(* 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 Relation 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.proc› 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) subsection ‹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) simproc_setup finite ("finite A") = ‹ let val finite_subset = @{thm finite_subset} val Eq_TrueI = @{thm Eq_TrueI} fun is_subset A th = case Thm.prop_of th of (_ $ (Const (\<^const_name>‹less_eq›, Type (\<^type_name>‹fun›, [Type (\<^type_name>‹set›, _), _])) $ A' $ B)) => if A aconv A' then SOME(B,th) else NONE | _ => NONE; fun is_finite th = case Thm.prop_of th of (_ $ (Const (\<^const_name>‹finite›, _) $ A)) => SOME(A,th) | _ => NONE; fun comb (A,sub_th) (A',fin_th) ths = if A aconv A' then (sub_th,fin_th) :: ths else ths fun proc ctxt ct = (let val _ $ A = Thm.term_of ct val prems = Simplifier.prems_of ctxt val fins = map_filter is_finite prems val subsets = map_filter (is_subset A) prems in case fold_product comb subsets fins [] of (sub_th,fin_th) :: _ => SOME((fin_th RS (sub_th RS finite_subset)) RS Eq_TrueI) | _ => NONE end) in K proc end › (* Needs to be used with care *) declare [[simproc del: finite]] 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 (⋃(B ` A)) ⟷ (∀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) 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 qed lemma all_subset_image: "(∀B. B ⊆ f ` A ⟶ P B) ⟷ (∀B. B ⊆ A ⟶ P(f ` B))" by (safe elim!: subset_imageE) (use image_mono in ‹blast+›) (* slow *) lemma all_finite_subset_image: "(∀B. finite B ∧ B ⊆ f ` A ⟶ P B) ⟷ (∀B. finite B ∧ B ⊆ A ⟶ P (f ` B))" proof safe fix B :: "'a set" assume B: "finite B" "B ⊆ f ` A" and P: "∀B. finite B ∧ B ⊆ A ⟶ P (f ` B)" show "P B" using finite_subset_image [OF B] P by blast qed blast lemma ex_finite_subset_image: "(∃B. finite B ∧ B ⊆ f ` A ∧ P B) ⟷ (∃B. finite B ∧ B ⊆ A ∧ P (f ` B))" proof safe fix B :: "'a set" assume B: "finite B" "B ⊆ f ` A" and "P B" show "∃B. finite B ∧ B ⊆ A ∧ P (f ` B)" using finite_subset_image [OF B] ‹P B› by blast qed blast lemma finite_vimage_IntI: "finite F ⟹ inj_on h A ⟹ finite (h -` F ∩ 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 "⋀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_inverse_image_gen: assumes "finite A" "inj_on f D" shows "finite {j∈D. f j ∈ A}" using finite_vimage_IntI [OF assms] by (simp add: Collect_conj_eq inf_commute vimage_def) lemma finite_inverse_image: assumes "finite A" "inj f" shows "finite {j. f j ∈ A}" using finite_inverse_image_gen [OF assms] by simp 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 lemma finite_converse [iff]: "finite (r¯) ⟷ finite r" unfolding converse_def conversep_iff using [[simproc add: finite_Collect]] by (auto elim: finite_imageD simp: inj_on_def) lemma finite_Domain: "finite r ⟹ finite (Domain r)" by (induct set: finite) auto lemma finite_Range: "finite r ⟹ finite (Range r)" by (induct set: finite) auto lemma finite_Field: "finite r ⟹ finite (Field r)" by (simp add: Field_def finite_Domain finite_Range) lemma finite_Image[simp]: "finite R ⟹ finite (R `` A)" by(rule finite_subset[OF _ finite_Range]) auto subsection ‹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''. The commutativity requirement is relativised to the carrier set ‹S›: › locale comp_fun_commute_on = fixes S :: "'a set" fixes f :: "'a ⇒ 'b ⇒ 'b" assumes comp_fun_commute_on: "x ∈ S ⟹ y ∈ S ⟹ f y ∘ f x = f x ∘ f y" begin lemma fun_left_comm: "x ∈ S ⟹ y ∈ S ⟹ f y (f x z) = f x (f y z)" using comp_fun_commute_on by (simp add: fun_eq_iff) lemma commute_left_comp: "x ∈ S ⟹ y ∈ S ⟹ f y ∘ (f x ∘ g) = f x ∘ (f y ∘ g)" by (simp add: o_assoc comp_fun_commute_on) 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 definition 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_on 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: assumes "A ⊆ S" assumes "fold_graph f z A y" "a ∈ A" shows "∃y'. y = f a y' ∧ fold_graph f z (A - {a}) y'" using assms(2-,1) 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 from insertI