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src/HOL/Finite_Set.thy

author | wenzelm |

Fri, 07 May 2021 16:49:08 +0200 | |

changeset 73647 | a037f01aedab |

parent 73620 | 58aed6f71f90 |

permissions | -rw-r--r-- |

tuned comments;

(* Title: HOL/Finite_Set.thy Author: Tobias Nipkow Author: Lawrence C Paulson Author: Markus Wenzel Author: Jeremy Avigad Author: Andrei Popescu *) section \<open>Finite sets\<close> theory Finite_Set imports Product_Type Sum_Type Fields begin subsection \<open>Predicate for finite sets\<close> context notes [[inductive_internals]] begin inductive finite :: "'a set \<Rightarrow> bool" where emptyI [simp, intro!]: "finite {}" | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)" end simproc_setup finite_Collect ("finite (Collect P)") = \<open>K Set_Comprehension_Pointfree.simproc\<close> declare [[simproc del: finite_Collect]] lemma finite_induct [case_names empty insert, induct set: finite]: \<comment> \<open>Discharging \<open>x \<notin> F\<close> entails extra work.\<close> assumes "finite F" assumes "P {}" and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)" shows "P F" using \<open>finite F\<close> proof induct show "P {}" by fact next fix x F assume F: "finite F" and P: "P F" show "P (insert x F)" proof cases assume "x \<in> F" then have "insert x F = F" by (rule insert_absorb) with P show ?thesis by (simp only:) next assume "x \<notin> F" from F this P show ?thesis by (rule insert) qed qed lemma infinite_finite_induct [case_names infinite empty insert]: assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A" and empty: "P {}" and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)" shows "P A" proof (cases "finite A") case False with infinite show ?thesis . next case True then show ?thesis by (induct A) (fact empty insert)+ qed subsubsection \<open>Choice principles\<close> lemma ex_new_if_finite: \<comment> \<open>does not depend on def of finite at all\<close> assumes "\<not> finite (UNIV :: 'a set)" and "finite A" shows "\<exists>a::'a. a \<notin> A" proof - from assms have "A \<noteq> UNIV" by blast then show ?thesis by blast qed text \<open>A finite choice principle. Does not need the SOME choice operator.\<close> lemma finite_set_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)" proof (induct rule: finite_induct) case empty then show ?case by simp next case (insert a A) then obtain f b where f: "\<forall>x\<in>A. P x (f x)" and ab: "P a b" by auto show ?case (is "\<exists>f. ?P f") proof show "?P (\<lambda>x. if x = a then b else f x)" using f ab by auto qed qed subsubsection \<open>Finite sets are the images of initial segments of natural numbers\<close> lemma finite_imp_nat_seg_image_inj_on: assumes "finite A" shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}" using assms proof induct case empty show ?case proof show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp qed next case (insert a A) have notinA: "a \<notin> A" by fact from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast then have "insert a A = f(n:=a) ` {i. i < Suc n}" and "inj_on (f(n:=a)) {i. i < Suc n}" using notinA by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq) then show ?case by blast qed lemma nat_seg_image_imp_finite: "A = f ` {i::nat. i < n} \<Longrightarrow> finite A" proof (induct n arbitrary: A) case 0 then show ?case by simp next case (Suc n) let ?B = "f ` {i. i < n}" have finB: "finite ?B" by (rule Suc.hyps[OF refl]) show ?case proof (cases "\<exists>k<n. f n = f k") case True then have "A = ?B" using Suc.prems by (auto simp:less_Suc_eq) then show ?thesis using finB by simp next case False then have "A = insert (f n) ?B" using Suc.prems by (auto simp:less_Suc_eq) then show ?thesis using finB by simp qed qed lemma finite_conv_nat_seg_image: "finite A \<longleftrightarrow> (\<exists>n f. A = f ` {i::nat. i < n})" by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on) lemma finite_imp_inj_to_nat_seg: assumes "finite A" shows "\<exists>f n. f ` A = {i::nat. i < n} \<and> inj_on f A" proof - from finite_imp_nat_seg_image_inj_on [OF \<open>finite A\<close>] obtain f and n :: nat where bij: "bij_betw f {i. i<n} A" by (auto simp: bij_betw_def) let ?f = "the_inv_into {i. i<n} f" have "inj_on ?f A \<and> ?f ` A = {i. i<n}" by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij]) then show ?thesis by blast qed lemma finite_Collect_less_nat [iff]: "finite {n::nat. n < k}" by (fastforce simp: finite_conv_nat_seg_image) lemma finite_Collect_le_nat [iff]: "finite {n::nat. n \<le> k}" by (simp add: le_eq_less_or_eq Collect_disj_eq) subsection \<open>Finiteness and common set operations\<close> lemma rev_finite_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A" proof (induct arbitrary: A rule: finite_induct) case empty then show ?case by simp next case (insert x F A) have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+ show "finite A" proof cases assume x: "x \<in> A" with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff) with r have "finite (A - {x})" . then have "finite (insert x (A - {x}))" .. also have "insert x (A - {x}) = A" using x by (rule insert_Diff) finally show ?thesis . next show ?thesis when "A \<subseteq> F" using that by fact assume "x \<notin> A" with A show "A \<subseteq> F" by (simp add: subset_insert_iff) qed qed lemma finite_subset: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A" by (rule rev_finite_subset) lemma finite_UnI: assumes "finite F" and "finite G" shows "finite (F \<union> G)" using assms by induct simp_all lemma finite_Un [iff]: "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G" by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"]) lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A" proof - have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un) then show ?thesis by simp qed lemma finite_Int [simp, intro]: "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)" by (blast intro: finite_subset) lemma finite_Collect_conjI [simp, intro]: "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}" by (simp add: Collect_conj_eq) lemma finite_Collect_disjI [simp]: "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}" by (simp add: Collect_disj_eq) lemma finite_Diff [simp, intro]: "finite A \<Longrightarrow> finite (A - B)" by (rule finite_subset, rule Diff_subset) lemma finite_Diff2 [simp]: assumes "finite B" shows "finite (A - B) \<longleftrightarrow> finite A" proof - have "finite A \<longleftrightarrow> finite ((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int) also have "\<dots> \<longleftrightarrow> finite (A - B)" using \<open>finite B\<close> by simp finally show ?thesis .. qed lemma finite_Diff_insert [iff]: "finite (A - insert a B) \<longleftrightarrow> finite (A - B)" proof - have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp moreover have "A - insert a B = A - B - {a}" by auto ultimately show ?thesis by simp qed lemma finite_compl [simp]: "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)" by (simp add: Compl_eq_Diff_UNIV) lemma finite_Collect_not [simp]: "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)" by (simp add: Collect_neg_eq) lemma finite_Union [simp, intro]: "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite (\<Union>A)" by (induct rule: finite_induct) simp_all lemma finite_UN_I [intro]: "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)" by (induct rule: finite_induct) simp_all lemma finite_UN [simp]: "finite A \<Longrightarrow> finite (\<Union>(B ` A)) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))" by (blast intro: finite_subset) lemma finite_Inter [intro]: "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)" by (blast intro: Inter_lower finite_subset) lemma finite_INT [intro]: "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)" by (blast intro: INT_lower finite_subset) lemma finite_imageI [simp, intro]: "finite F \<Longrightarrow> finite (h ` F)" by (induct rule: finite_induct) simp_all lemma finite_image_set [simp]: "finite {x. P x} \<Longrightarrow> finite {f x |x. P x}" by (simp add: image_Collect [symmetric]) lemma finite_image_set2: "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y |x y. P x \<and> Q y}" by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto lemma finite_imageD: assumes "finite (f ` A)" and "inj_on f A" shows "finite A" using assms proof (induct "f ` A" arbitrary: A) case empty then show ?case by simp next case (insert x B) then have B_A: "insert x B = f ` A" by simp then obtain y where "x = f y" and "y \<in> A" by blast from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}" by