(* Title: HOL/Finite_Set.thy Author: Tobias Nipkow Author: Lawrence C Paulson Author: Markus Wenzel Author: Jeremy Avigad Author: Andrei Popescu *) section ‹Finite sets› theory Finite_Set imports Product_Type Sum_Type Fields begin subsection ‹Predicate for finite sets› context notes [[inductive_internals]] begin inductive finite :: "'a set ⇒ bool" where emptyI [simp, intro!]: "finite {}" | insertI [simp, intro!]: "finite A ⟹ finite (insert a A)" end simproc_setup finite_Collect ("finite (Collect P)") = ‹K Set_Comprehension_Pointfree.simproc› declare [[simproc del: finite_Collect]] lemma finite_induct [case_names empty insert, induct set: finite]: ― ‹Discharging ‹x ∉ F› entails extra work.› assumes "finite F" assumes "P {}" and insert: "⋀x F. finite F ⟹ x ∉ F ⟹ P F ⟹ P (insert x F)" shows "P F" using ‹finite F› 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 ∈ F" then have "insert x F = F" by (rule insert_absorb) with P show ?thesis by (simp only:) next assume "x ∉ F" from F this P show ?thesis by (rule insert) qed qed lemma infinite_finite_induct [case_names infinite empty insert]: assumes infinite: "⋀A. ¬ finite A ⟹ P A" and empty: "P {}" and insert: "⋀x F. finite F ⟹ x ∉ F ⟹ P F ⟹ 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 ‹Choice principles› lemma ex_new_if_finite: ― ‹does not depend on def of finite at all› assumes "¬ finite (UNIV :: 'a set)" and "finite A" shows "∃a::'a. a ∉ A" proof - from assms have "A ≠ UNIV" by blast then show ?thesis by blast qed text ‹A finite choice principle. Does not need the SOME choice operator.› lemma finite_set_choice: "finite A ⟹ ∀x∈A. ∃y. P x y ⟹ ∃f. ∀x∈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: "∀x∈A. P x (f x)" and ab: "P a b" by auto show ?case (is "∃f. ?P f") proof show "?P (λx. if x = a then b else f x)" using f ab by auto qed qed subsubsection ‹Finite sets are the images of initial segments of natural numbers› lemma finite_imp_nat_seg_image_inj_on: assumes "finite A" shows "∃(n::nat) f. A = f ` {i. i < n} ∧ inj_on f {i. i < n}" using assms proof induct case empty show ?case proof show "∃f. {} = f ` {i::nat. i < 0} ∧ inj_on f {i. i < 0}" by simp qed next case (insert a A) have notinA: "a ∉ 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} ⟹ 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 "∃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 ⟷ (∃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 "∃f n. f ` A = {i::nat. i < n} ∧ inj_on f A" proof - from finite_imp_nat_seg_image_inj_on [OF ‹finite A›] 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 ∧ ?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 ≤ k}" by (simp add: le_eq_less_or_eq Collect_disj_eq) subsubsection ‹Finiteness and common set operations› lemma rev_finite_subset: "finite B ⟹ A ⊆ B ⟹ 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 ⊆ insert x F" and r: "A - {x} ⊆ F ⟹ finite (A - {x})" by fact+ show "finite A" proof cases assume x: "x ∈ A" with A have "A - {x} ⊆ 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 ⊆ F" using that by fact assume "x ∉ A" with A show "A ⊆ F" by (simp add: subset_insert_iff) qed qed lemma finite_subset: "A ⊆ B ⟹ finite B ⟹ finite A" by (rule rev_finite_subset) lemma finite_UnI: assumes "finite F" and "finite G" shows "finite (F ∪ G)" using assms by induct simp_all lemma finite_Un [iff]: "finite (F ∪ G) ⟷ finite F ∧ finite G" by (blast intro: finite_UnI finite_subset [of _ "F ∪ G"]) lemma finite_insert [simp]: "finite (insert a A) ⟷ finite A" proof - have "finite {a} ∧ finite A ⟷ finite A" by simp then have "finite ({a} ∪ A) ⟷ finite A" by (simp only: finite_Un) then show ?thesis by simp qed lemma finite_Int [simp, intro]: "finite F ∨ finite G ⟹ finite (F ∩ G)" by (blast intro: finite_subset) lemma finite_Collect_conjI [simp, intro]: "finite {x. P x} ∨ finite {x. Q x} ⟹ finite {x. P x ∧ Q x}" by (simp add: Collect_conj_eq) lemma finite_Collect_disjI [simp]: "finite {x. P x ∨ Q x} ⟷ finite {x. P x} ∧ finite {x. Q x}" by (simp add: Collect_disj_eq) lemma finite_Diff [simp, intro]: "finite A ⟹ finite (A - B)" by (rule finite_subset, rule Diff_subset) lemma finite_Diff2 [simp]: assumes "finite B" shows "finite (A - B) ⟷ finite A" proof - have "finite A ⟷ finite ((A - B) ∪ (A ∩ B))" by (simp add: Un_Diff_Int) also have "… ⟷ finite (A - B)" using ‹finite B› by simp finally show ?thesis .. qed lemma finite_Diff_insert [iff]: "finite (A - insert a B) ⟷ finite (A - B)" proof - have "finite (A - B) ⟷ 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) ⟹ finite (- A) ⟷ finite (UNIV :: 'a set)" by (simp add: Compl_eq_Diff_UNIV) lemma finite_Collect_not [simp]: "finite {x :: 'a. P x} ⟹ finite {x. ¬ P x} ⟷ finite (UNIV :: 'a set)" by (simp add: Collect_neg_eq) lemma finite_Union [simp, intro]: "finite A ⟹ (⋀M. M ∈ A ⟹ finite M) ⟹ finite (⋃A)" by (induct rule: finite_induct) simp_all lemma finite_UN_I [intro]: "finite A ⟹ (⋀a. a ∈ A ⟹ finite (B a)) ⟹ finite (⋃a∈A. B a)" by (induct rule: finite_induct) simp_all lemma finite_UN [simp]: "finite A ⟹ finite (UNION A B) ⟷ (∀x∈A. finite (B x))" by (blast intro: finite_subset) lemma finite_Inter [intro]: "∃A∈M. finite A ⟹ finite (⋂M)" by (blast intro: Inter_lower finite_subset) lemma finite_INT [intro]: "∃x∈I. finite (A x) ⟹ finite (⋂x∈I. A x)" by (blast intro: INT_lower finite_subset) lemma finite_imageI [simp, intro]: "finite F ⟹ finite (h ` F)" by (induct rule: finite_induct) simp_all lemma finite_image_set [simp]: "finite {x. P x} ⟹ finite {f x |x. P x}" by (simp add: image_Collect [symmetric]) lemma finite_image_set2: "finite {x. P x} ⟹ finite {y. Q y} ⟹ finite {f x y |x y. P x ∧ Q y}" by (rule finite_subset [where B = "⋃x ∈ {x. P x}. ⋃y ∈ {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 ∈ A" by blast from B_A ‹x ∉ B› have "B = f ` A - {x}" by blast with B_A ‹x ∉ B› ‹x = f y› ‹inj_on f A› ‹y ∈ A› have "B = f ` (A - {y})" by (simp add: inj_on_image_set_diff Set.Diff_subset) moreover from ‹inj_on f A› 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 ⟹ finite (f ` A) ⟷ finite A" using finite_imageD by blast lemma finite_surj: "finite A ⟹ B ⊆ f ` A ⟹ finite B" by (erule finite_subset) (rule finite_imageI) lemma finite_range_imageI: "finite (range g) ⟹ finite (range (λx. f (g x)))" by (drule finite_imageI) (simp add: range_composition) lemma finite_subset_image: assumes "finite B" shows "B ⊆ f ` A ⟹ ∃C⊆A. finite C ∧ 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 (* slow *) qed lemma finite_vimage_IntI: "finite F ⟹ inj_on h A ⟹ finite (h -` F ∩ A)" apply (induct rule: finite_induct) apply simp_all apply (subst vimage_insert) apply (simp add: finite_subset [OF inj_on_vimage_singleton] Int_Un_distrib2) done lemma finite_finite_vimage_IntI: assumes "finite F" and "⋀y. y ∈ F ⟹ finite ((h -` {y}) ∩ A)" shows "finite (h -` F ∩ A)" proof - have *: "h -` F ∩ A = (⋃ y∈F. (h -` {y}) ∩ A)" by blast show ?thesis by (simp only: * assms finite_UN_I) qed lemma finite_vimageI: "finite F ⟹ inj h ⟹ finite (h -` F)" using finite_vimage_IntI[of F h UNIV] by auto lemma finite_vimageD': "finite (f -` A) ⟹ A ⊆ range f ⟹ finite A" by (auto simp add: subset_image_iff intro: finite_subset[rotated]) lemma finite_vimageD: "finite (h -` F) ⟹ surj h ⟹ finite F" by (auto dest: finite_vimageD') lemma finite_vimage_iff: "bij h ⟹ finite (h -` F) ⟷ finite F" unfolding bij_def by (auto elim: finite_vimageD finite_vimageI) lemma finite_Collect_bex [simp]: assumes "finite A" shows "finite {x. ∃y∈A. Q x y} ⟷ (∀y∈A. finite {x. Q x y})" proof - have "{x. ∃y∈A. Q x y} = (⋃y∈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. ∃y. P y ∧ Q x y} ⟷ (∀y. P y ⟶ finite {x. Q x y})" proof - have "{x. ∃y. P y ∧ Q x y} = (⋃y∈{y. P y}. {x. Q x y})" by auto with assms show ?thesis by simp qed lemma finite_Plus: "finite A ⟹ finite B ⟹ 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 ⊆ 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 ⊆ 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) ⟷ finite A ∧ finite B" by (auto intro: finite_PlusD finite_Plus) lemma finite_Plus_UNIV_iff [simp]: "finite (UNIV :: ('a + 'b) set) ⟷ finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set)" by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff) lemma finite_SigmaI [simp, intro]: "finite A ⟹ (⋀a. a∈A ⟹ finite (B a)) ⟹ finite (SIGMA a:A. B a)" unfolding Sigma_def by blast lemma finite_SigmaI2: assumes "finite {x∈A. B x ≠ {}}" and "⋀a. a ∈ A ⟹ finite (B a)" shows "finite (Sigma A B)" proof - from assms have "finite (Sigma {x∈A. B x ≠ {}} B)" by auto also have "Sigma {x:A. B x ≠ {}} B = Sigma A B" by auto finally show ?thesis . qed lemma finite_cartesian_product: "finite A ⟹ finite B ⟹ finite (A × B)" by (rule finite_SigmaI) lemma finite_Prod_UNIV: "finite (UNIV :: 'a set) ⟹ finite (UNIV :: 'b set) ⟹ finite (UNIV :: ('a × 'b) set)" by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product) lemma finite_cartesian_productD1: assumes "finite (A × B)" and "B ≠ {}" shows "finite A" proof - from assms obtain n f where "A × B = f ` {i::nat. i < n}" by (auto simp add: finite_conv_nat_seg_image) then have "fst ` (A × B) = fst ` f ` {i::nat. i < n}" by simp with ‹B ≠ {}› have "A = (fst ∘ f) ` {i::nat. i < n}" by (simp add: image_comp) then have "∃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 × B)" and "A ≠ {}" shows "finite B" proof - from assms obtain n f where "A × B = f ` {i::nat. i < n}" by (auto simp add: finite_conv_nat_seg_image) then have "snd ` (A × B) = snd ` f ` {i::nat. i < n}" by simp with ‹A ≠ {}› have "B = (snd ∘ f) ` {i::nat. i < n}" by (simp add: image_comp) then have "∃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 × B) ⟷ (A = {} ∨ B = {} ∨ (finite A ∧ finite B))" by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product) lemma finite_prod: "finite (UNIV :: ('a × 'b) set) ⟷ finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set)" using finite_cartesian_product_iff[of UNIV UNIV] by simp lemma finite_Pow_iff [iff]: "finite (Pow A) ⟷ finite A" proof assume "finite (Pow A)" then have "finite ((λx. {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 ⟹ finite {B. B ⊆ A}" by (simp add: Pow_def [symmetric]) lemma finite_set: "finite (UNIV :: 'a set set) ⟷ finite (UNIV :: 'a set)" by (simp only: finite_Pow_iff Pow_UNIV[symmetric]) lemma finite_UnionD: "finite (⋃A) ⟹ finite A" by (blast intro: finite_subset [OF subset_Pow_Union]) lemma finite_bind: assumes "finite S" assumes "∀x ∈ S. finite (f x)" shows "finite (Set.bind S f)" using assms by (simp add: bind_UNION) lemma finite_filter [simp]: "finite S ⟹ 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. ∀x. (x ∈ A ⟶ f x ∈ B) ∧ (x ∉ A ⟶ f x = d)}" (is "finite ?S") proof - let ?F = "λf. {(a,b). a ∈ A ∧ b = f a}" have "?F ` ?S ⊆ Pow(A × 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 "¬ finite {a. P a}" shows "∃a. P a" proof (rule classical) assume "¬ ?thesis" with assms show ?thesis by auto qed subsubsection ‹Further induction rules on finite sets› lemma finite_ne_induct [case_names singleton insert, consumes 2]: assumes "finite F" and "F ≠ {}" assumes "⋀x. P {x}" and "⋀x F. finite F ⟹ F ≠ {} ⟹ x ∉ F ⟹ P F ⟹ 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 ⊆ A" and empty: "P {}" and insert: "⋀a F. finite F ⟹ a ∈ A ⟹ a ∉ F ⟹ P F ⟹ P (insert a F)" shows "P F" using ‹finite F› ‹F ⊆ A› proof induct show "P {}" by fact next fix x F assume "finite F" and "x ∉ F" and P: "F ⊆ A ⟹ P F" and i: "insert x F ⊆ A" show "P (insert x F)" proof (rule insert) from i show "x ∈ A" by blast from i have "F ⊆ A" by blast with P show "P F" . show "finite F" by fact show "x ∉ F" by fact qed qed lemma finite_empty_induct: assumes "finite A" and "P A" and remove: "⋀a A. finite A ⟹ a ∈ A ⟹ P A ⟹ P (A - {a})" shows "P {}" proof - have "P (A - B)" if "B ⊆ A" for B :: "'a set" proof - from ‹finite A› that have "finite B" by (rule rev_finite_subset) from this ‹B ⊆ A› show "P (A - B)" proof induct case empty from ‹P A› show ?case by simp next case (insert b B) have "P (A - B - {b})" proof (rule remove) from ‹finite A› show "finite (A - B)" by induct auto from insert show "b ∈ 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 ≠ c}" and const: "P (λa. c)" and update: "⋀a b f. finite {a. f a ≠ c} ⟹ f a = c ⟹ b ≠ c ⟹ P f ⟹ P (f(a := b))" shows "P f" using finite proof (induct "{a. f a ≠ c}" arbitrary: f) case empty with const show ?case by simp next case (insert a A) then have "A = {a'. (f(a := c)) a' ≠ c}" and "f a ≠ c" by auto with ‹finite A› have "finite {a'. (f(a := c)) a' ≠ c}" by simp have "(f(a := c)) a = c" by simp from insert ‹A = {a'. (f(a := c)) a' ≠ c}› have "P (f(a := c))" by simp with ‹finite {a'. (f(a := c)) a' ≠ c}› ‹(f(a := c)) a = c› ‹f a ≠ c› 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 ⊆ A" and empty: "P {}" and insert: "⋀a F. ⟦finite F; a ∈ A; F ⊆ A; a ∉ F; P F ⟧ ⟹ 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 ∉ F" and P: "F ⊆ A ⟹ P F" and i: "insert x F ⊆ A" show "P (insert x F)" proof (rule insert) from i show "x ∈ A" by blast from i have "F ⊆ A" by blast with P show "P F" . show "finite F" by fact show "x ∉ F" by fact show "F ⊆ A" by fact qed qed subsection ‹Class ‹finite›› 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) ⟷ 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 (λf. {(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 ⇒ 'b) set)" proof (rule finite_imageD) let ?graph = "λf::'a ⇒ 'b. {(x, y). y = f x}" have "range ?graph ⊆ 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 ‹A basic fold functional for finite sets› text ‹The intended behaviour is ‹fold f z {x⇩_{1}, …, x⇩_{n}} = f x⇩_{1}(… (f x⇩_{n}z)…)› if ‹f› is ``left-commutative'': › locale comp_fun_commute = fixes f :: "'a ⇒ 'b ⇒ 'b" assumes comp_fun_commute: "f y ∘ f x = f x ∘ 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 ∘ (f x ∘ g) = f x ∘ (f y ∘ g)" by (simp add: o_assoc comp_fun_commute) end inductive fold_graph :: "('a ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a set ⇒ 'b ⇒ bool" for f :: "'a ⇒ 'b ⇒ 'b" and z :: 'b where emptyI [intro]: "fold_graph f z {} z" | insertI [intro]: "x ∉ A ⟹ fold_graph f z A y ⟹ 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 ∧ x ∈ B" if "fold_graph g z A x" "⋀a b. a ∈ A ⟹ b ∈ B ⟹ f a b = g a b" "⋀a b. a ∈ A ⟹ b ∈ B ⟹ g a b ∈ B" "z ∈ B" using that(1-3) proof (induction rule: fold_graph.induct) case (insertI x A y) have "fold_graph f z A y" "y ∈ B" unfolding atomize_conj by (rule insertI.IH) (auto intro: insertI.prems) then have "g x y ∈ 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 "⋀a b. a ∈ A ⟹ b ∈ B ⟹ f a b = g a b" "⋀a b. a ∈ A ⟹ b ∈ B ⟹ g a b ∈ B" "z ∈ 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 ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a set ⇒ '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 "⋀a b. a ∈ A ⟹ b ∈ B ⟹ f a b = g a b" "⋀a b. a ∈ A ⟹ b ∈ B ⟹ g a b ∈ B" "z ∈ B" unfolding Finite_Set.fold_def by (subst fold_graph_closed_eq[where