have "x ∈ S" "a ∈ S" by auto then have "f x y = f a (f x y')" unfolding y by (intro fun_left_comm; simp) 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 "insert x A ⊆ S" 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 ‹insert x A ⊆ S› _ insertI1]) lemma fold_graph_determ: assumes "A ⊆ S" assumes "fold_graph f z A x" "fold_graph f z A y" shows "y = x" using assms(2-,1) proof (induct arbitrary: y set: fold_graph) case emptyI then show ?case by fast next case (insertI x A y v) from ‹insert x A ⊆ S› and ‹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'› insertI have "y' = y" by simp with ‹v = f x y'› show "v = f x y" by simp qed lemma fold_equality: "A ⊆ S ⟹ 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 "A ⊆ S" assumes "finite A" shows "fold_graph f z A (fold f z A)" proof - from ‹finite A› have "∃x. fold_graph f z A x" by (rule finite_imp_fold_graph) moreover note fold_graph_determ[OF ‹A ⊆ S›] 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 "insert x A ⊆ S" assumes "finite A" and "x ∉ A" shows "fold f z (insert x A) = f x (fold f z A)" proof (rule fold_equality[OF ‹insert x A ⊆ S›]) fix z from ‹insert x A ⊆ S› ‹finite A› have "fold_graph f z A (fold f z A)" by (blast intro: 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: assumes "insert x A ⊆ S" "finite A" shows "f x (fold f z A) = fold f (f x z) A" using assms(2,1) proof (induct rule: finite_induct) case empty then show ?case by simp next case (insert y F) then have "fold f (f x z) (insert y F) = f y (fold f (f x z) F)" by simp also have "… = f x (f y (fold f z F))" using insert by (simp add: fun_left_comm[where ?y=x]) also have "… = f x (fold f z (insert y F))" proof - from insert have "insert y F ⊆ S" by simp from fold_insert[OF this] insert show ?thesis by simp qed finally show ?case .. qed lemma fold_insert2: "insert x A ⊆ S ⟹ 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 "A ⊆ S" 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) (use assms in ‹auto›) finally show ?thesis . qed lemma fold_insert_remove: assumes "insert x A ⊆ S" 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}))" using ‹insert x A ⊆ S› by (blast intro: fold_rec) then show ?thesis by simp qed lemma fold_set_union_disj: assumes "A ⊆ S" "B ⊆ S" 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 ‹finite B› assms(1,2,3,5) proof induct case (insert x F) have "fold f z (A ∪ insert x F) = f x (fold f (fold f z A) F)" using insert by auto also have "… = fold f (fold f z A) (insert x F)" using insert by (blast intro: fold_insert[symmetric]) finally show ?case . qed simp end text ‹Other properties of \<^const>‹fold›:› lemma finite_set_fold_single [simp]: "Finite_Set.fold f z {x} = f x z" proof - have "fold_graph f z {x} (f x z)" by (auto intro: fold_graph.intros) moreover { fix X y have "fold_graph f z X y ⟹ (X = {} ⟶ y = z) ∧ (X = {x} ⟶ y = f x z)" by (induct rule: fold_graph.induct) auto } ultimately have "(THE y. fold_graph f z {x} y) = f x z" by blast thus ?thesis by (simp add: Finite_Set.fold_def) qed lemma fold_graph_image: assumes "inj_on g A" shows "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 o g) z A w" proof assume "fold_graph f z (g ` A) w" then show "fold_graph (f ∘ g) z A w" 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 "fold_graph (f ∘ g) z A w" then show "fold_graph f z (g ` A) w" 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 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 then show ?thesis by (auto simp add: fold_def fold_graph_image[OF assms]) qed lemma fold_cong: assumes "comp_fun_commute_on S f" "comp_fun_commute_on S g" and "A ⊆ S" "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› ‹A ⊆ S› cong proof (induct A) case empty then show ?case by simp next case insert interpret f: comp_fun_commute_on S f by (fact ‹comp_fun_commute_on S f›) interpret g: comp_fun_commute_on S g by (fact ‹comp_fun_commute_on S 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_on = comp_fun_commute_on + assumes comp_fun_idem_on: "x ∈ S ⟹ f x ∘ f x = f x" begin lemma fun_left_idem: "x ∈ S ⟹ f x (f x z) = f x z" using comp_fun_idem_on by (simp add: fun_eq_iff) lemma fold_insert_idem: assumes "insert x A ⊆ S" 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_on fun_left_idem) next assume "x ∉ A" then show ?thesis using assms by auto qed declare fold_insert [simp del] fold_insert_idem [simp] lemma fold_insert_idem2: "insert x A ⊆ S ⟹ 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_on› etc.