lemma permutations_of_multisetD: "xs \<in> permutations_of_multiset A \<Longrightarrow> mset xs = A" + by (simp add: permutations_of_multiset_def) + +lemma permutations_of_multiset_Cons_iff: + "x # xs \<in> permutations_of_multiset A \<longleftrightarrow> x \<in># A \<and> xs \<in> permutations_of_multiset (A - {#x#})" + by (auto simp: permutations_of_multiset_def) + +lemma permutations_of_multiset_empty [simp]: "permutations_of_multiset {#} = {[]}" + unfolding permutations_of_multiset_def by simp + +lemma permutations_of_multiset_nonempty: + assumes nonempty: "A \<noteq> {#}" + shows "permutations_of_multiset A = + (\<Union>x\<in>set_mset A. (op # x) ` permutations_of_multiset (A - {#x#}))" (is "_ = ?rhs") +proof safe + fix xs assume "xs \<in> permutations_of_multiset A" + hence mset_xs: "mset xs = A" by (simp add: permutations_of_multiset_def) + hence "xs \<noteq> []" by (auto simp: nonempty) + then obtain x xs' where xs: "xs = x # xs'" by (cases xs) simp_all + with mset_xs have "x \<in> set_mset A" "xs' \<in> permutations_of_multiset (A - {#x#})" + by (auto simp: permutations_of_multiset_def) + with xs show "xs \<in> ?rhs" by auto +qed (auto simp: permutations_of_multiset_def) + +lemma permutations_of_multiset_singleton [simp]: "permutations_of_multiset {#x#} = {[x]}" + by (simp add: permutations_of_multiset_nonempty) + +lemma permutations_of_multiset_doubleton: + "permutations_of_multiset {#x,y#} = {[x,y], [y,x]}" + by (simp add: permutations_of_multiset_nonempty insert_commute) + +lemma rev_permutations_of_multiset [simp]: + "rev ` permutations_of_multiset A = permutations_of_multiset A" +proof + have "rev ` rev ` permutations_of_multiset A \<subseteq> rev ` permutations_of_multiset A" + unfolding permutations_of_multiset_def by auto + also have "rev ` rev ` permutations_of_multiset A = permutations_of_multiset A" + by (simp add: image_image) + finally show "permutations_of_multiset A \<subseteq> rev ` permutations_of_multiset A" . +next + show "rev ` permutations_of_multiset A \<subseteq> permutations_of_multiset A" + unfolding permutations_of_multiset_def by auto +qed + +lemma length_finite_permutations_of_multiset: + "xs \<in> permutations_of_multiset A \<Longrightarrow> length xs = size A" + by (auto simp: permutations_of_multiset_def) + +lemma permutations_of_multiset_lists: "permutations_of_multiset A \<subseteq> lists (set_mset A)" + by (auto simp: permutations_of_multiset_def) + +lemma finite_permutations_of_multiset [simp]: "finite (permutations_of_multiset A)" +proof (rule finite_subset) + show "permutations_of_multiset A \<subseteq> {xs. set xs \<subseteq> set_mset A \<and> length xs = size A}" + by (auto simp: permutations_of_multiset_def) + show "finite {xs. set xs \<subseteq> set_mset A \<and> length xs = size A}" + by (rule finite_lists_length_eq) simp_all +qed + +lemma permutations_of_multiset_not_empty [simp]: "permutations_of_multiset A \<noteq> {}" +proof - + from ex_mset[of A] guess xs .. + thus ?thesis by (auto simp: permutations_of_multiset_def) +qed + +lemma permutations_of_multiset_image: + "permutations_of_multiset (image_mset f A) = map f ` permutations_of_multiset A" +proof safe + fix xs assume A: "xs \<in> permutations_of_multiset (image_mset f A)" + from ex_mset[of A] obtain ys where ys: "mset ys = A" .. + with A have "mset xs = mset (map f ys)" + by (simp add: permutations_of_multiset_def) + from mset_eq_permutation[OF this] guess \<sigma> . note \<sigma> = this + with ys have "xs = map f (permute_list \<sigma> ys)" + by (simp add: permute_list_map) + moreover from \<sigma> ys have "permute_list \<sigma> ys \<in> permutations_of_multiset A" + by (simp add: permutations_of_multiset_def) + ultimately show "xs \<in> map f ` permutations_of_multiset