blast with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff) moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})" by (rule inj_on_diff) ultimately have "finite (A - {y})" by (rule insert.hyps) then show "finite A" by simp qed lemma finite_image_iff: "inj_on f A \<Longrightarrow> finite (f ` A) \<longleftrightarrow> finite A" using finite_imageD by blast lemma finite_surj: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B" by (erule finite_subset) (rule finite_imageI) lemma finite_range_imageI: "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))" by (drule finite_imageI) (simp add: range_composition) lemma finite_subset_image: assumes "finite B" shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C" using assms proof induct case empty then show ?case by simp next case insert then show ?case by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast qed lemma all_subset_image: "(\<forall>B. B \<subseteq> f ` A \<longrightarrow> P B) \<longleftrightarrow> (\<forall>B. B \<subseteq> A \<longrightarrow> P(f ` B))" by (safe elim!: subset_imageE) (use image_mono in \<open>blast+\<close>) (* slow *) lemma all_finite_subset_image: "(\<forall>B. finite B \<and> B \<subseteq> f ` A \<longrightarrow> P B) \<longleftrightarrow> (\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B))" proof safe fix B :: "'a set" assume B: "finite B" "B \<subseteq> f ` A" and P: "\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B)" show "P B" using finite_subset_image [OF B] P by blast qed blast lemma ex_finite_subset_image: "(\<exists>B. finite B \<and> B \<subseteq> f ` A \<and> P B) \<longleftrightarrow> (\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B))" proof safe fix B :: "'a set" assume B: "finite B" "B \<subseteq> f ` A" and "P B" show "\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B)" using finite_subset_image [OF B] \<open>P B\<close> by blast qed blast lemma finite_vimage_IntI: "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)" proof (induct rule: finite_induct) case (insert x F) then show ?case by (simp add: vimage_insert [of h x F] finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2) qed simp lemma finite_finite_vimage_IntI: assumes "finite F" and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)" shows "finite (h -` F \<inter> A)" proof - have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)" by blast show ?thesis by (simp only: * assms finite_UN_I) qed lemma finite_vimageI: "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)" using finite_vimage_IntI[of F h UNIV] by auto lemma finite_vimageD': "finite (f -` A) \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> finite A" by (auto simp add: subset_image_iff intro: finite_subset[rotated]) lemma finite_vimageD: "finite (h -` F) \<Longrightarrow> surj h \<Longrightarrow> finite F" by (auto dest: finite_vimageD') lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F" unfolding bij_def by (auto elim: finite_vimageD finite_vimageI) lemma finite_Collect_bex [simp]: assumes "finite A" shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})" proof - have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto with assms show ?thesis by simp qed lemma finite_Collect_bounded_ex [simp]: assumes "finite {y. P y}" shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})" proof - have "{x. \<exists>y. P y \<and> Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto with assms show ?thesis by simp qed lemma finite_Plus: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)" by (simp add: Plus_def) lemma finite_PlusD: fixes A :: "'a set" and B :: "'b set" assumes fin: "finite (A <+> B)" shows "finite A" "finite B" proof - have "Inl ` A \<subseteq> A <+> B" by auto then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset) then show "finite A" by (rule finite_imageD) (auto intro: inj_onI) next have "Inr ` B \<subseteq> A <+> B" by auto then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset) then show "finite B" by (rule finite_imageD) (auto intro: inj_onI) qed lemma finite_Plus_iff [simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B" by (auto intro: finite_PlusD finite_Plus) lemma finite_Plus_UNIV_iff [simp]: "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)" by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff) lemma finite_SigmaI [simp, intro]: "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (SIGMA a:A. B a)" unfolding Sigma_def by blast lemma finite_SigmaI2: assumes "finite {x\<in>A. B x \<noteq> {}}" and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)" proof - from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto finally show ?thesis . qed lemma finite_cartesian_product: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)" by (rule finite_SigmaI) lemma finite_Prod_UNIV: "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)" by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product) lemma finite_cartesian_productD1: assumes "finite (A \<times> B)" and "B \<noteq> {}" shows "finite A" proof - from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}" by (auto simp add: finite_conv_nat_seg_image) then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}" by (simp add: image_comp) then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast then show ?thesis by (auto simp add: finite_conv_nat_seg_image) qed lemma finite_cartesian_productD2: assumes "finite (A \<times> B)" and "A \<noteq> {}" shows "finite B" proof - from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}" by (auto simp add: finite_conv_nat_seg_image) then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}" by (simp add: image_comp) then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast then show ?thesis by (auto simp add: finite_conv_nat_seg_image) qed lemma finite_cartesian_product_iff: "finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))" by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product) lemma finite_prod: "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)" using finite_cartesian_product_iff[of UNIV UNIV] by simp lemma finite_Pow_iff [iff]: "finite (Pow A) \<longleftrightarrow> finite A" proof assume "finite (Pow A)" then have "finite ((\<lambda>x. {x}) ` A)" by (blast intro: finite_subset) (* somewhat slow *) then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp next assume "finite A" then show "finite (Pow A)" by induct (simp_all add: Pow_insert) qed corollary finite_Collect_subsets [simp, intro]: "finite A \<Longrightarrow> finite {B. B \<subseteq> A}" by (simp add: Pow_def [symmetric]) lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)" by (simp only: finite_Pow_iff Pow_UNIV[symmetric]) lemma finite_UnionD: "finite (\<Union>A) \<Longrightarrow> finite A" by (blast intro: finite_subset [OF subset_Pow_Union]) lemma finite_bind: assumes "finite S" assumes "\<forall>x \<in> S. finite (f x)" shows "finite (Set.bind S f)" using assms by (simp add: bind_UNION) lemma finite_filter [simp]: "finite S \<Longrightarrow> finite (Set.filter P S)" unfolding Set.filter_def by simp lemma finite_set_of_finite_funs: assumes "finite A" "finite B" shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S") proof - let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}" have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp have 2: "inj_on ?F ?S" by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff) (* somewhat slow *) show ?thesis by (rule finite_imageD [OF 1 2]) qed lemma not_finite_existsD: assumes "\<not> finite {a. P a}" shows "\<exists>a. P a" proof (rule classical) assume "\<not> ?thesis" with assms show ?thesis by auto qed subsection \<open>Further induction rules on finite sets\<close> lemma finite_ne_induct [case_names singleton insert, consumes 2]: assumes "finite F" and "F \<noteq> {}" assumes "\<And>x. P {x}" and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)" shows "P F" using assms proof induct case empty then show ?case by simp next case (insert x F) then show ?case by cases auto qed lemma finite_subset_induct [consumes 2, case_names empty insert]: assumes "finite F" and "F \<subseteq> A" and empty: "P {}" and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)" shows "P F" using \<open>finite F\<close> \<open>F \<subseteq> A\<close> proof induct show "P {}" by fact next fix x F assume "finite F" and "x \<notin> F" and P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A" show "P (insert x F)" proof (rule insert) from i show "x \<in> A" by blast from i have "F \<subseteq> A" by blast with