› lemma (in comp_fun_commute_on) comp_comp_fun_commute_on: "range g ⊆ S ⟹ comp_fun_commute_on R (f ∘ g)" by standard (force intro: comp_fun_commute_on) lemma (in comp_fun_idem_on) comp_comp_fun_idem_on: assumes "range g ⊆ S" shows "comp_fun_idem_on R (f ∘ g)" proof interpret f_g: comp_fun_commute_on R "f o g" by (fact comp_comp_fun_commute_on[OF ‹range g ⊆ S›]) show "x ∈ R ⟹ y ∈ R ⟹ (f ∘ g) y ∘ (f ∘ g) x = (f ∘ g) x ∘ (f ∘ g) y" for x y by (fact f_g.comp_fun_commute_on) qed (use ‹range g ⊆ S› in ‹force intro: comp_fun_idem_on›) lemma (in comp_fun_commute_on) comp_fun_commute_on_funpow: "comp_fun_commute_on S (λx. f x ^^ g x)" proof fix x y assume "x ∈ S" "y ∈ S" show "f y ^^ g y ∘ f x ^^ g x = f x ^^ g x ∘ f y ^^ g 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 with ‹x ∈ S› ‹y ∈ S› show ?case by (simp add: comp_assoc hyp) (simp add: o_assoc comp_fun_commute_on) 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 ‹\<^term>‹UNIV› as carrier set› locale comp_fun_commute = fixes f :: "'a ⇒ 'b ⇒ 'b" assumes comp_fun_commute: "f y ∘ f x = f x ∘ f y" begin lemma (in -) comp_fun_commute_def': "comp_fun_commute f = comp_fun_commute_on UNIV f" unfolding comp_fun_commute_def comp_fun_commute_on_def by blast text ‹ We abuse the ‹rewrites› functionality of locales to remove trivial assumptions that result from instantiating the carrier set to \<^term>‹UNIV›. › sublocale comp_fun_commute_on UNIV f rewrites "⋀X. (X ⊆ UNIV) ≡ True" and "⋀x. x ∈ UNIV ≡ True" and "⋀P. (True ⟹ P) ≡ Trueprop P" and "⋀P Q. (True ⟹ PROP P ⟹ PROP Q) ≡ (PROP P ⟹ True ⟹ PROP Q)" proof - show "comp_fun_commute_on UNIV f" by standard (simp add: comp_fun_commute) qed simp_all end lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f o g)" unfolding comp_fun_commute_def' by (fact comp_comp_fun_commute_on) lemma (in comp_fun_commute) comp_fun_commute_funpow: "comp_fun_commute (λx. f x ^^ g x)" unfolding comp_fun_commute_def' by (fact comp_fun_commute_on_funpow) locale comp_fun_idem = comp_fun_commute + assumes comp_fun_idem: "f x o f x = f x" begin lemma (in -) comp_fun_idem_def': "comp_fun_idem f = comp_fun_idem_on UNIV f" unfolding comp_fun_idem_on_def comp_fun_idem_def comp_fun_commute_def' unfolding comp_fun_idem_axioms_def comp_fun_idem_on_axioms_def by blast text ‹ Again, we abuse the ‹rewrites› functionality of locales to remove trivial assumptions that result from instantiating the carrier set to \<^term>‹UNIV›. › sublocale comp_fun_idem_on UNIV f rewrites "⋀X. (X ⊆ UNIV) ≡ True" and "⋀x. x ∈ UNIV ≡ True" and "⋀P. (True ⟹ P) ≡ Trueprop P" and "⋀P Q. (True ⟹ PROP P ⟹ PROP Q) ≡ (PROP P ⟹ True ⟹ PROP Q)" proof - show "comp_fun_idem_on UNIV f" by standard (simp_all add: comp_fun_idem comp_fun_commute) qed simp_all end lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f o g)" unfolding comp_fun_idem_def' by (fact comp_comp_fun_idem_on) subsubsection ‹Expressing set operations via \<^const>‹fold›› lemma comp_fun_commute_const: "comp_fun_commute (λ_. f)" by standard (rule refl) 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 proof - interpret commute_insert: comp_fun_commute "(λx A'. if P x then Set.insert x A' else A')" by (fact comp_fun_commute_filter_fold) from ‹finite A› show ?thesis by induct (auto simp add: Set.filter_def) qed 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 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" proof - interpret commute_product: comp_fun_commute "(λx z. fold (λy. Set.insert (x, y)) z B)" by (fact comp_fun_commute_product_fold[OF ‹finite B›]) from assms