A" by blast +qed (auto simp: permutations_of_multiset_def) + + +subsection \<open>Cardinality of permutations\<close> + +text \<open> + In this section, we prove some basic facts about the number of permutations of a multiset. +\<close> + +context +begin + +private lemma multiset_setprod_fact_insert: + "(\<Prod>y\<in>set_mset (A+{#x#}). fact (count (A+{#x#}) y)) = + (count A x + 1) * (\<Prod>y\<in>set_mset A. fact (count A y))" +proof - + have "(\<Prod>y\<in>set_mset (A+{#x#}). fact (count (A+{#x#}) y)) = + (\<Prod>y\<in>set_mset (A+{#x#}). (if y = x then count A x + 1 else 1) * fact (count A y))" + by (intro setprod.cong) simp_all + also have "\<dots> = (count A x + 1) * (\<Prod>y\<in>set_mset (A+{#x#}). fact (count A y))" + by (simp add: setprod.distrib setprod.delta) + also have "(\<Prod>y\<in>set_mset (A+{#x#}). fact (count A y)) = (\<Prod>y\<in>set_mset A. fact (count A y))" + by (intro setprod.mono_neutral_right) (auto simp: not_in_iff) + finally show ?thesis . +qed + +private lemma multiset_setprod_fact_remove: + "x \<in># A \<Longrightarrow> (\<Prod>y\<in>set_mset A. fact (count A y)) = + count A x * (\<Prod>y\<in>set_mset (A-{#x#}). fact (count (A-{#x#}) y))" + using multiset_setprod_fact_insert[of "A - {#x#}" x] by simp + +lemma card_permutations_of_multiset_aux: + "card (permutations_of_multiset A) * (\<Prod>x\<in>set_mset A. fact (count A x)) = fact (size A)" +proof (induction A rule: multiset_remove_induct) + case (remove A) + have "card (permutations_of_multiset A) = + card (\<Union>x\<in>set_mset A. op # x ` permutations_of_multiset (A - {#x#}))" + by (simp add: permutations_of_multiset_nonempty remove.hyps) + also have "\<dots> = (\<Sum>x\<in>set_mset A. card (permutations_of_multiset (A - {#x#})))" + by (subst card_UN_disjoint) (auto simp: card_image) + also have "\<dots> * (\<Prod>x\<in>set_mset A. fact (count A x)) = + (\<Sum>x\<in>set_mset A. card (permutations_of_multiset (A - {#x#})) * + (\<Prod>y\<in>set_mset A. fact (count A y)))" + by (subst setsum_distrib_right) simp_all + also have "\<dots> = (\<Sum>x\<in>set_mset A. count A x * fact (size A - 1))" + proof (intro setsum.cong refl) + fix x assume x: "x \<in># A" + have "card (permutations_of_multiset (A - {#x#})) * (\<Prod>y\<in>set_mset A. fact (count A y)) = + count A x * (card (permutations_of_multiset (A - {#x#})) * + (\<Prod>y\<in>set_mset (A - {#x#}). fact (count (A - {#x#}) y)))" (is "?lhs = _") + by (subst multiset_setprod_fact_remove[OF x]) simp_all + also note remove.IH[OF x] + also from x have "size (A - {#x#}) = size A - 1" by (simp add: size_Diff_submset) + finally show "?lhs = count A x * fact (size A - 1)" . + qed + also have "(\<Sum>x\<in>set_mset A. count A x * fact (size A - 1)) = + size A * fact (size A - 1)" + by (simp add: setsum_distrib_right size_multiset_overloaded_eq) + also from remove.hyps have "\<dots> = fact (size A)" + by (cases "size A") auto + finally show ?case . +qed simp_all + +theorem card_permutations_of_multiset: + "card (permutations_of_multiset A) = fact (size A) div (\<Prod>x\<in>set_mset A. fact (count A x))" + "(\<Prod>x\<in>set_mset A. fact (count A x) :: nat) dvd fact (size A)" + by (simp_all add: card_permutations_of_multiset_aux[of A, symmetric]) + +lemma card_permutations_of_multiset_insert_aux: + "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = + (size A + 1) * card (permutations_of_multiset A)" +proof - + note card_permutations_of_multiset_aux[of "A + {#x#}"] + also have "fact (size (A + {#x#})) = (size A + 1) * fact (size A)" by simp + also note multiset_setprod_fact_insert[of A x] + also note card_permutations_of_multiset_aux[of A, symmetric] + finally have "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) * + (\<Prod>y\<in>set_mset