P show "P F" . show "finite F" by fact show "x \<notin> F" by fact qed qed lemma finite_empty_induct: assumes "finite A" and "P A" and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})" shows "P {}" proof - have "P (A - B)" if "B \<subseteq> A" for B :: "'a set" proof - from \<open>finite A\<close> that have "finite B" by (rule rev_finite_subset) from this \<open>B \<subseteq> A\<close> show "P (A - B)" proof induct case empty from \<open>P A\<close> show ?case by simp next case (insert b B) have "P (A - B - {b})" proof (rule remove) from \<open>finite A\<close> show "finite (A - B)" by induct auto from insert show "b \<in> A - B" by simp from insert show "P (A - B)" by simp qed also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric]) finally show ?case . qed qed then have "P (A - A)" by blast then show ?thesis by simp qed lemma finite_update_induct [consumes 1, case_names const update]: assumes finite: "finite {a. f a \<noteq> c}" and const: "P (\<lambda>a. c)" and update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))" shows "P f" using finite proof (induct "{a. f a \<noteq> c}" arbitrary: f) case empty with const show ?case by simp next case (insert a A) then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c" by auto with \<open>finite A\<close> have "finite {a'. (f(a := c)) a' \<noteq> c}" by simp have "(f(a := c)) a = c" by simp from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))" by simp with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close> have "P ((f(a := c))(a := f a))" by (rule update) then show ?case by simp qed lemma finite_subset_induct' [consumes 2, case_names empty insert]: assumes "finite F" and "F \<subseteq> A" and empty: "P {}" and insert: "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F" using assms(1,2) proof induct show "P {}" by fact next fix x F assume "finite F" and "x \<notin> F" and P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A" show "P (insert x F)" proof (rule insert) from i show "x \<in> A" by blast from i have "F \<subseteq> A" by blast with P show "P F" . show "finite F" by fact show "x \<notin> F" by fact show "F \<subseteq> A" by fact qed qed subsection \<open>Class \<open>finite\<close>\<close> class finite = assumes finite_UNIV: "finite (UNIV :: 'a set)" begin lemma finite [simp]: "finite (A :: 'a set)" by (rule subset_UNIV finite_UNIV finite_subset)+ lemma finite_code [code]: "finite (A :: 'a set) \<longleftrightarrow> True" by simp end instance prod :: (finite, finite) finite by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite) lemma inj_graph: "inj (\<lambda>f. {(x, y). y = f x})" by (rule inj_onI) (auto simp add: set_eq_iff fun_eq_iff) instance "fun" :: (finite, finite) finite proof show "finite (UNIV :: ('a \<Rightarrow> 'b) set)" proof (rule finite_imageD) let ?graph = "\<lambda>f::'a \<Rightarrow> 'b. {(x, y). y = f x}" have "range ?graph \<subseteq> Pow UNIV" by simp moreover have "finite (Pow (UNIV :: ('a * 'b) set))" by (simp only: finite_Pow_iff finite) ultimately show "finite (range ?graph)" by (rule finite_subset) show "inj ?graph" by (rule inj_graph) qed qed instance bool :: finite by standard (simp add: UNIV_bool) instance set :: (finite) finite by standard (simp only: Pow_UNIV [symmetric] finite_Pow_iff finite) instance unit :: finite by standard (simp add: UNIV_unit) instance sum :: (finite, finite) finite by standard (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite) subsection \<open>A basic fold functional for finite sets\<close> text \<open>The intended behaviour is \<open>fold f z {x\<^sub>1, \<dots>, x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close> if \<open>f\<close> is ``left-commutative'': \<close> locale comp_fun_commute = fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y" begin lemma fun_left_comm: "f y (f x z) = f x (f y z)" using comp_fun_commute by (simp add: fun_eq_iff) lemma commute_left_comp: "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)" by (simp add: o_assoc comp_fun_commute) end inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where emptyI [intro]: "fold_graph f z {} z" | insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)" inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x" lemma fold_graph_closed_lemma: "fold_graph f z A x \<and> x \<in> B" if "fold_graph g z A x" "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> f a b = g a b" "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> g a b \<in> B" "z \<in> B" using that(1-3) proof (induction rule: fold_graph.induct) case (insertI x A y) have "fold_graph f z A y" "y \<in> B" unfolding atomize_conj by (rule insertI.IH) (auto intro: insertI.prems) then have "g x y \<in> B" and f_eq: "f x y = g x y" by (auto simp: insertI.prems) moreover have "fold_graph f z (insert x A) (f x y)" by (rule fold_graph.insertI; fact) ultimately show ?case by (simp add: f_eq) qed (auto intro!: that) lemma fold_graph_closed_eq: "fold_graph f z A = fold_graph g z A" if "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> f a b = g a b" "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> g a b \<in> B" "z \<in> B" using fold_graph_closed_lemma[of f z A _ B g] fold_graph_closed_lemma[of g z A _ B f] that by auto definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)" lemma fold_closed_eq: "fold f z A = fold g z A" if "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> f a b = g a b" "\<And>a b. a \<in> A \<Longrightarrow> b \<in> B \<Longrightarrow> g a b \<in> B" "z \<in> B" unfolding Finite_Set.fold_def by (subst fold_graph_closed_eq[where B=B and g=g]) (auto simp: that) text \<open> A tempting alternative for the definiens is \<^term>\<open>if finite A then THE y. fold_graph f z A y else e\<close>. It allows the removal of finiteness assumptions from the theorems \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>. The proofs become ugly. It is not worth the effort. (???) \<close> lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x" by (induct rule: finite_induct) auto subsubsection \<open>From \<^const>\<open>fold_graph\<close> to \<^term>\<open>fold\<close>\<close> context comp_fun_commute begin lemma fold_graph_finite: assumes "fold_graph f z A y" shows "finite A" using assms by induct simp_all lemma fold_graph_insertE_aux: "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'" proof (induct set: fold_graph) case emptyI then show ?case by simp next case (insertI x A y) show ?case proof (cases "x = a") case True with insertI show ?thesis by auto next case False then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'" using insertI by auto have "f x y = f a (f x y')" unfolding y by (rule fun_left_comm) moreover have "fold_graph f z (insert x A - {a}) (f x y')" using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close> by (simp add: insert_Diff_if fold_graph.insertI) ultimately show ?thesis by fast qed qed lemma fold_graph_insertE: assumes "fold_graph f z (insert x A) v" and "x \<notin> A" obtains y where "v = f x y" and "fold_graph f z A y" using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1]) lemma fold_graph_determ: "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x" proof (induct arbitrary: y set: fold_graph) case emptyI then show ?case by fast next case (insertI x A y v) from \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close> obtain y' where "v = f x y'" and "fold_graph f z A y'" by (rule fold_graph_insertE) from \<open>fold_graph f z A y'\<close> have "y' = y" by (rule insertI) with \<open>v = f x y'\<close> show "v = f x y" by simp qed lemma fold_equality: "fold_graph f z A y \<Longrightarrow> fold f z A = y" by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite) lemma fold_graph_fold: assumes "finite A" shows "fold_graph f z A (fold f z A)" proof - from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph) moreover note fold_graph_determ ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I) then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI') with assms show ?thesis by (simp add: fold_def) qed text \<open>The base case for \<open>fold\<close>:\<close> lemma (in -) fold_infinite [simp]: "\<not> finite A \<Longrightarrow> fold f z A = z" by (auto simp: fold_def) lemma (in -) fold_empty [simp]: "fold f z {} = z" by (auto simp: fold_def) text \<open>The various recursion equations for \<^const>\<open>fold\<close>:\<close> lemma fold_insert [simp]: assumes "finite A" and "x \<notin> A" shows "fold f z (insert x A) = f x (fold f z A)" proof (rule fold_equality) fix z from \<open>finite A\<close> have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold) with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI) then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp qed declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del] \<comment> \<open>No more proofs involve these.