B=B and g=g]) (auto simp: that) text ‹ A tempting alternative for the definiens is @{term "if finite A then THE y. fold_graph f z A y else e"}. It allows the removal of finiteness assumptions from the theorems ‹fold_comm›, ‹fold_reindex› and ‹fold_distrib›. The proofs become ugly. It is not worth the effort. (???) › lemma finite_imp_fold_graph: "finite A ⟹ ∃x. fold_graph f z A x" by (induct rule: finite_induct) auto subsubsection ‹From @{const fold_graph} to @{term fold}› 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 ⟹ a ∈ A ⟹ ∃y'. y = f a y' ∧ 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 ‹x ≠ a› and ‹x ∉ A› 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 ∉ 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 ⟹ fold_graph f z A y ⟹ y = x" proof (induct arbitrary: y set: fold_graph) case emptyI then show ?case by fast next case (insertI x A y v) from ‹fold_graph f z (insert x A) v› and ‹x ∉ A› obtain y' where "v = f x y'" and "fold_graph f z A y'" by (rule fold_graph_insertE) from ‹fold_graph f z A y'› have "y' = y" by (rule insertI) with ‹v = f x y'› show "v = f x y" by simp qed lemma fold_equality: "fold_graph f z A y ⟹ 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 "∃x. fold_graph f z A x" by (rule finite_imp_fold_graph) moreover note fold_graph_determ ultimately have "∃!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 ‹The base case for ‹fold›:› lemma (in -) fold_infinite [simp]: "¬ finite A ⟹ 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 ‹The various recursion equations for @{const fold}:› lemma fold_insert [simp]: assumes "finite A" and "x ∉ A" shows "fold f z (insert x A) = f x (fold f z A)" proof (rule fold_equality) fix z from ‹finite A› have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold) with ‹x ∉ A› 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] ― ‹No more proofs involve these.› lemma fold_fun_left_comm: "finite A ⟹ 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 ⟹ x ∉ A ⟹ 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 ∈ A" shows "fold f z A = f x (fold f z (A - {x}))" proof - have A: "A = insert x (A - {x})" using ‹x ∈ A› by blast then have "fold f z A = fold f z (insert x (A - {x}))" by simp also have "… = f x (fold f z (A - {x}))" by (rule fold_insert) (simp add: ‹finite A›)+ 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 ‹finite A› have "finite (insert x A)" by auto moreover have "x ∈ 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 ∩ B = {}" shows "Finite_Set.fold f z (A ∪ B) = Finite_Set.fold f (Finite_Set.fold f z A) B" using assms(2,1,3) by induct simp_all end text ‹Other properties of @{const fold}:› lemma fold_image: assumes "inj_on g A" shows "fold f z (g ` A) = fold (f ∘ 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 ∘ g) z A" proof fix w show "fold_graph f z (g ` A) w ⟷ fold_graph (f ∘ g) z A w" (is "?P ⟷ ?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 ‹inj_on g B› ‹x ∉ A› ‹insert x A = image g B› obtain x' A' where "x' ∉ 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 ∘ g) z A' r" by (auto intro: insertI.hyps) with ‹x' ∉ A'› have "fold_graph (f ∘ g) z (insert x' A') ((f ∘ 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 ‹x ∉ A› insertI.prems have "g x ∉ 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: "⋀x. x ∈ A ⟹ 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 ‹finite A› cong proof (induct A) case empty then show ?case by simp next case insert interpret f: comp_fun_commute f by (fact ‹comp_fun_commute f›) interpret g: comp_fun_commute g by (fact ‹comp_fun_commute g›) from insert show ?case by simp qed with assms show ?thesis by simp qed text ‹A simplified version for idempotent functions:› locale comp_fun_idem = comp_fun_commute + assumes comp_fun_idem: "f x ∘ 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 ∈ A" then obtain B where "A = insert x B" and "x ∉ B" by (rule set_insert) then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem) next assume "x ∉ A" then show ?thesis using assms by simp qed declare fold_insert [simp del] fold_insert_idem [simp] lemma fold_insert_idem2: "finite A ⟹ fold f z (insert x A) = fold f (f x z) A" by (simp add: fold_fun_left_comm) end subsubsection ‹Liftings to ‹comp_fun_commute› etc.› lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f ∘ g)" by standard (simp_all add: comp_fun_commute) lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f ∘ 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 (λx. f x ^^ g x)" proof show "f y ^^ g y ∘ f x ^^ g x = f x ^^ g x ∘ 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 ∘ f x = f x ∘ 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 ∘ f x = f x ∘ 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 ∘ f x ^^ h x = f x ^^ h x ∘ f y ^^ h y" by auto with False h_def have hyp2: "f y ^^ g y ∘ f x ^^ h x = f x ^^ h x ∘ 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 ‹Expressing set operations via @{const fold}› lemma comp_fun_commute_const: "comp_fun_commute (λ_. 