A. fact (count A y)) = + (size A + 1) * card (permutations_of_multiset A) * + (\<Prod>x\<in>set_mset A. fact (count A x))" by (simp only: mult_ac) + thus ?thesis by (subst (asm) mult_right_cancel) simp_all +qed + +lemma card_permutations_of_multiset_remove_aux: + assumes "x \<in># A" + shows "card (permutations_of_multiset A) * count A x = + size A * card (permutations_of_multiset (A - {#x#}))" +proof - + from assms have A: "A - {#x#} + {#x#} = A" by simp + from assms have B: "size A = size (A - {#x#}) + 1" + by (subst A [symmetric], subst size_union) simp + show ?thesis + using card_permutations_of_multiset_insert_aux[of "A - {#x#}" x, unfolded A] assms + by (simp add: B) +qed + +lemma real_card_permutations_of_multiset_remove: + assumes "x \<in># A" + shows "real (card (permutations_of_multiset (A - {#x#}))) = + real (card (permutations_of_multiset A) * count A x) / real (size A)" + using assms by (subst card_permutations_of_multiset_remove_aux[OF assms]) auto + +lemma real_card_permutations_of_multiset_remove': + assumes "x \<in># A" + shows "real (card (permutations_of_multiset A)) = + real (size A * card (permutations_of_multiset (A - {#x#}))) / real (count A x)" + using assms by (subst card_permutations_of_multiset_remove_aux[OF assms, symmetric]) simp + +end + + + +subsection \<open>Permutations of a set\<close> + +definition permutations_of_set :: "'a set \<Rightarrow> 'a list set" where + "permutations_of_set A = {xs. set xs = A \<and> distinct xs}" + +lemma permutations_of_set_altdef: + "finite A \<Longrightarrow> permutations_of_set A = permutations_of_multiset (mset_set A)" + by (auto simp add: permutations_of_set_def permutations_of_multiset_def mset_set_set + in_multiset_in_set [symmetric] mset_eq_mset_set_imp_distinct) + +lemma permutations_of_setI [intro]: + assumes "set xs = A" "distinct xs" + shows "xs \<in> permutations_of_set A" + using assms unfolding permutations_of_set_def by simp + +lemma permutations_of_setD: + assumes "xs \<in> permutations_of_set A" + shows "set xs = A" "distinct xs" + using assms unfolding permutations_of_set_def by simp_all + +lemma permutations_of_set_lists: "permutations_of_set A \<subseteq> lists A" + unfolding permutations_of_set_def by auto + +lemma permutations_of_set_empty [simp]: "permutations_of_set {} = {[]}" + by (auto simp: permutations_of_set_def) + +lemma UN_set_permutations_of_set [simp]: + "finite A \<Longrightarrow> (\<Union>xs\<in>permutations_of_set A. set xs) = A" + using finite_distinct_list by (auto simp: permutations_of_set_def) + +lemma permutations_of_set_infinite: + "\<not>finite A \<Longrightarrow> permutations_of_set A = {}" + by (auto simp: permutations_of_set_def) + +lemma permutations_of_set_nonempty: + "A \<noteq> {} \<Longrightarrow> permutations_of_set A = + (\<Union>x\<in>A. (\<lambda>xs. x # xs) ` permutations_of_set (A - {x}))" + by (cases "finite A") + (simp_all add: permutations_of_multiset_nonempty mset_set_empty_iff mset_set_Diff + permutations_of_set_altdef permutations_of_set_infinite) + +lemma permutations_of_set_singleton [simp]: "permutations_of_set {x} = {[x]}" + by (subst permutations_of_set_nonempty) auto + +lemma permutations_of_set_doubleton: + "x \<noteq> y \<Longrightarrow> permutations_of_set {x,y} = {[x,y], [y,x]}" + by (subst permutations_of_set_nonempty) + (simp_all add: insert_Diff_if insert_commute) + +lemma rev_permutations_of_set [simp]: + "rev ` permutations_of_set A = permutations_of_set A" + by (cases "finite A") (simp_all add: permutations_of_set_altdef permutations_of_set_infinite) + +lemma length_finite_permutations_of_set: + "xs \<in> permutations_of_set A \<Longrightarrow> length xs = card A" + by (auto simp: permutations_of_set_def distinct_card) + +lemma