\<close> lemma fold_fun_left_comm: "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A" proof (induct rule: finite_induct) case empty then show ?case by simp next case insert then show ?case by (simp add: fun_left_comm [of x]) qed lemma fold_insert2: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A" by (simp add: fold_fun_left_comm) lemma fold_rec: assumes "finite A" and "x \<in> A" shows "fold f z A = f x (fold f z (A - {x}))" proof - have A: "A = insert x (A - {x})" using \<open>x \<in> A\<close> by blast then have "fold f z A = fold f z (insert x (A - {x}))" by simp also have "\<dots> = f x (fold f z (A - {x}))" by (rule fold_insert) (simp add: \<open>finite A\<close>)+ finally show ?thesis . qed lemma fold_insert_remove: assumes "finite A" shows "fold f z (insert x A) = f x (fold f z (A - {x}))" proof - from \<open>finite A\<close> have "finite (insert x A)" by auto moreover have "x \<in> insert x A" by auto ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))" by (rule fold_rec) then show ?thesis by simp qed lemma fold_set_union_disj: assumes "finite A" "finite B" "A \<inter> B = {}" shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B" using assms(2,1,3) by induct simp_all end text \<open>Other properties of \<^const>\<open>fold\<close>:\<close> lemma fold_image: assumes "inj_on g A" shows "fold f z (g ` A) = fold (f \<circ> g) z A" proof (cases "finite A") case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def) next case True have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A" proof fix w show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q") proof assume ?P then show ?Q using assms proof (induct "g ` A" w arbitrary: A) case emptyI then show ?case by (auto intro: fold_graph.emptyI) next case (insertI x A r B) from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A' where "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'" by (rule inj_img_insertE) from insertI.prems have "fold_graph (f \<circ> g) z A' r" by (auto intro: insertI.hyps) with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)" by (rule fold_graph.insertI) then show ?case by simp qed next assume ?Q then show ?P using assms proof induct case emptyI then show ?case by (auto intro: fold_graph.emptyI) next case (insertI x A r) from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A" by auto moreover from insertI have "fold_graph f z (g ` A) r" by simp ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)" by (rule fold_graph.insertI) then show ?case by simp qed qed qed with True assms show ?thesis by (auto simp add: fold_def) qed lemma fold_cong: assumes "comp_fun_commute f" "comp_fun_commute g" and "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x" and "s = t" and "A = B" shows "fold f s A = fold g t B" proof - have "fold f s A = fold g s A" using \<open>finite A\<close> cong proof (induct A) case empty then show ?case by simp next case insert interpret f: comp_fun_commute f by (fact \<open>comp_fun_commute f\<close>) interpret g: comp_fun_commute g by (fact \<open>comp_fun_commute g\<close>) from insert show ?case by simp qed with assms show ?thesis by simp qed text \<open>A simplified version for idempotent functions:\<close> locale comp_fun_idem = comp_fun_commute + assumes comp_fun_idem: "f x \<circ> f x = f x" begin lemma fun_left_idem: "f x (f x z) = f x z" using comp_fun_idem by (simp add: fun_eq_iff) lemma fold_insert_idem: assumes fin: "finite A" shows "fold f z (insert x A) = f x (fold f z A)" proof cases assume "x \<in> A" then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert) then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem) next assume "x \<notin> A" then show ?thesis using assms by simp qed declare fold_insert [simp del] fold_insert_idem [simp] lemma fold_insert_idem2: "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A" by (simp add: fold_fun_left_comm) end subsubsection \<open>Liftings to \<open>comp_fun_commute\<close> etc.\<close> lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f \<circ> g)" by standard (simp_all add: comp_fun_commute) lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f \<circ> g)" by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales) (simp_all add: comp_fun_idem) lemma (in comp_fun_commute) comp_fun_commute_funpow: "comp_fun_commute (\<lambda>x. f x ^^ g x)" proof show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y" for x y proof (cases "x = y") case True then show ?thesis by simp next case False show ?thesis proof (induct "g x" arbitrary: g) case 0 then show ?case by simp next case (Suc n g) have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y" proof (induct "g y" arbitrary: g) case 0 then show ?case by simp next case (Suc n g) define h where "h z = g z - 1" for z with Suc have "n = h y" by simp with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y" by auto from Suc h_def have "g y = Suc (h y)" by simp then show ?case by (simp add: comp_assoc hyp) (simp add: o_assoc comp_fun_commute) qed define h where "h z = (if z = x then g x - 1 else g z)" for z with Suc have "n = h x" by simp with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y" by auto with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp from Suc h_def have "g x = Suc (h x)" by simp then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2) (simp add: comp_assoc hyp1) qed qed qed subsubsection \<open>Expressing set operations via \<^const>\<open>fold\<close>\<close> lemma comp_fun_commute_const: "comp_fun_commute (\<lambda>_. f)" by standard rule lemma comp_fun_idem_insert: "comp_fun_idem insert" by standard auto lemma comp_fun_idem_remove: "comp_fun_idem Set.remove" by standard auto lemma (in semilattice_inf) comp_fun_idem_inf: "comp_fun_idem inf" by standard (auto simp add: inf_left_commute) lemma (in semilattice_sup) comp_fun_idem_sup: "comp_fun_idem sup" by standard (auto simp add: sup_left_commute) lemma union_fold_insert: assumes "finite A" shows "A \<union> B = fold insert B A" proof - interpret comp_fun_idem insert by (fact comp_fun_idem_insert) from \<open>finite A\<close> show ?thesis by (induct A arbitrary: B) simp_all qed lemma minus_fold_remove: assumes "finite A" shows "B - A = fold Set.remove B A" proof - interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove) from \<open>finite A\<close> have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto (* slow *) then show ?thesis .. qed lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')" proof - interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) show ?thesis by standard (auto simp: fun_eq_iff) qed lemma Set_filter_fold: assumes "finite A" shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A" using assms by induct (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold]) lemma inter_Set_filter: assumes "finite B" shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B" using assms by induct (auto simp: Set.filter_def) lemma image_fold_insert: assumes "finite A" shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A" proof - interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by standard auto show ?thesis using assms by (induct A) auto qed lemma Ball_fold: assumes "finite A" shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A" proof - interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by standard auto show ?thesis using assms by (induct A) auto qed lemma Bex_fold: assumes "finite A" shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A" proof - interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by standard auto show ?thesis using assms by (induct A) auto qed lemma comp_fun_commute_Pow_fold: "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast (* somewhat slow *) lemma Pow_fold: assumes "finite A" shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A" proof - interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold) show ?thesis using assms by (induct A) (auto simp: Pow_insert) qed lemma fold_union_pair: assumes "finite B" shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B" proof - interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by standard auto show ?thesis using assms by (induct arbitrary: A) simp_all qed lemma comp_fun_commute_product_fold: "finite B \<Longrightarrow> comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)" by