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 ∪ B = fold insert B A" proof - interpret comp_fun_idem insert by (fact comp_fun_idem_insert) from ‹finite A› 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 ‹finite A› 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 (λx 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 (λx 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 ∩ B = Set.filter (λx. x ∈ A) B" using assms by induct (auto simp: Set.filter_def) lemma image_fold_insert: assumes "finite A" shows "image f A = fold (λk A. Set.insert (f k) A) {} A" proof - interpret comp_fun_commute "λk 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 (λk s. s ∧ P k) True A" proof - interpret comp_fun_commute "λk s. s ∧ 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 (λk s. s ∨ P k) False A" proof - interpret comp_fun_commute "λk s. s ∨ P k" by standard auto show ?thesis using assms by (induct A) auto qed lemma comp_fun_commute_Pow_fold: "comp_fun_commute (λx A. A ∪ 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 (λx A. A ∪ Set.insert x ` A) {{}} A" proof - interpret comp_fun_commute "λx A. A ∪ 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 "(⋃y∈B. {(x, y)}) ∪ A = fold (λy. Set.insert (x, y)) A B" proof - interpret comp_fun_commute "λy. 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 ⟹ comp_fun_commute (λx z. fold (λy. Set.insert (x, y)) z B)" by standard (auto simp: fold_union_pair [symmetric]) lemma product_fold: assumes "finite A" "finite B" shows "A × B = fold (λx z. fold (λy. 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 ‹finite A› 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 ‹finite A› 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 ⟹ 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 ⟹ 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 (INFIMUM A f) = fold (inf ∘ f) B A" (is "?inf = ?fold") proof - interpret comp_fun_idem inf by (fact comp_fun_idem_inf) interpret comp_fun_idem "inf ∘ f" by (fact comp_comp_fun_idem) from ‹finite A› 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 (SUPREMUM A f) = fold (sup ∘ f) B A" (is "?sup = ?fold") proof - interpret comp_fun_idem sup by (fact comp_fun_idem_sup) interpret comp_fun_idem "sup ∘ f" by (fact comp_comp_fun_idem) from ‹finite A› have "?fold = ?sup" by (induct A arbitrary: B) (simp_all add: sup_left_commute) then show ?thesis .. qed lemma INF_fold_inf: "finite A ⟹ INFIMUM A f = fold (inf ∘ f) top A" using inf_INF_fold_inf [of A top] by simp lemma SUP_fold_sup: "finite A ⟹ SUPREMUM A f = fold (sup ∘ f) bot A" using sup_SUP_fold_sup [of A bot] by simp end subsection ‹Locales as mini-packages for fold operations› subsubsection ‹The natural case› locale folding = fixes f :: "'a ⇒ 'b ⇒ 'b" and z :: "'b" assumes comp_fun_commute: "f y ∘ f x = f x ∘ f y" begin interpretation fold?: comp_fun_commute f by standard (use comp_fun_commute in ‹simp add: fun_eq_iff›) definition F :: "'a set ⇒ 'b" where eq_fold: "F A = fold f z A" lemma empty [simp]:"F {} = z" by (simp add: eq_fold) lemma infinite [simp]: "¬ finite A ⟹ F A = z" by (simp add: eq_fold) lemma insert [simp]: assumes "finite A" and "x ∉ 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 ‹finite A› show ?thesis by (simp add: eq_fold fun_eq_iff) qed lemma remove: assumes "finite A" and "x ∈ A" shows "F A = f x (F (A - {x}))" proof - from ‹x ∈ A› obtain B where A: "A = insert x B" and "x ∉ B" by (auto dest: mk_disjoint_insert) moreover from ‹finite A› A have "finite B" by simp ultimately show ?thesis by simp qed lemma insert_remove: "finite A ⟹ F (insert x A) = f x (F (A - {x}))" by (cases "x ∈ A") (simp_all add: remove insert_absorb) end subsubsection ‹With idempotency› locale folding_idem = folding + assumes comp_fun_idem: "f x ∘ 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 ‹finite A› show ?thesis by (simp add: eq_fold fun_eq_iff) qed end subsection ‹Finite cardinality› text ‹ The traditional definition @{prop "card A ≡ LEAST n. ∃f. A = {f i |i. i < n}"} is ugly to work with. But now that we have @{const fold} things are easy: › global_interpretation card: folding "λ_. Suc" 0 defines card = "folding.F (λ_. Suc) 0" by standard rule lemma card_infinite: "¬ finite A ⟹ card A = 0" by (fact card.infinite) lemma card_empty: "card {} = 0" by (fact card.empty) lemma card_insert_disjoint: "finite A ⟹ x ∉ A ⟹ card (insert x A) = Suc (card A)" by (fact card.insert) lemma card_insert_if: "finite A ⟹ card (insert x A) = (if x ∈ 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 ⟹ finite A" by (rule ccontr) simp lemma card_0_eq [simp]: "finite A ⟹ card A = 0 ⟷ A = {}" by (auto dest: mk_disjoint_insert) lemma finite_UNIV_card_ge_0: "finite (UNIV :: 'a set) ⟹ card (UNIV :: 'a set) > 0" by (rule ccontr) simp lemma card_eq_0_iff: "card A = 0 ⟷ A = {} ∨ ¬ finite A" by auto lemma card_range_greater_zero: "finite (range f) ⟹ card (range f) > 0" by (rule