finite_permutations_of_set [simp]: "finite (permutations_of_set A)" + by (cases "finite A") (simp_all add: permutations_of_set_infinite permutations_of_set_altdef) + +lemma permutations_of_set_empty_iff [simp]: + "permutations_of_set A = {} \<longleftrightarrow> \<not>finite A" + unfolding permutations_of_set_def using finite_distinct_list[of A] by auto + +lemma card_permutations_of_set [simp]: + "finite A \<Longrightarrow> card (permutations_of_set A) = fact (card A)" + by (simp add: permutations_of_set_altdef card_permutations_of_multiset del: One_nat_def) + +lemma permutations_of_set_image_inj: + assumes inj: "inj_on f A" + shows "permutations_of_set (f ` A) = map f ` permutations_of_set A" + by (cases "finite A") + (simp_all add: permutations_of_set_infinite permutations_of_set_altdef + permutations_of_multiset_image mset_set_image_inj inj finite_image_iff) + +lemma permutations_of_set_image_permutes: + "\<sigma> permutes A \<Longrightarrow> map \<sigma> ` permutations_of_set A = permutations_of_set A" + by (subst permutations_of_set_image_inj [symmetric]) + (simp_all add: permutes_inj_on permutes_image) + + +subsection \<open>Code generation\<close> + +text \<open> + First, we give code an implementation for permutations of lists. +\<close> + +declare length_remove1 [termination_simp] + +fun permutations_of_list_impl where + "permutations_of_list_impl xs = (if xs = [] then [[]] else + List.bind (remdups xs) (\<lambda>x. map (op # x) (permutations_of_list_impl (remove1 x xs))))" + +fun permutations_of_list_impl_aux where + "permutations_of_list_impl_aux acc xs = (if xs = [] then [acc] else + List.bind (remdups xs) (\<lambda>x. permutations_of_list_impl_aux (x#acc) (remove1 x xs)))" + +declare permutations_of_list_impl_aux.simps [simp del] +declare permutations_of_list_impl.simps [simp del] + +lemma permutations_of_list_impl_Nil [simp]: + "permutations_of_list_impl [] = [[]]" + by (simp add: permutations_of_list_impl.simps) + +lemma permutations_of_list_impl_nonempty: + "xs \<noteq> [] \<Longrightarrow> permutations_of_list_impl xs = + List.bind (remdups xs) (\<lambda>x. map (op # x) (permutations_of_list_impl (remove1 x xs)))" + by (subst permutations_of_list_impl.simps) simp_all + +lemma set_permutations_of_list_impl: + "set (permutations_of_list_impl xs) = permutations_of_multiset (mset xs)" + by (induction xs rule: permutations_of_list_impl.induct) + (subst permutations_of_list_impl.simps, + simp_all add: permutations_of_multiset_nonempty set_list_bind) + +lemma distinct_permutations_of_list_impl: + "distinct (permutations_of_list_impl xs)" + by (induction xs rule: permutations_of_list_impl.induct, + subst permutations_of_list_impl.simps) + (auto intro!: distinct_list_bind simp: distinct_map o_def disjoint_family_on_def) + +lemma permutations_of_list_impl_aux_correct': + "permutations_of_list_impl_aux acc xs = + map (\<lambda>xs. rev xs @ acc) (permutations_of_list_impl xs)" + by (induction acc xs rule: permutations_of_list_impl_aux.induct, + subst permutations_of_list_impl_aux.simps, subst permutations_of_list_impl.simps) + (auto simp: map_list_bind intro!: list_bind_cong) + +lemma permutations_of_list_impl_aux_correct: + "permutations_of_list_impl_aux [] xs = map rev (permutations_of_list_impl xs)" + by (simp add: permutations_of_list_impl_aux_correct') + +lemma distinct_permutations_of_list_impl_aux: + "distinct (permutations_of_list_impl_aux acc xs)" + by (simp add: permutations_of_list_impl_aux_correct' distinct_map + distinct_permutations_of_list_impl inj_on_def) + +lemma set_permutations_of_list_impl_aux: + "set (permutations_of_list_impl_aux [] xs) = permutations_of_multiset (mset xs)" + by (simp add: permutations_of_list_impl_aux_correct