standard (auto simp: fold_union_pair [symmetric]) lemma product_fold: assumes "finite A" "finite B" shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A" using assms unfolding Sigma_def by (induct A) (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair) context complete_lattice begin lemma inf_Inf_fold_inf: assumes "finite A" shows "inf (Inf A) B = fold inf B A" proof - interpret comp_fun_idem inf by (fact comp_fun_idem_inf) from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B) (simp_all add: inf_commute fun_eq_iff) qed lemma sup_Sup_fold_sup: assumes "finite A" shows "sup (Sup A) B = fold sup B A" proof - interpret comp_fun_idem sup by (fact comp_fun_idem_sup) from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B) (simp_all add: sup_commute fun_eq_iff) qed lemma Inf_fold_inf: "finite A \<Longrightarrow> Inf A = fold inf top A" using inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2) lemma Sup_fold_sup: "finite A \<Longrightarrow> Sup A = fold sup bot A" using sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2) lemma inf_INF_fold_inf: assumes "finite A" shows "inf B (\<Sqinter>(f ` A)) = fold (inf \<circ> f) B A" (is "?inf = ?fold") proof - interpret comp_fun_idem inf by (fact comp_fun_idem_inf) interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem) from \<open>finite A\<close> have "?fold = ?inf" by (induct A arbitrary: B) (simp_all add: inf_left_commute) then show ?thesis .. qed lemma sup_SUP_fold_sup: assumes "finite A" shows "sup B (\<Squnion>(f ` A)) = fold (sup \<circ> f) B A" (is "?sup = ?fold") proof - interpret comp_fun_idem sup by (fact comp_fun_idem_sup) interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem) from \<open>finite A\<close> have "?fold = ?sup" by (induct A arbitrary: B) (simp_all add: sup_left_commute) then show ?thesis .. qed lemma INF_fold_inf: "finite A \<Longrightarrow> \<Sqinter>(f ` A) = fold (inf \<circ> f) top A" using inf_INF_fold_inf [of A top] by simp lemma SUP_fold_sup: "finite A \<Longrightarrow> \<Squnion>(f ` A) = fold (sup \<circ> f) bot A" using sup_SUP_fold_sup [of A bot] by simp lemma finite_Inf_in: assumes "finite A" "A\<noteq>{}" and inf: "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> inf x y \<in> A" shows "Inf A \<in> A" proof - have "Inf B \<in> A" if "B \<le> A" "B\<noteq>{}" for B using finite_subset [OF \<open>B \<subseteq> A\<close> \<open>finite A\<close>] that by (induction B) (use inf in \<open>force+\<close>) then show ?thesis by (simp add: assms) qed lemma finite_Sup_in: assumes "finite A" "A\<noteq>{}" and sup: "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> sup x y \<in> A" shows "Sup A \<in> A" proof - have "Sup B \<in> A" if "B \<le> A" "B\<noteq>{}" for B using finite_subset [OF \<open>B \<subseteq> A\<close> \<open>finite A\<close>] that by (induction B) (use sup in \<open>force+\<close>) then show ?thesis by (simp add: assms) qed end subsection \<open>Locales as mini-packages for fold operations\<close> subsubsection \<open>The natural case\<close> locale folding = fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: "'b" assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y" begin interpretation fold?: comp_fun_commute f by standard (use comp_fun_commute in \<open>simp add: fun_eq_iff\<close>) definition F :: "'a set \<Rightarrow> 'b" where eq_fold: "F A = fold f z A" lemma empty [simp]:"F {} = z" by (simp add: eq_fold) lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z" by (simp add: eq_fold) lemma insert [simp]: assumes "finite A" and "x \<notin> A" shows "F (insert x A) = f x (F A)" proof - from fold_insert assms have "fold f z (insert x A) = f x (fold f z A)" by simp with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff) qed lemma remove: assumes "finite A" and "x \<in> A" shows "F A = f x (F (A - {x}))" proof - from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B" by (auto dest: mk_disjoint_insert) moreover from \<open>finite A\<close> A have "finite B" by simp ultimately show ?thesis by simp qed lemma insert_remove: "finite A \<Longrightarrow> F (insert x A) = f x (F (A - {x}))" by (cases "x \<in> A") (simp_all add: remove insert_absorb) end subsubsection \<open>With idempotency\<close> locale folding_idem = folding + assumes comp_fun_idem: "f x \<circ> f x = f x" begin declare insert [simp del] interpretation fold?: comp_fun_idem f by standard (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff) lemma insert_idem [simp]: assumes "finite A" shows "F (insert x A) = f x (F A)" proof - from fold_insert_idem assms have "fold f z (insert x A) = f x (fold f z A)" by simp with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff) qed end subsection \<open>Finite cardinality\<close> text \<open> The traditional definition \<^prop>\<open>card A \<equiv> LEAST n. \<exists>f. A = {f i |i. i < n}\<close> is ugly to work with. But now that we have \<^const>\<open>fold\<close> things are easy: \<close> global_interpretation card: folding "\<lambda>_. Suc" 0 defines card = "folding.F (\<lambda>_. Suc) 0" by standard rule lemma card_insert_disjoint: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)" by (fact card.insert) lemma card_insert_if: "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))" by auto (simp add: card.insert_remove card.remove) lemma card_ge_0_finite: "card A > 0 \<Longrightarrow> finite A" by (rule ccontr) simp lemma card_0_eq [simp]: "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}" by (auto dest: mk_disjoint_insert) lemma finite_UNIV_card_ge_0: "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0" by (rule ccontr) simp lemma card_eq_0_iff: "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A" by auto lemma card_range_greater_zero: "finite (range f) \<Longrightarrow> card (range f) > 0" by (rule ccontr) (simp add: card_eq_0_iff) lemma card_gt_0_iff: "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A" by (simp add: neq0_conv [symmetric] card_eq_0_iff) lemma card_Suc_Diff1: assumes "finite A" "x \<in> A" shows "Suc (card (A - {x})) = card A" proof - have "Suc (card (A - {x})) = card (insert x (A - {x}))" using assms by (simp add: card.insert_remove) also have "... = card A" using assms by (simp add: card_insert_if) finally show ?thesis . qed lemma card_insert_le_m1: assumes "n > 0" "card y \<le> n - 1" shows "card (insert x y) \<le> n" using assms by (cases "finite y") (auto simp: card_insert_if) lemma card_Diff_singleton: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1" by (simp add: card_Suc_Diff1 [symmetric]) lemma card_Diff_singleton_if: "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)" by (simp add: card_Diff_singleton) lemma card_Diff_insert[simp]: assumes "finite A" and "a \<in> A" and "a \<notin> B" shows "card (A - insert a B) = card (A - B) - 1" proof - have "A - insert a B = (A - B) - {a}" using assms by blast then show ?thesis using assms by (simp add: card_Diff_singleton) qed lemma card_insert_le: "finite A \<Longrightarrow> card A \<le> card (insert x A)" by (simp add: card_insert_if) lemma card_Collect_less_nat[simp]: "card {i::nat. i < n} = n" by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq) lemma card_Collect_le_nat[simp]: "card {i::nat. i \<le> n} = Suc n" using card_Collect_less_nat[of "Suc n"] by (simp add: less_Suc_eq_le) lemma card_mono: assumes "finite B" and "A \<subseteq> B" shows "card A \<le> card B" proof - from assms have "finite A" by (auto intro: finite_subset) then show ?thesis using assms proof (induct A arbitrary: B) case empty then show ?case by simp next case (insert x A) then have "x \<in> B" by simp from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto with insert.hyps have "card A \<le> card (B - {x})" by auto with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case by simp (simp only: card.remove) qed qed lemma card_seteq: assumes "finite B" and A: "A \<subseteq> B" "card B \<le> card A" shows "A = B" using assms proof (induction arbitrary: A rule: finite_induct) case (insert b B) then have A: "finite A" "A - {b} \<subseteq> B" by force+ then have "card B \<le> card (A - {b})" using insert by (auto simp add: card_Diff_singleton_if) then have "A - {b} = B" using A insert.IH by auto then show ?case using insert.hyps insert.prems by auto qed auto lemma psubset_card_mono: "finite B \<Longrightarrow> A < B \<Longrightarrow> card A < card B" using card_seteq [of B A] by (auto simp add: psubset_eq) lemma card_Un_Int: assumes "finite A" "finite B" shows "card A + card B = card (A \<union> B) + card (A \<inter> B)" using