ccontr) (simp add: card_eq_0_iff) lemma card_gt_0_iff: "0 < card A ⟷ A ≠ {} ∧ finite A" by (simp add: neq0_conv [symmetric] card_eq_0_iff) lemma card_Suc_Diff1: "finite A ⟹ x ∈ A ⟹ Suc (card (A - {x})) = card A" apply (rule insert_Diff [THEN subst, where t = A]) apply assumption apply (simp del: insert_Diff_single) done lemma card_insert_le_m1: "n > 0 ⟹ card y ≤ n - 1 ⟹ card (insert x y) ≤ n" apply (cases "finite y") apply (cases "x ∈ y") apply (auto simp: insert_absorb) done lemma card_Diff_singleton: "finite A ⟹ x ∈ A ⟹ card (A - {x}) = card A - 1" by (simp add: card_Suc_Diff1 [symmetric]) lemma card_Diff_singleton_if: "finite A ⟹ card (A - {x}) = (if x ∈ A then card A - 1 else card A)" by (simp add: card_Diff_singleton) lemma card_Diff_insert[simp]: assumes "finite A" and "a ∈ A" and "a ∉ 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: "finite A ⟹ card (insert x A) = Suc (card (A - {x}))" by (fact card.insert_remove) lemma card_insert_le: "finite A ⟹ card A ≤ 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 ≤ 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 ⊆ B" shows "card A ≤ 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 ∈ B" by simp from insert have "A ⊆ B - {x}" and "finite (B - {x})" by auto with insert.hyps have "card A ≤ card (B - {x})" by auto with ‹finite A› ‹x ∉ A› ‹finite B› ‹x ∈ B› show ?case by simp (simp only: card.remove) qed qed lemma card_seteq: "finite B ⟹ (⋀A. A ⊆ B ⟹ card B ≤ card A ⟹ A = B)" apply (induct rule: finite_induct) apply simp apply clarify apply (subgoal_tac "finite A ∧ A - {x} ⊆ F") prefer 2 apply (blast intro: finite_subset, atomize) apply (drule_tac x = "A - {x}" in spec) apply (simp add: card_Diff_singleton_if split: if_split_asm) apply (case_tac "card A", auto) done lemma psubset_card_mono: "finite B ⟹ A < B ⟹ card A < card B" apply (simp add: psubset_eq linorder_not_le [symmetric]) apply (blast dest: card_seteq) done lemma card_Un_Int: assumes "finite A" "finite B" shows "card A + card B = card (A ∪ B) + card (A ∩ 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 ⟹ finite B ⟹ A ∩ B = {} ⟹ card (A ∪ B) = card A + card B" using card_Un_Int [of A B] by simp lemma card_Un_le: "card (A ∪ B) ≤ card A + card B" apply (cases "finite A") apply (cases "finite B") apply (use le_iff_add card_Un_Int in blast) apply simp apply simp done lemma card_Diff_subset: assumes "finite B" and "B ⊆ 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 ∩ B)" shows "card (A - B) = card A - card (A ∩ B)" proof - have "A - B = A - A ∩ 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 ≤ card (A - B)" proof - have "card A - card B ≤ card A - card (A ∩ B)" using card_mono[OF assms Int_lower2, of A] by arith also have "… = card (A - B)" using assms by (simp add: card_Diff_subset_Int) finally show ?thesis . qed lemma card_Diff1_less: "finite A ⟹ x ∈ A ⟹ card (A - {x}) < card A" by (rule Suc_less_SucD) (simp add: card_Suc_Diff1 del: card_Diff_insert) lemma card_Diff2_less: "finite A ⟹ x ∈ A ⟹ y ∈ A ⟹ card (A - {x} - {y}) < card A" apply (cases "x = y") apply (simp add: card_Diff1_less del:card_Diff_insert) apply (rule less_trans) prefer 2 apply (auto intro!: card_Diff1_less simp del: card_Diff_insert) done lemma card_Diff1_le: "finite A ⟹ card (A - {x}) ≤ card A" by (cases "x ∈ A") (simp_all add: card_Diff1_less less_imp_le) lemma card_psubset: "finite B ⟹ A ⊆ B ⟹ card A < card B ⟹ A < B" by (erule psubsetI) blast lemma card_le_inj: assumes fA: "finite A" and fB: "finite B" and c: "card A ≤ card B" shows "∃f. f ` A ⊆ B ∧ 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 ≤ card t" by simp from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where "f ` s ⊆ t" "inj_on f s" by blast with "2.prems"(2) "2.hyps"(2) show ?case apply - apply (rule exI[where x = "λz. if z = x then y else f z"]) apply (auto simp add: inj_on_def) done qed qed lemma card_subset_eq: assumes fB: "finite B" and AB: "A ⊆ 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 ∩ (B - A) = {}" by blast have eq: "A ∪ (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 ∉ F ⟹ ∀c1 ∈ insert x F. ∀c2 ∈ insert x F. c1 ≠ c2 ⟶ c1 ∩ c2 = {} ⟹ x ∩ ⋃F = {}" by auto (* somewhat slow *) lemma finite_psubset_induct [consumes 1, case_names psubset]: assumes finite: "finite A" and major: "⋀A. finite A ⟹ (⋀B. B ⊂ A ⟹ P B) ⟹ 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 ⟹ finite B ⟹ P B" for B by fact have "P B" if "B ⊂ A" for B proof - from that have "card B < card A" using psubset_card_mono fin by blast moreover from that have "B ⊆ 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: "⋀T. T ⊂ S ⟹ P T ⟹ ∃s∈S - T. P (insert s T)" shows "P S" proof - have "0 ≤ card S" by simp then have "∃T ⊆ S. card T = card S ∧ P T" proof (induct rule: dec_induct) case base with ‹P {}› show ?case by (intro exI[of _ "{}"]) auto next case (step n) then obtain T where T: "T ⊆ S" "card T = n" "P T" by auto with ‹n < card S› have "T ⊂ S" "P T" by auto with select[of T] obtain s where "s ∈ S" "s ∉ T" "P (insert s T)" by auto with step(2) T ‹finite S› show ?case by (intro exI[of _ "insert s T"]) (auto dest: finite_subset) qed with ‹finite S› 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: "¬ finite B ⟹ P B" and remove: "⋀A. finite A ⟹ A ≠ {} ⟹ A ⊆ B ⟹ (⋀x. x ∈ A ⟹ P (A - {x})) ⟹ 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 ⊆ 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 ‹A ⊆ B› and ‹finite B› have "finite A" by (rule finite_subset) moreover from Suc.hyps have "A ≠ {}" by auto moreover note ‹A ⊆ B› moreover have "P (A - {x})" if x: "x ∈ A" for x using x Suc.prems ‹Suc n = card A› 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 ⇒ bool" assumes "finite B" and "P {}" and "⋀A. finite A ⟹ A ≠ {} ⟹ A ⊆ B ⟹ (⋀x. x ∈ A ⟹ P (A - {x})) ⟹ P A" defines "B' ≡ B" shows "P B'" by (induct B' rule: remove_induct) (simp_all add: assms) text ‹Main cardinality theorem.› lemma card_partition [rule_format]: "finite C ⟹ finite (⋃C) ⟹ (∀c∈C. card c = k) ⟹ (∀c1 ∈ C. ∀c2 ∈ C. c1 ≠ c2 ⟶ c1 ∩ c2 = {}) ⟹ k * card C = card (⋃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 _ "⋃(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 ⊆ UNIV" by simp show "UNIV ⊆ A" proof show "x ∈ A" for x proof (rule ccontr) assume "x ∉ A" then have "A ⊂ 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 ‹The form of a finite set of given cardinality› lemma card_eq_SucD: assumes "card A = Suc k" shows "∃b B. A = insert b B ∧ b ∉ B ∧ card B = k ∧ (k = 0 ⟶ B = {})" proof - have fin: "finite A" using assms by (auto intro: ccontr) moreover have "card A ≠ 0" using assms by auto ultimately obtain b where b: "b ∈ A" by auto show ?thesis proof (intro exI conjI) show "A = insert b (A - {b})" using b by blast show "b ∉ A - {b}" by blast show "card (A - {b}) = k" and "k = 0 ⟶ A - {b} = {}" using assms b fin by (fastforce dest: mk_disjoint_insert)+ qed qed lemma card_Suc_eq: "card A = Suc k ⟷ (∃b B. A = insert b B ∧ b ∉ B ∧ card B = k ∧ (k = 0 ⟶ B = {}))" apply (auto elim!: card_eq_SucD) apply (subst card.insert) apply (auto simp add: intro:ccontr) done 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 ⟷ card A = 1" unfolding is_singleton_def by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def) lemma card_le_Suc_iff: "finite A ⟹ Suc n ≤ card A = (∃a B. A = insert a B ∧ a ∉ B ∧ n ≤ card B ∧ finite B)" by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff dest: subset_singletonD split: nat.splits if_splits) lemma finite_fun_UNIVD2: assumes fin: "finite (UNIV :: ('a ⇒ 'b) set)" shows "finite (UNIV :: 'b set)" proof - from fin have "finite (range (λf :: 'a ⇒ 'b. f arbitrary))" for arbitrary by (rule finite_imageI) moreover have "UNIV = range (λf :: 'a ⇒ '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 "¬ finite A" shows "∃B. finite B ∧ card B = n ∧ B ⊆ A" proof (induction n) case 0 show ?case by (intro exI[of _ "{}"]) auto next case (Suc n) then obtain B where B: "finite B ∧ card B = n ∧ B ⊆ A" .. with ‹¬ finite A› have "A ≠ B" by auto with B have "B ⊂ A" by auto then have "∃x. x ∈ A - B" by (elim psubset_imp_ex_mem) then obtain x where x: "x ∈ A - B" .. with B have "finite (insert x B) ∧ card (insert x B) = Suc n ∧ insert x B ⊆ A" by auto then show "∃B. finite B ∧ card B = Suc n ∧ B ⊆ A" .. qed text ‹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.› lemma finite_if_finite_subsets_card_bdd: assumes "⋀G. G ⊆ F ⟹ finite G ⟹ card G ≤ C" shows "finite F ∧ card F ≤ C" proof (cases "finite F") case False obtain n::nat where n: "n > max C 0" by auto obtain G where G: "G ⊆ F" "card G = n" using infinite_arbitrarily_large[OF False] by auto hence "finite G" using ‹n > max C 0› 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 ‹Cardinality of image› lemma card_image_le: "finite A ⟹ card (f ` A) ≤ card A" by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if) lemma card_image: "inj_on f A ⟹ card (f ` A) = card A" proof (induct A rule: infinite_finite_induct) case (infinite A) then have "¬ 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 ⟹ card A = card B" by (auto simp: card_image bij_betw_def) lemma endo_inj_surj: "finite A ⟹ f ` A ⊆ A ⟹ inj_on f A ⟹ 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 ⟹ inj_on f A ⟷ 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 ⊆ B" "finite B" shows "card A ≤ 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 surj_card_le: "finite A ⟹ B ⊆ f ` A ⟹ card B ≤ card A" by (blast intro: card_image_le card_mono le_trans) lemma card_bij_eq: "inj_on f A ⟹ f ` A ⊆ B ⟹ inj_on g B ⟹ g ` B ⊆ A ⟹ finite A ⟹ finite B ⟹ card A = card B" by (auto intro: le_antisym card_inj_on_le) lemma bij_betw_finite: "bij_betw f A B ⟹ finite A ⟷ finite B" unfolding bij_betw_def using finite_imageD [of f A] by auto lemma inj_on_finite: "inj_on f A ⟹ f ` A ≤ B ⟹ finite B ⟹ finite A" using finite_imageD finite_subset by blast lemma card_vimage_inj: "inj f ⟹ A ⊆ range f ⟹ 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 ‹Pigeonhole