set_permutations_of_list_impl) + +declare set_permutations_of_list_impl_aux [symmetric, code] + +value [code] "permutations_of_multiset {#1,2,3,4::int#}" + + + +text \<open> + Now we turn to permutations of sets. We define an auxiliary version with an + accumulator to avoid having to map over the results. +\<close> +function permutations_of_set_aux where + "permutations_of_set_aux acc A = + (if \<not>finite A then {} else if A = {} then {acc} else + (\<Union>x\<in>A. permutations_of_set_aux (x#acc) (A - {x})))" +by auto +termination by (relation "Wellfounded.measure (card \<circ> snd)") (simp_all add: card_gt_0_iff) + +lemma permutations_of_set_aux_altdef: + "permutations_of_set_aux acc A = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A" +proof (cases "finite A") + assume "finite A" + thus ?thesis + proof (induction A arbitrary: acc rule: finite_psubset_induct) + case (psubset A acc) + show ?case + proof (cases "A = {}") + case False + note [simp del] = permutations_of_set_aux.simps + from psubset.hyps False + have "permutations_of_set_aux acc A = + (\<Union>y\<in>A. permutations_of_set_aux (y#acc) (A - {y}))" + by (subst permutations_of_set_aux.simps) simp_all + also have "\<dots> = (\<Union>y\<in>A. (\<lambda>xs. rev xs @ acc) ` (\<lambda>xs. y # xs) ` permutations_of_set (A - {y}))" + by (intro SUP_cong refl, subst psubset) (auto simp: image_image) + also from False have "\<dots> = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A" + by (subst (2) permutations_of_set_nonempty) (simp_all add: image_UN) + finally show ?thesis . + qed simp_all + qed +qed (simp_all add: permutations_of_set_infinite) + +declare permutations_of_set_aux.simps [simp del] + +lemma permutations_of_set_aux_correct: + "permutations_of_set_aux [] A = permutations_of_set A" + by (simp add: permutations_of_set_aux_altdef) + + +text \<open> + In another refinement step, we define a version on lists. +\<close> +declare length_remove1 [termination_simp] + +fun permutations_of_set_aux_list where + "permutations_of_set_aux_list acc xs = + (if xs = [] then [acc] else + List.bind xs (\<lambda>x. permutations_of_set_aux_list (x#acc) (List.remove1 x xs)))" + +definition permutations_of_set_list where + "permutations_of_set_list xs = permutations_of_set_aux_list [] xs" + +declare permutations_of_set_aux_list.simps [simp del] + +lemma permutations_of_set_aux_list_refine: + assumes "distinct xs" + shows "set (permutations_of_set_aux_list acc xs) = permutations_of_set_aux acc (set xs)" + using assms + by (induction acc xs rule: permutations_of_set_aux_list.induct) + (subst permutations_of_set_aux_list.simps, + subst permutations_of_set_aux.simps, + simp_all add: set_list_bind cong: SUP_cong) + + +text \<open> + The permutation lists contain no duplicates if the inputs contain no duplicates. + Therefore, these functions can easily be used when working with a representation of + sets by distinct lists. + The same approach should generalise to any kind of set implementation that supports + a monadic bind operation, and since the results are disjoint, merging should be cheap. +\<close> +lemma distinct_permutations_of_set_aux_list: + "distinct xs \<Longrightarrow> distinct (permutations_of_set_aux_list acc xs)" + by (induction acc xs rule: permutations_of_set_aux_list.induct) + (subst permutations_of_set_aux_list.simps, + auto intro!: distinct_list_bind simp: disjoint_family_on_def + permutations_of_set_aux_list_refine permutations_of_set_aux_altdef) + +lemma distinct_permutations_of_set_list: + "distinct xs \<Longrightarrow> distinct (permutations_of_set_list xs)" + by (simp add: permutations_of_set_list_def distinct_permutations_of_set_aux_list) + +lemma permutations_of_list: + "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))" + by (simp add: permutations_of_set_aux_correct [symmetric] + permutations_of_set_aux_list_refine permutations_of_set_list_def) + +lemma permutations_of_list_code [code]: + "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))" + "permutations_of_set (List.coset xs) = + Code.abort (STR ''Permutation of set complement not supported'') + (\<lambda>_. permutations_of_set (List.coset xs))" + by (simp_all add: permutations_of_list) + +value [code] "permutations_of_set (set ''abcd'')" + +end \ No newline at end of file