assms proof (induct A) case empty then show ?case by simp next case insert then show ?case by (auto simp add: insert_absorb Int_insert_left) qed lemma card_Un_disjoint: "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> card (A \<union> B) = card A + card B" using card_Un_Int [of A B] by simp lemma card_Un_disjnt: "\<lbrakk>finite A; finite B; disjnt A B\<rbrakk> \<Longrightarrow> card (A \<union> B) = card A + card B" by (simp add: card_Un_disjoint disjnt_def) lemma card_Un_le: "card (A \<union> B) \<le> card A + card B" proof (cases "finite A \<and> finite B") case True then show ?thesis using le_iff_add card_Un_Int [of A B] by auto qed auto lemma card_Diff_subset: assumes "finite B" and "B \<subseteq> A" shows "card (A - B) = card A - card B" using assms proof (cases "finite A") case False with assms show ?thesis by simp next case True with assms show ?thesis by (induct B arbitrary: A) simp_all qed lemma card_Diff_subset_Int: assumes "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)" proof - have "A - B = A - A \<inter> B" by auto with assms show ?thesis by (simp add: card_Diff_subset) qed lemma diff_card_le_card_Diff: assumes "finite B" shows "card A - card B \<le> card (A - B)" proof - have "card A - card B \<le> card A - card (A \<inter> B)" using card_mono[OF assms Int_lower2, of A] by arith also have "\<dots> = card (A - B)" using assms by (simp add: card_Diff_subset_Int) finally show ?thesis . qed lemma card_le_sym_Diff: assumes "finite A" "finite B" "card A \<le> card B" shows "card(A - B) \<le> card(B - A)" proof - have "card(A - B) = card A - card (A \<inter> B)" using assms(1,2) by(simp add: card_Diff_subset_Int) also have "\<dots> \<le> card B - card (A \<inter> B)" using assms(3) by linarith also have "\<dots> = card(B - A)" using assms(1,2) by(simp add: card_Diff_subset_Int Int_commute) finally show ?thesis . qed lemma card_less_sym_Diff: assumes "finite A" "finite B" "card A < card B" shows "card(A - B) < card(B - A)" proof - have "card(A - B) = card A - card (A \<inter> B)" using assms(1,2) by(simp add: card_Diff_subset_Int) also have "\<dots> < card B - card (A \<inter> B)" using assms(1,3) by (simp add: card_mono diff_less_mono) also have "\<dots> = card(B - A)" using assms(1,2) by(simp add: card_Diff_subset_Int Int_commute) finally show ?thesis . qed lemma card_Diff1_less_iff: "card (A - {x}) < card A \<longleftrightarrow> finite A \<and> x \<in> A" proof (cases "finite A \<and> x \<in> A") case True then show ?thesis by (auto simp: card_gt_0_iff intro: diff_less) qed auto lemma card_Diff1_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) < card A" unfolding card_Diff1_less_iff by auto lemma card_Diff2_less: assumes "finite A" "x \<in> A" "y \<in> A" shows "card (A - {x} - {y}) < card A" proof (cases "x = y") case True with assms show ?thesis by (simp add: card_Diff1_less del: card_Diff_insert) next case False then have "card (A - {x} - {y}) < card (A - {x})" "card (A - {x}) < card A" using assms by (intro card_Diff1_less; simp)+ then show ?thesis by (blast intro: less_trans) qed lemma card_Diff1_le: "finite A \<Longrightarrow> card (A - {x}) \<le> card A" by (cases "x \<in> A") (simp_all add: card_Diff1_less less_imp_le) lemma card_psubset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> card A < card B \<Longrightarrow> A < B" by (erule psubsetI) blast lemma card_le_inj: assumes fA: "finite A" and fB: "finite B" and c: "card A \<le> card B" shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A" using fA fB c proof (induct arbitrary: B rule: finite_induct) case empty then show ?case by simp next case (insert x s t) then show ?case proof (induct rule: finite_induct [OF insert.prems(1)]) case 1 then show ?case by simp next case (2 y t) from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t" by simp from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where "f ` s \<subseteq> t" "inj_on f s" by blast with "2.prems"(2) "2.hyps"(2) show ?case unfolding inj_on_def by (rule_tac x = "\<lambda>z. if z = x then y else f z" in exI) auto qed qed lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and c: "card A = card B" shows "A = B" proof - from fB AB have fA: "finite A" by (auto intro: finite_subset) from fA fB have fBA: "finite (B - A)" by auto have e: "A \<inter> (B - A) = {}" by blast have eq: "A \<union> (B - A) = B" using AB by blast from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0" by arith then have "B - A = {}" unfolding card_eq_0_iff using fA fB by simp with AB show "A = B" by blast qed lemma insert_partition: "x \<notin> F \<Longrightarrow> \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<Longrightarrow> x \<inter> \<Union>F = {}" by auto (* somewhat slow *) lemma finite_psubset_induct [consumes 1, case_names psubset]: assumes finite: "finite A" and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" shows "P A" using finite proof (induct A taking: card rule: measure_induct_rule) case (less A) have fin: "finite A" by fact have ih: "card B < card A \<Longrightarrow> finite B \<Longrightarrow> P B" for B by fact have "P B" if "B \<subset> A" for B proof - from that have "card B < card A" using psubset_card_mono fin by blast moreover from that have "B \<subseteq> A" by auto then have "finite B" using fin finite_subset by blast ultimately show ?thesis using ih by simp qed with fin show "P A" using major by blast qed lemma finite_induct_select [consumes 1, case_names empty select]: assumes "finite S" and "P {}" and select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)" shows "P S" proof - have "0 \<le> card S" by simp then have "\<exists>T \<subseteq> S. card T = card S \<and> P T" proof (induct rule: dec_induct) case base with \<open>P {}\<close> show ?case by (intro exI[of _ "{}"]) auto next case (step n) then obtain T where T: "T \<subseteq> S" "card T = n" "P T" by auto with \<open>n < card S\<close> have "T \<subset> S" "P T" by auto with select[of T] obtain s where "s \<in> S" "s \<notin> T" "P (insert s T)" by auto with step(2) T \<open>finite S\<close> show ?case by (intro exI[of _ "insert s T"]) (auto dest: finite_subset) qed with \<open>finite S\<close> show "P S" by (auto dest: card_subset_eq) qed lemma remove_induct [case_names empty infinite remove]: assumes empty: "P ({} :: 'a set)" and infinite: "\<not> finite B \<Longrightarrow> P B" and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A" shows "P B" proof (cases "finite B") case False then show ?thesis by (rule infinite) next case True define A where "A = B" with True have "finite A" "A \<subseteq> B" by simp_all then show "P A" proof (induct "card A" arbitrary: A) case 0 then have "A = {}" by auto with empty show ?case by simp next case (Suc n A) from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A" by (rule finite_subset) moreover from Suc.hyps have "A \<noteq> {}" by auto moreover note \<open>A \<subseteq> B\<close> moreover have "P (A - {x})" if x: "x \<in> A" for x using x Suc.prems \<open>Suc n = card A\<close> by (intro Suc) auto ultimately show ?case by (rule remove) qed qed lemma finite_remove_induct [consumes 1, case_names empty remove]: fixes P :: "'a set \<Rightarrow> bool" assumes "finite B" and "P {}" and "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A" defines "B' \<equiv> B" shows "P B'" by (induct B' rule: remove_induct) (simp_all add: assms) text \<open>Main cardinality theorem.