show ?thesis unfolding Sigma_def by (induct A) (simp_all add: fold_union_pair) qed 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 (⨅(f ` A)) = 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 (⨆(f ` A)) = 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 ⟹ ⨅(f ` A) = fold (inf ∘ f) top A" using inf_INF_fold_inf [of A top] by simp lemma SUP_fold_sup: "finite A ⟹ ⨆(f ` A) = fold (sup ∘ f) bot A" using sup_SUP_fold_sup [of A bot] by simp lemma finite_Inf_in: assumes "finite A" "A≠{}" and inf: "⋀x y. ⟦x ∈ A; y ∈ A⟧ ⟹ inf x y ∈ A" shows "Inf A ∈ A" proof - have "Inf B ∈ A" if "B ≤ A" "B≠{}" for B using finite_subset [OF ‹B ⊆ A› ‹finite A›] that by (induction B) (use inf in ‹force+›) then show ?thesis by (simp add: assms) qed lemma finite_Sup_in: assumes "finite A" "A≠{}" and sup: "⋀x y. ⟦x ∈ A; y ∈ A⟧ ⟹ sup x y ∈ A" shows "Sup A ∈ A" proof - have "Sup B ∈ A" if "B ≤ A" "B≠{}" for B using finite_subset [OF ‹B ⊆ A› ‹finite A›] that by (induction B) (use sup in ‹force+›) then show ?thesis by (simp add: assms) qed end subsubsection ‹Expressing relation operations via \<^const>‹fold›› lemma Id_on_fold: assumes "finite A" shows "Id_on A = Finite_Set.fold (λx. Set.insert (Pair x x)) {} A" proof - interpret comp_fun_commute "λx. Set.insert (Pair x x)" by standard auto from assms show ?thesis unfolding Id_on_def by (induct A) simp_all qed lemma comp_fun_commute_Image_fold: "comp_fun_commute (λ(x,y) A. if x ∈ S then Set.insert y 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 comp_fun_commute split: prod.split) qed lemma Image_fold: assumes "finite R" shows "R `` S = Finite_Set.fold (λ(x,y) A. if x ∈ S then Set.insert y A else A) {} R" proof - interpret comp_fun_commute "(λ(x,y) A. if x ∈ S then Set.insert y A else A)" by (rule comp_fun_commute_Image_fold) have *: "⋀x F. Set.insert x F `` S = (if fst x ∈ S then Set.insert (snd x) (F `` S) else (F `` S))" by (force intro: rev_ImageI) show ?thesis using assms by (induct R) (auto simp: * ) qed lemma insert_relcomp_union_fold: assumes "finite S" shows "{x} O S ∪ X = Finite_Set.fold (λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S" proof - interpret comp_fun_commute "λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'" proof - interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) show "comp_fun_commute (λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')" by standard (auto simp add: fun_eq_iff split: prod.split) qed have *: "{x} O S = {(x', z). x' = fst x ∧ (snd x, z) ∈ S}" by (auto simp: relcomp_unfold intro!: exI) show ?thesis unfolding * using ‹finite S› by (induct S) (auto split: prod.split) qed lemma insert_relcomp_fold: assumes "finite S" shows "Set.insert x R O S = Finite_Set.fold (λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S" proof - have "Set.insert x R O S = ({x} O S) ∪ (R O S)" by auto then show ?thesis by (auto simp: insert_relcomp_union_fold [OF assms]) qed lemma comp_fun_commute_relcomp_fold: assumes "finite S" shows "comp_fun_commute (λ(x,y) A. Finite_Set.fold (λ(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)" proof - have *: "⋀a b A. Finite_Set.fold (λ(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S ∪ A" by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong) show ?thesis by standard (auto simp: * ) qed lemma relcomp_fold: assumes "finite R" "finite S" shows "R O S = Finite_Set.fold (λ(x,y) A. Finite_Set.fold (λ(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R" proof - interpret commute_relcomp_fold: comp_fun_commute "(λ(x, y) A. Finite_Set.fold (λ(w, z) A'. if y = w then insert (x, z) A' else A') A S)" by (fact comp_fun_commute_relcomp_fold[OF ‹finite S›]) from assms show ?thesis by (induct R) (auto simp: comp_fun_commute_relcomp_fold insert_relcomp_fold cong: if_cong) qed subsection ‹Locales as mini-packages for fold operations› subsubsection ‹The natural case› locale folding_on = fixes S :: "'a set" fixes f :: "'a ⇒ 'b ⇒ 'b" and z :: "'b" assumes comp_fun_commute_on: "x ∈ S ⟹ y ∈ S ⟹ f y o f x = f x o f y" begin interpretation fold?: comp_fun_commute_on S f by standard (simp add: comp_fun_commute_on) definition F :: "'a set ⇒ 'b" where eq_fold: "F A = Finite_Set.