--- a/src/HOL/Library/Set_Permutations.thy Thu Sep 29 11:24:36 2016 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,250 +0,0 @@ -(* - Title: Set_Permutations.thy - Author: Manuel Eberl, TU München - - The set of permutations of a finite set, i.e. the set of all - lists that contain every element of the set once. -*) - -section \<open>Set Permutations\<close> - -theory Set_Permutations -imports - Complex_Main - "~~/src/HOL/Library/Disjoint_Sets" - "~~/src/HOL/Library/Permutations" -begin - -subsection \<open>Definition and general facts\<close> - -definition permutations_of_set :: "'a set \<Rightarrow> 'a list set" where - "permutations_of_set A = {xs. set xs = A \<and> distinct xs}" - -lemma permutations_of_setI [intro]: - assumes "set xs = A" "distinct xs" - shows "xs \<in> permutations_of_set A" - using assms unfolding permutations_of_set_def by simp - -lemma permutations_of_setD: - assumes "xs \<in> permutations_of_set A" - shows "set xs = A" "distinct xs" - using assms unfolding permutations_of_set_def by simp_all - -lemma permutations_of_set_lists: "permutations_of_set A \<subseteq> lists A" - unfolding permutations_of_set_def by auto - -lemma permutations_of_set_empty [simp]: "permutations_of_set {} = {[]}" - by (auto simp: permutations_of_set_def) - -lemma UN_set_permutations_of_set [simp]: - "finite A \<Longrightarrow> (\<Union>xs\<in>permutations_of_set A. set xs) = A" - using finite_distinct_list by (auto simp: permutations_of_set_def) - -lemma permutations_of_set_nonempty: - assumes "A \<noteq> {}" - shows "permutations_of_set A = - (\<Union>x\<in>A. (\<lambda>xs. x # xs) ` permutations_of_set (A - {x}))" (is "?lhs = ?rhs") -proof (intro equalityI subsetI) - fix ys assume ys: "ys \<in> permutations_of_set A" - with assms have "ys \<noteq> []" by (auto simp: permutations_of_set_def) - then obtain x xs where xs: "ys = x # xs" by (cases ys) simp_all - from xs ys have "x \<in> A" "xs \<in> permutations_of_set (A - {x})" - by (auto simp: permutations_of_set_def) - with xs show "ys \<in> ?rhs" by auto -next - fix ys assume ys: "ys \<in> ?rhs" - then obtain x xs where xs: "ys = x # xs" "x \<in> A" "xs \<in> permutations_of_set (A - {x})" - by auto - with ys show "ys \<in> ?lhs" by (auto simp: permutations_of_set_def) -qed - -lemma permutations_of_set_singleton [simp]: "permutations_of_set {x} = {[x]}" - by (subst permutations_of_set_nonempty) auto - -lemma permutations_of_set_doubleton: - "x \<noteq> y \<Longrightarrow> permutations_of_set {x,y} = {[x,y], [y,x]}" - by (subst permutations_of_set_nonempty) - (simp_all add: insert_Diff_if insert_commute) - -lemma rev_permutations_of_set [simp]: - "rev ` permutations_of_set A = permutations_of_set A" -proof - have "rev ` rev ` permutations_of_set A \<subseteq> rev ` permutations_of_set A" - unfolding permutations_of_set_def by auto - also have "rev ` rev ` permutations_of_set A = permutations_of_set A" - by (simp add: image_image) - finally show "permutations_of_set A \<subseteq> rev ` permutations_of_set A" . -next - show "rev ` permutations_of_set A \<subseteq> permutations_of_set A" - unfolding permutations_of_set_def by auto -qed - -lemma length_finite_permutations_of_set: - "xs \<in> permutations_of_set