\<close> lemma card_partition [rule_format]: "finite C \<Longrightarrow> finite (\<Union>C) \<Longrightarrow> (\<forall>c\<in>C. card c = k) \<Longrightarrow> (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<Longrightarrow> k * card C = card (\<Union>C)" proof (induct rule: finite_induct) case empty then show ?case by simp next case (insert x F) then show ?case by (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert _ _)"]) qed lemma card_eq_UNIV_imp_eq_UNIV: assumes fin: "finite (UNIV :: 'a set)" and card: "card A = card (UNIV :: 'a set)" shows "A = (UNIV :: 'a set)" proof show "A \<subseteq> UNIV" by simp show "UNIV \<subseteq> A" proof show "x \<in> A" for x proof (rule ccontr) assume "x \<notin> A" then have "A \<subset> UNIV" by auto with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono) with card show False by simp qed qed qed text \<open>The form of a finite set of given cardinality\<close> lemma card_eq_SucD: assumes "card A = Suc k" shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})" proof - have fin: "finite A" using assms by (auto intro: ccontr) moreover have "card A \<noteq> 0" using assms by auto ultimately obtain b where b: "b \<in> A" by auto show ?thesis proof (intro exI conjI) show "A = insert b (A - {b})" using b by blast show "b \<notin> A - {b}" by blast show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}" using assms b fin by (fastforce dest: mk_disjoint_insert)+ qed qed lemma card_Suc_eq: "card A = Suc k \<longleftrightarrow> (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))" by (auto simp: card_insert_if card_gt_0_iff elim!: card_eq_SucD) lemma card_Suc_eq_finite: "card A = Suc k \<longleftrightarrow> (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> finite B)" unfolding card_Suc_eq using card_gt_0_iff by fastforce lemma card_1_singletonE: assumes "card A = 1" obtains x where "A = {x}" using assms by (auto simp: card_Suc_eq) lemma is_singleton_altdef: "is_singleton A \<longleftrightarrow> card A = 1" unfolding is_singleton_def by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def) lemma card_1_singleton_iff: "card A = Suc 0 \<longleftrightarrow> (\<exists>x. A = {x})" by (simp add: card_Suc_eq) lemma card_le_Suc0_iff_eq: assumes "finite A" shows "card A \<le> Suc 0 \<longleftrightarrow> (\<forall>a1 \<in> A. \<forall>a2 \<in> A. a1 = a2)" (is "?C = ?A") proof assume ?C thus ?A using assms by (auto simp: le_Suc_eq dest: card_eq_SucD) next assume ?A show ?C proof cases assume "A = {}" thus ?C using \<open>?A\<close> by simp next assume "A \<noteq> {}" then obtain a where "A = {a}" using \<open>?A\<close> by blast thus ?C by simp qed qed lemma card_le_Suc_iff: "Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)" proof (cases "finite A") case True then show ?thesis by (fastforce simp: card_Suc_eq less_eq_nat.simps split: nat.splits) qed auto lemma finite_fun_UNIVD2: assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)" shows "finite (UNIV :: 'b set)" proof - from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))" for arbitrary by (rule finite_imageI) moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)" for arbitrary by (rule UNIV_eq_I) auto ultimately show "finite (UNIV :: 'b set)" by simp qed lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1" unfolding UNIV_unit by simp lemma infinite_arbitrarily_large: assumes "\<not> finite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A" proof (induction n) case 0 show ?case by (intro exI[of _ "{}"]) auto next case (Suc n) then obtain B where B: "finite B \<and> card B = n \<and> B \<subseteq> A" .. with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto with B have "B \<subset> A" by auto then have "\<exists>x. x \<in> A - B" by (elim psubset_imp_ex_mem) then obtain x where x: "x \<in> A - B" .. with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A" by auto then show "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" .. qed text \<open>Sometimes, to prove that a set is finite, it is convenient to work with finite subsets and to show that their cardinalities are uniformly bounded. This possibility is formalized in the next criterion.\<close> lemma finite_if_finite_subsets_card_bdd: assumes "\<And>G. G \<subseteq> F \<Longrightarrow> finite G \<Longrightarrow> card G \<le> C" shows "finite F \<and> card F \<le> C" proof (cases "finite F") case False obtain n::nat where n: "n > max C 0" by auto obtain G where G: "G \<subseteq> F" "card G = n" using infinite_arbitrarily_large[OF False] by auto hence "finite G" using \<open>n > max C 0\<close> using card.infinite gr_implies_not0 by blast hence False using assms G n not_less by auto thus ?thesis .. next case True thus ?thesis using assms[of F] by auto qed subsubsection \<open>Cardinality of image\<close> lemma card_image_le: "finite A \<Longrightarrow> card (f ` A) \<le> card A" by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if) lemma card_image: "inj_on f A \<Longrightarrow> card (f ` A) = card A" proof (induct A rule: infinite_finite_induct) case (infinite A) then have "\<not> finite (f ` A)" by (auto dest: finite_imageD) with infinite show ?case by simp qed simp_all lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B" by (auto simp: card_image bij_betw_def) lemma endo_inj_surj: "finite A \<Longrightarrow> f ` A \<subseteq> A \<Longrightarrow> inj_on f A \<Longrightarrow> f ` A = A" by (simp add: card_seteq card_image) lemma eq_card_imp_inj_on: assumes "finite A" "card(f ` A) = card A" shows "inj_on f A" using assms proof (induct rule:finite_induct) case empty show ?case by simp next case (insert x A) then show ?case using card_image_le [of A f] by (simp add: card_insert_if split: if_splits) qed lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card (f ` A) = card A" by (blast intro: card_image eq_card_imp_inj_on) lemma card_inj_on_le: assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B" proof - have "finite A" using assms by (blast intro: finite_imageD dest: finite_subset) then show ?thesis using assms by (force intro: card_mono simp: card_image [symmetric]) qed lemma inj_on_iff_card_le: "\<lbrakk> finite A; finite B \<rbrakk> \<Longrightarrow> (\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)" using card_inj_on_le[of _ A B] card_le_inj[of A B] by blast lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A" by (blast intro: card_image_le card_mono le_trans) lemma card_bij_eq: "inj_on f A \<Longrightarrow> f ` A \<subseteq> B \<Longrightarrow> inj_on g B \<Longrightarrow> g ` B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B" by (auto intro: le_antisym card_inj_on_le) lemma bij_betw_finite: "bij_betw f A B \<Longrightarrow> finite A \<longleftrightarrow> finite B" unfolding bij_betw_def using finite_imageD [of f A] by auto lemma inj_on_finite: "inj_on f A \<Longrightarrow> f ` A \<le> B \<Longrightarrow> finite B \<Longrightarrow> finite A" using finite_imageD finite_subset by blast lemma card_vimage_inj: "inj f \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> card (f -` A) = card A" by (auto 4 3 simp: subset_image_iff inj_vimage_image_eq intro: card_image[symmetric, OF subset_inj_on]) subsubsection \<open>Pigeonhole Principles\<close> lemma pigeonhole: "card A > card (f ` A) \<Longrightarrow> \<not> inj_on f A " by (auto dest: card_image less_irrefl_nat) lemma pigeonhole_infinite: assumes "\<not> finite A" and "finite (f`A)" shows "\<exists>a0\<in>A. \<not> finite {a\<in>A. f a = f a0}" using assms(2,1) proof (induct "f`A" arbitrary: A rule: finite_induct) case empty then show ?case by simp next case (insert b F) show ?case proof (cases "finite {a\<in>A. f a = b}") case True with \<open>\<not> finite A\<close> have "\<not> finite (A - {a\<in>A. f a = b})" by simp also have "A - {a\<in>A. f a = b} = {a\<in>A. f a \<noteq> b}" by blast finally have "\<not> finite {a\<in>A. f a \<noteq> b}" . from insert(3)[OF _ this] insert(2,4) show ?thesis by simp (blast intro: rev_finite_subset) next case False then have "{a \<in> A. f a = b} \<noteq> {}" by force with False show ?thesis by blast qed qed lemma pigeonhole_infinite_rel: assumes "\<not> finite A" and "finite B" and "\<forall>a\<in>A. \<exists>b\<in>B. R a b" shows "\<exists>b\<in>B. \<not> finite {a:A. R a b}" proof - let ?F = "\<lambda>a. {b\<in>B. R a b}" from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>] have "finite (?F ` A)" by (blast intro: rev_finite_subset) from pigeonhole_infinite [where f = ?F, OF assms(1) this] obtain a0 where "a0 \<in> A" and infinite: "\<not> finite {a\<in>A. ?F a = ?F a0}" .. obtain b0 where "b0 \<in> B" and "R a0 b0" using \<open>a0 \<in> A\<close> assms(3) by blast have "finite {a\<in>A. ?F a = ?F a0}" if "finite {a\<in>A. R a b0}" using \<open>b0 \<in> B\<close> \<open>R a0 b0\<close> that by (blast intro: rev_finite_subset) with infinite \<open>b0 \<in> B\<close> show ?thesis by blast qed subsubsection \<open>Cardinality of sums\<close> lemma card_Plus: assumes "finite A" "finite B" shows "card (A <+> B) = card A + card B" proof - have "Inl`A \<inter> Inr`B = {}" by fast with assms show ?thesis by (simp add: Plus_def card_Un_disjoint card_image) qed lemma card_Plus_conv_if: "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)" by (auto simp add: card_Plus) text \<open>Relates to equivalence classes. Based on a theorem of F. KammÃ¼ller.