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 "insert x A ⊆ S" and "finite A" and "x ∉ A" shows "F (insert x A) = f x (F A)" proof - from fold_insert assms have "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)" by simp with ‹finite A› show ?thesis by (simp add: eq_fold fun_eq_iff) qed lemma remove: assumes "A ⊆ S" and "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 using ‹A ⊆ S› by auto qed lemma insert_remove: assumes "insert x A ⊆ S" and "finite A" shows "F (insert x A) = f x (F (A - {x}))" using assms by (cases "x ∈ A") (simp_all add: remove insert_absorb) end subsubsection ‹With idempotency› locale folding_idem_on = folding_on + assumes comp_fun_idem_on: "x ∈ S ⟹ y ∈ S ⟹ f x ∘ f x = f x" begin declare insert [simp del] interpretation fold?: comp_fun_idem_on S f by standard (simp_all add: comp_fun_commute_on comp_fun_idem_on) lemma insert_idem [simp]: assumes "insert x A ⊆ S" and "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 subsubsection ‹\<^term>‹UNIV› as the carrier set› locale folding = fixes f :: "'a ⇒ 'b ⇒ 'b" and z :: "'b" assumes comp_fun_commute: "f y ∘ f x = f x ∘ f y" begin lemma (in -) folding_def': "folding f = folding_on UNIV f" unfolding folding_def folding_on_def by blast text ‹ Again, we abuse the ‹rewrites› functionality of locales to remove trivial assumptions that result from instantiating the carrier set to \<^term>‹UNIV›. › sublocale folding_on UNIV f rewrites "⋀X. (X ⊆ UNIV) ≡ True" and "⋀x. x ∈ UNIV ≡ True" and "⋀P. (True ⟹ P) ≡ Trueprop P" and "⋀P Q. (True ⟹ PROP P ⟹ PROP Q) ≡ (PROP P ⟹ True ⟹ PROP Q)" proof - show "folding_on UNIV f" by standard (simp add: comp_fun_commute) qed simp_all end locale folding_idem = folding + assumes comp_fun_idem: "f x ∘ f x = f x" begin lemma (in -) folding_idem_def': "folding_idem f = folding_idem_on UNIV f" unfolding folding_idem_def folding_def' folding_idem_on_def unfolding folding_idem_axioms_def folding_idem_on_axioms_def by blast text ‹ Again, we abuse the ‹rewrites› functionality of locales to remove trivial assumptions that result from instantiating the carrier set to \<^term>‹UNIV›. › sublocale folding_idem_on UNIV f rewrites "⋀X. (X ⊆ UNIV) ≡ True" and "⋀x. x ∈ UNIV ≡ True" and "⋀P. (True ⟹ P) ≡ Trueprop P" and "⋀P Q. (True ⟹ PROP P ⟹ PROP Q) ≡ (PROP P ⟹ True ⟹ PROP Q)" proof - show "folding_idem_on UNIV f" by standard (simp add: comp_fun_idem) qed simp_all 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_on.F (λ_. Suc) 0" by standard (rule refl) 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: assumes "finite A" "x ∈ 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 ≤ n - 1" shows "card (insert x y) ≤ n" using assms by (cases "finite y") (auto simp: card_insert_if) lemma card_Diff_singleton: assumes "x ∈ A" shows "card (A - {x}) = card A - 1" proof (cases "finite A") case True with assms show ?thesis by (simp add: card_Suc_Diff1 [symmetric]) qed auto lemma card_Diff_singleton_if: "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 "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_le: "card A ≤ card (insert x A)" proof (cases "finite A") case True then show ?thesis by (simp add: card_insert_if) qed auto 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: assumes "finite B" and A: "A ⊆ B" "card B ≤ 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} ⊆ B" by force+ then have "card B ≤ 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 ⟹ A < B ⟹ 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 ∪ 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_disjnt: "⟦finite A; finite B; disjnt A B⟧ ⟹ card (A ∪ B) = card A + card B" by (simp add: card_Un_disjoint disjnt_def) lemma card_Un_le: "card (A ∪ B) ≤ card A + card B" proof (cases "finite A ∧ 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 ⊆ 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