Principles› lemma pigeonhole: "card A > card (f ` A) ⟹ ¬ inj_on f A " by (auto dest: card_image less_irrefl_nat) lemma pigeonhole_infinite: assumes "¬ finite A" and "finite (f`A)" shows "∃a0∈A. ¬ finite {a∈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∈A. f a = b}") case True with ‹¬ finite A› have "¬ finite (A - {a∈A. f a = b})" by simp also have "A - {a∈A. f a = b} = {a∈A. f a ≠ b}" by blast finally have "¬ finite {a∈A. f a ≠ b}" . from insert(3)[OF _ this] insert(2,4) show ?thesis by simp (blast intro: rev_finite_subset) next case False then have "{a ∈ A. f a = b} ≠ {}" by force with False show ?thesis by blast qed qed lemma pigeonhole_infinite_rel: assumes "¬ finite A" and "finite B" and "∀a∈A. ∃b∈B. R a b" shows "∃b∈B. ¬ finite {a:A. R a b}" proof - let ?F = "λa. {b∈B. R a b}" from finite_Pow_iff[THEN iffD2, OF ‹finite B›] have "finite (?F ` A)" by (blast intro: rev_finite_subset) from pigeonhole_infinite [where f = ?F, OF assms(1) this] obtain a0 where "a0 ∈ A" and infinite: "¬ finite {a∈A. ?F a = ?F a0}" .. obtain b0 where "b0 ∈ B" and "R a0 b0" using ‹a0 ∈ A› assms(3) by blast have "finite {a∈A. ?F a = ?F a0}" if "finite {a∈A. R a b0}" using ‹b0 ∈ B› ‹R a0 b0› that by (blast intro: rev_finite_subset) with infinite ‹b0 ∈ B› show ?thesis by blast qed subsubsection ‹Cardinality of sums› lemma card_Plus: assumes "finite A" "finite B" shows "card (A <+> B) = card A + card B" proof - have "Inl`A ∩ 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 ∧ finite B then card A + card B else 0)" by (auto simp add: card_Plus) text ‹Relates to equivalence classes. Based on a theorem of F. KammÃ¼ller.› lemma dvd_partition: assumes f: "finite (⋃C)" and "∀c∈C. k dvd card c" "∀c1∈C. ∀c2∈C. c1 ≠ c2 ⟶ c1 ∩ c2 = {}" shows "k dvd card (⋃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 then show ?case apply simp apply (subst card_Un_disjoint) apply (auto simp add: disjoint_eq_subset_Compl) done qed qed subsubsection ‹Relating injectivity and surjectivity› lemma finite_surj_inj: assumes "finite A" "A ⊆ 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) ⟹ surj f ⟹ inj f" for f :: "'a ⇒ 'a" by (blast intro: finite_surj_inj subset_UNIV) lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) ⟹ inj f ⟹ surj f" for f :: "'a ⇒ 'a" by (fastforce simp:surj_def dest!: endo_inj_surj) corollary infinite_UNIV_nat [iff]: "¬ finite (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: "¬ finite (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 ⇒ 'a)" by (simp add: inj_on_def) ultimately have "finite (UNIV :: nat set)" by (rule finite_imageD) then show False by simp qed hide_const (open) Finite_Set.fold subsection ‹Infinite Sets› text ‹ Some elementary facts about infinite sets, mostly by Stephan Merz. Beware! Because "infinite" merely abbreviates a negation, these lemmas may not work well with ‹blast›. › abbreviation infinite :: "'a set ⇒ bool" where "infinite S ≡ ¬ finite S" text ‹ Infinite sets are non-empty, and if we remove some elements from an infinite set, the result is still infinite. › lemma infinite_imp_nonempty: "infinite S ⟹ S ≠ {}" by auto lemma infinite_remove: "infinite S ⟹ infinite (S - {a})" by simp lemma Diff_infinite_finite: assumes "finite T" "infinite S" shows "infinite (S - T)" using ‹finite T› proof induct from ‹infinite S› 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 ⟹ infinite (S ∪ T)" by simp lemma infinite_Un: "infinite (S ∪ T) ⟷ infinite S ∨ infinite T" by simp lemma infinite_super: assumes "S ⊆ T" and "infinite S" shows "infinite T" proof assume "finite T" with ‹S ⊆ T› have "finite S" by (simp add: finite_subset) with ‹infinite S› show False by simp qed proposition infinite_coinduct [consumes 1, case_names infinite]: assumes "X A" and step: "⋀A. X A ⟹ ∃x∈A. X (A - {x}) ∨ infinite (A - {x})" shows "infinite A" proof assume "finite A" then show False using ‹X A› proof (induction rule: finite_psubset_induct) case (psubset A) then obtain x where "x ∈ A" "X (A - {x}) ∨ infinite (A - {x})" using local.step psubset.prems by blast then have "X (A - {x})" using psubset.hyps by blast show False apply (rule psubset.IH [where B = "A - {x}"]) apply (use ‹x ∈ A› in blast) apply (simp add: ‹X (A - {x})›) done qed qed text ‹ 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. › lemma inf_img_fin_dom': assumes img: "finite (f ` A)" and dom: "infinite A" shows "∃y ∈ f ` A. infinite (f -` {y} ∩ A)" proof (rule ccontr) have "A ⊆ (⋃y∈f ` A. f -` {y} ∩ A)" by auto moreover assume "¬ ?thesis" with img have "finite (⋃y∈f ` A. f -` {y} ∩ 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 ∈ f`A" and "infinite (f -` {y} ∩ 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 "∃y ∈ 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 ∈ f`A" and "infinite (f -` {y})" using assms by (blast dest: inf_img_fin_dom) proposition finite_image_absD: "finite (abs ` S) ⟹ finite S" for S :: "'a::linordered_ring set" by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom) end