\<close> lemma dvd_partition: assumes f: "finite (\<Union>C)" and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}" shows "k dvd card (\<Union>C)" proof - have "finite C" by (rule finite_UnionD [OF f]) then show ?thesis using assms proof (induct rule: finite_induct) case empty show ?case by simp next case (insert c C) then have "c \<inter> \<Union>C = {}" by auto with insert show ?case by (simp add: card_Un_disjoint) qed qed subsubsection \<open>Finite orders\<close> context order begin lemma finite_has_maximal: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. \<forall> b \<in> A. m \<le> b \<longrightarrow> m = b" proof (induction rule: finite_psubset_induct) case (psubset A) from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by auto let ?B = "{b \<in> A. a < b}" show ?case proof cases assume "?B = {}" hence "\<forall> b \<in> A. a \<le> b \<longrightarrow> a = b" using le_neq_trans by blast thus ?thesis using \<open>a \<in> A\<close> by blast next assume "?B \<noteq> {}" have "a \<notin> ?B" by auto hence "?B \<subset> A" using \<open>a \<in> A\<close> by blast from psubset.IH[OF this \<open>?B \<noteq> {}\<close>] show ?thesis using order.strict_trans2 by blast qed qed lemma finite_has_maximal2: "\<lbrakk> finite A; a \<in> A \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. a \<le> m \<and> (\<forall> b \<in> A. m \<le> b \<longrightarrow> m = b)" using finite_has_maximal[of "{b \<in> A. a \<le> b}"] by fastforce lemma finite_has_minimal: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. \<forall> b \<in> A. b \<le> m \<longrightarrow> m = b" proof (induction rule: finite_psubset_induct) case (psubset A) from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by auto let ?B = "{b \<in> A. b < a}" show ?case proof cases assume "?B = {}" hence "\<forall> b \<in> A. b \<le> a \<longrightarrow> a = b" using le_neq_trans by blast thus ?thesis using \<open>a \<in> A\<close> by blast next assume "?B \<noteq> {}" have "a \<notin> ?B" by auto hence "?B \<subset> A" using \<open>a \<in> A\<close> by blast from psubset.IH[OF this \<open>?B \<noteq> {}\<close>] show ?thesis using order.strict_trans1 by blast qed qed lemma finite_has_minimal2: "\<lbrakk> finite A; a \<in> A \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. m \<le> a \<and> (\<forall> b \<in> A. b \<le> m \<longrightarrow> m = b)" using finite_has_minimal[of "{b \<in> A. b \<le> a}"] by fastforce end subsubsection \<open>Relating injectivity and surjectivity\<close> lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A" proof - have "f ` A = A" by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le) then show ?thesis using assms by (simp add: eq_card_imp_inj_on) qed lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f" for f :: "'a \<Rightarrow> 'a" by (blast intro: finite_surj_inj subset_UNIV) lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f" for f :: "'a \<Rightarrow> 'a" by (fastforce simp:surj_def dest!: endo_inj_surj) lemma surjective_iff_injective_gen: assumes fS: "finite S" and fT: "finite T" and c: "card S = card T" and ST: "f ` S \<subseteq> T" shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs") proof assume h: "?lhs" { fix x y assume x: "x \<in> S" assume y: "y \<in> S" assume f: "f x = f y" from x fS have S0: "card S \<noteq> 0" by auto have "x = y" proof (rule ccontr) assume xy: "\<not> ?thesis" have th: "card S \<le> card (f ` (S - {y}))" unfolding c proof (rule card_mono) show "finite (f ` (S - {y}))" by (simp add: fS) have "\<lbrakk>x \<noteq> y; x \<in> S; z \<in> S; f x = f y\<rbrakk> \<Longrightarrow> \<exists>x \<in> S. x \<noteq> y \<and> f z = f x" for z by (case_tac "z = y \<longrightarrow> z = x") auto then show "T \<subseteq> f ` (S - {y})" using h xy x y f by fastforce qed also have " \<dots> \<le> card (S - {y})" by (simp add: card_image_le fS) also have "\<dots> \<le> card S - 1" using y fS by simp finally show False using S0 by arith qed } then show ?rhs unfolding inj_on_def by blast next assume h: ?rhs have "f ` S = T" by (simp add: ST c card_image card_subset_eq fT h) then show ?lhs by blast qed hide_const (open) Finite_Set.fold subsection \<open>Infinite Sets\<close> text \<open> Some elementary facts about infinite sets, mostly by Stephan Merz. Beware! Because "infinite" merely abbreviates a negation, these lemmas may not work well with \<open>blast\<close>. \<close> abbreviation infinite :: "'a set \<Rightarrow> bool" where "infinite S \<equiv> \<not> finite S" text \<open> Infinite sets are non-empty, and if we remove some elements from an infinite set, the result is still infinite. \<close> lemma infinite_UNIV_nat [iff]: "infinite (UNIV :: nat set)" proof assume "finite (UNIV :: nat set)" with finite_UNIV_inj_surj [of Suc] show False by simp (blast dest: Suc_neq_Zero surjD) qed lemma infinite_UNIV_char_0: "infinite (UNIV :: 'a::semiring_char_0 set)" proof assume "finite (UNIV :: 'a set)" with subset_UNIV have "finite (range of_nat :: 'a set)" by (rule finite_subset) moreover have "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def) ultimately have "finite (UNIV :: nat set)" by (rule finite_imageD) then show False by simp qed lemma infinite_imp_nonempty: "infinite S \<Longrightarrow> S \<noteq> {}" by auto lemma infinite_remove: "infinite S \<Longrightarrow> infinite (S - {a})" by simp lemma Diff_infinite_finite: assumes "finite T" "infinite S" shows "infinite (S - T)" using \<open>finite T\<close> proof induct from \<open>infinite S\<close> show "infinite (S - {})" by auto next fix T x assume ih: "infinite (S - T)" have "S - (insert x T) = (S - T) - {x}" by (rule Diff_insert) with ih show "infinite (S - (insert x T))" by (simp add: infinite_remove) qed lemma Un_infinite: "infinite S \<Longrightarrow> infinite (S \<union> T)" by simp lemma infinite_Un: "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T" by simp lemma infinite_super: assumes "S \<subseteq> T" and "infinite S" shows "infinite T" proof assume "finite T" with \<open>S \<subseteq> T\<close> have "finite S" by (simp add: finite_subset) with \<open>infinite S\<close> show False by simp qed proposition infinite_coinduct [consumes 1, case_names infinite]: assumes "X A" and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})" shows "infinite A" proof assume "finite A" then show False using \<open>X A\<close> proof (induction rule: finite_psubset_induct) case (psubset A) then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})" using local.step psubset.prems by blast then have "X (A - {x})" using psubset.hyps by blast show False proof (rule psubset.IH [where B = "A - {x}"]) show "A - {x} \<subset> A" using \<open>x \<in> A\<close> by blast qed fact qed qed text \<open> For any function with infinite domain and finite range there is some element that is the image of infinitely many domain elements. In particular, any infinite sequence of elements from a finite set contains some element that occurs infinitely often. \<close> lemma inf_img_fin_dom': assumes img: "finite (f ` A)" and dom: "infinite A" shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)" proof (rule ccontr) have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto moreover assume "\<not> ?thesis" with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast ultimately have "finite A" by (rule finite_subset) with dom show False by contradiction qed lemma inf_img_fin_domE': assumes "finite (f ` A)" and "infinite A" obtains y where "y \<in> f`A" and "infinite (f -` {y} \<inter> A)" using assms by (blast dest: inf_img_fin_dom') lemma inf_img_fin_dom: assumes img: "finite (f`A)" and dom: "infinite A" shows "\<exists>y \<in> f`A. infinite (f -` {y})" using inf_img_fin_dom'[OF assms] by auto lemma inf_img_fin_domE: assumes "finite (f`A)" and "infinite A" obtains y where "y \<in> f`A" and "infinite (f -` {y})" using assms by (blast dest: inf_img_fin_dom) proposition finite_image_absD: "finite (abs ` S) \<Longrightarrow> finite S" for S :: "'a::linordered_ring set" by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom) subsection \<open>The finite powerset operator\<close> definition Fpow :: "'a set \<Rightarrow> 'a set set" where "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}" lemma Fpow_mono: "A \<subseteq> B \<Longrightarrow> Fpow A \<subseteq> Fpow B" unfolding Fpow_def by auto lemma empty_in_Fpow: "{} \<in> Fpow A" unfolding Fpow_def by auto lemma Fpow_not_empty: "Fpow A \<noteq> {}" using empty_in_Fpow by blast lemma Fpow_subset_Pow: "Fpow A \<subseteq> Pow A" unfolding Fpow_def by auto lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}" unfolding Fpow_def Pow_def by blast lemma inj_on_image_Fpow: assumes "inj_on f A" shows "inj_on (image f) (Fpow A)" using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"] inj_on_image_Pow by blast lemma image_Fpow_mono: assumes "f ` A \<subseteq> B" shows "(image f) ` (Fpow A) \<subseteq> Fpow B" using assms by(unfold Fpow_def, auto) end