A \<Longrightarrow> length xs = card A" - by (auto simp: permutations_of_set_def distinct_card) - -lemma permutations_of_set_infinite: - "\<not>finite A \<Longrightarrow> permutations_of_set A = {}" - by (auto simp: permutations_of_set_def) - -lemma finite_permutations_of_set [simp]: "finite (permutations_of_set A)" -proof (cases "finite A") - assume fin: "finite A" - have "permutations_of_set A \<subseteq> {xs. set xs \<subseteq> A \<and> length xs = card A}" - unfolding permutations_of_set_def by (auto simp: distinct_card) - moreover from fin have "finite \<dots>" using finite_lists_length_eq by blast - ultimately show ?thesis by (rule finite_subset) -qed (simp_all add: permutations_of_set_infinite) - -lemma permutations_of_set_empty_iff [simp]: - "permutations_of_set A = {} \<longleftrightarrow> \<not>finite A" - unfolding permutations_of_set_def using finite_distinct_list[of A] by auto - -lemma card_permutations_of_set [simp]: - "finite A \<Longrightarrow> card (permutations_of_set A) = fact (card A)" -proof (induction A rule: finite_remove_induct) - case (remove A) - hence "card (permutations_of_set A) = - card (\<Union>x\<in>A. op # x ` permutations_of_set (A - {x}))" - by (simp add: permutations_of_set_nonempty) - also from remove.hyps have "\<dots> = (\<Sum>i\<in>A. card (op # i ` permutations_of_set (A - {i})))" - by (intro card_UN_disjoint) auto - also have "\<dots> = (\<Sum>i\<in>A. card (permutations_of_set (A - {i})))" - by (intro setsum.cong) (simp_all add: card_image) - also from remove have "\<dots> = card A * fact (card A - 1)" by simp - also from remove.hyps have "\<dots> = fact (card A)" - by (cases "card A") simp_all - finally show ?case . -qed simp_all - -lemma permutations_of_set_image_inj: - assumes inj: "inj_on f A" - shows "permutations_of_set (f ` A) = map f ` permutations_of_set A" -proof (cases "finite A") - assume "\<not>finite A" - with inj show ?thesis - by (auto simp add: permutations_of_set_infinite dest: finite_imageD) -next - assume finite: "finite A" - show ?thesis - proof (rule sym, rule card_seteq) - from inj show "map f ` permutations_of_set A \<subseteq> permutations_of_set (f ` A)" - by (auto simp: permutations_of_set_def distinct_map) - - from inj have "card (map f ` permutations_of_set A) = card (permutations_of_set A)" - by (intro card_image inj_on_mapI) (auto simp: permutations_of_set_def) - also from finite inj have "\<dots> = card (permutations_of_set (f ` A))" - by (simp add: card_image) - finally show "card (permutations_of_set (f ` A)) \<le> - card (map f ` permutations_of_set A)" by simp - qed simp_all -qed - -lemma permutations_of_set_image_permutes: - "\<sigma> permutes A \<Longrightarrow> map \<sigma> ` permutations_of_set A = permutations_of_set A" - by (subst permutations_of_set_image_inj [symmetric]) - (simp_all add: permutes_inj_on permutes_image) - - -subsection \<open>Code generation\<close> - -text \<open> - We define an auxiliary version with an accumulator to avoid - having to map over the results. -\<close> -function permutations_of_set_aux where - "permutations_of_set_aux acc A = - (if \<not>finite A then {} else if A = {} then {acc} else - (\<Union>x\<in>A. permutations_of_set_aux (x#acc) (A - {x})))" -by auto -termination by (relation "Wellfounded.measure (card \<circ> snd)") (simp_all add: card_gt_0_iff) - -lemma permutations_of_set_aux_altdef: - "permutations_of_set_aux acc A = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A" -proof (cases "finite A") - assume "finite A" - thus ?thesis - proof (induction A arbitrary: acc rule: finite_psubset_induct) - case (psubset A acc) - show ?case - proof (cases "A = {}") - case False - note [simp del] = permutations_of_set_aux.simps - from psubset.hyps False - have "permutations_of_set_aux acc A = - (\<Union>y\<in>A. permutations_of_set_aux (y#acc) (A - {y}))" - by (subst permutations_of_set_aux.simps) simp_all - also have "\<dots> = (\<Union>y\<in>A. (\<lambda>xs. rev xs @ acc) ` (\<lambda>xs. y # xs) ` permutations_of_set (A - {y}))" - by (intro SUP_cong refl, subst psubset) (auto simp: image_image) - also from False have "\<dots> = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A" - by (subst (2) permutations_of_set_nonempty) (simp_all add: image_UN) - finally show ?thesis . - qed simp_all - qed -qed (simp_all add: permutations_of_set_infinite) - -declare permutations_of_set_aux.simps [simp del] - -lemma permutations_of_set_aux_correct: - "permutations_of_set_aux [] A = permutations_of_set A" - by (simp add: permutations_of_set_aux_altdef) - - -text \<open> - In another refinement step, we define a version on lists. -\<close> -declare length_remove1 [termination_simp] - -fun permutations_of_set_aux_list where - "permutations_of_set_aux_list acc xs = - (if xs = [] then [acc] else - List.bind xs (\<lambda>x. permutations_of_set_aux_list (x#acc) (List.remove1 x xs)))" - -definition permutations_of_set_list where - "permutations_of_set_list xs = permutations_of_set_aux_list [] xs" - -declare permutations_of_set_aux_list.simps [simp del] - -lemma permutations_of_set_aux_list_refine: - assumes "distinct xs" - shows "set (permutations_of_set_aux_list acc xs) = permutations_of_set_aux acc (set xs)" - using assms - by (induction acc xs rule: permutations_of_set_aux_list.induct) - (subst permutations_of_set_aux_list.simps, - subst permutations_of_set_aux.simps, - simp_all add: set_list_bind cong: SUP_cong) - - -text \<open> - The permutation lists contain no duplicates if the inputs contain no duplicates. - Therefore, these functions can easily be used when working with a representation of - sets by distinct lists. - The same approach should generalise to any kind of set implementation that supports - a monadic bind operation, and since the results are disjoint, merging should be cheap. -\<close> -lemma distinct_permutations_of_set_aux_list: - "distinct xs \<Longrightarrow> distinct (permutations_of_set_aux_list acc xs)" - by (induction acc xs rule: permutations_of_set_aux_list.induct) - (subst permutations_of_set_aux_list.simps, - auto intro!: distinct_list_bind simp: disjoint_family_on_def - permutations_of_set_aux_list_refine permutations_of_set_aux_altdef) - -lemma distinct_permutations_of_set_list: - "distinct xs \<Longrightarrow> distinct (permutations_of_set_list xs)" - by (simp add: permutations_of_set_list_def distinct_permutations_of_set_aux_list) - -lemma permutations_of_list: - "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))" - by (simp add: permutations_of_set_aux_correct [symmetric] - permutations_of_set_aux_list_refine permutations_of_set_list_def) - -lemma permutations_of_list_code [code]: - "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))" - "permutations_of_set (List.coset xs) = - Code.abort (STR ''Permutation of set complement not supported'') - (\<lambda>_. permutations_of_set (List.coset xs))" - by (simp_all add: permutations_of_list) - -value [code] "permutations_of_set (set ''abcd'')" - -end \ No newline at end of file

--- a/src/HOL/Probability/Random_Permutations.thy Thu Sep 29 11:24:36 2016 +0100 +++ b/src/HOL/Probability/Random_Permutations.thy Thu Sep 29 16:49:42 2016 +0200 @@ -11,7 +11,9 @@ section \<open>Random Permutations\<close> theory Random_Permutations -imports "~~/src/HOL/Probability/Probability_Mass_Function" "~~/src/HOL/Library/Set_Permutations" +imports + "~~/src/HOL/Probability/Probability_Mass_Function" + "~~/src/HOL/Library/Multiset_Permutations" begin text \<open>