(* Title: HOL/Finite_Set.thy Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel with contributions by Jeremy Avigad *) header {* Finite sets *} theory Finite_Set imports Power Option begin subsection {* Predicate for finite sets *} inductive finite :: "'a set => bool" where emptyI [simp, intro!]: "finite {}" | insertI [simp, intro!]: "finite A ==> finite (insert a A)" 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 thus ?thesis by blast qed lemma finite_induct [case_names empty insert, induct set: finite]: "finite F ==> P {} ==> (!!x F. finite F ==> x \ F ==> P F ==> P (insert x F)) ==> P F" -- {* Discharging @{text "x \ F"} entails extra work. *} proof - assume "P {}" and insert: "!!x F. finite F ==> x \ F ==> P F ==> P (insert x F)" assume "finite F" thus "P F" proof induct show "P {}" by fact fix x F assume F: "finite F" and P: "P F" show "P (insert x F)" proof cases assume "x \ F" hence "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 qed lemma finite_ne_induct[case_names singleton insert, consumes 2]: assumes fin: "finite F" shows "F \ {} \ \ \x. P{x}; \x F. \ finite F; F \ {}; x \ F; P F \ \ P (insert x F) \ \ P F" using fin proof induct case empty thus ?case by simp next case (insert x F) show ?case proof cases assume "F = {}" thus ?thesis using `P {x}` by simp next assume "F \ {}" thus ?thesis using insert by blast qed 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" proof - from `finite F` and `F \ A` show ?thesis 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 qed text{* A finite choice principle. Does not need the SOME choice operator. *} lemma finite_set_choice: "finite A \ ALL x:A. (EX y. P x y) \ EX f. ALL x:A. P x (f x)" proof (induct set: finite) case empty thus ?case by simp next case (insert a A) then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto show ?case (is "EX f. ?P f") proof show "?P(%x. if x = a then b else f x)" using f ab by auto qed qed text{* Finite sets are the images of initial segments of natural numbers: *} lemma finite_imp_nat_seg_image_inj_on: assumes fin: "finite A" shows "\ (n::nat) f. A = f ` {i. if. {} = 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 hence "insert a A = f(n:=a) ` {i. i < Suc n}" "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) thus ?case by blast qed lemma nat_seg_image_imp_finite: "!!f A. A = f ` {i::nat. i finite A" proof (induct n) case 0 thus ?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 assume "\k(\ k (n::nat) f. A = f ` {i::nat. i finite G ==> finite (F Un G)" by (induct set: finite) simp_all lemma finite_subset: "A \ B ==> finite B ==> finite A" -- {* Every subset of a finite set is finite. *} proof - assume "finite B" thus "!!A. A \ B ==> finite A" proof induct case empty thus ?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})" . hence "finite (insert x (A - {x}))" .. also have "insert x (A - {x}) = A" using x by (rule insert_Diff) finally show ?thesis . next show "A \ F ==> ?thesis" by fact assume "x \ A" with A show "A \ F" by (simp add: subset_insert_iff) qed qed qed lemma rev_finite_subset: "finite B ==> A \ B ==> finite A" by (rule finite_subset) lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) 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_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" -- {* The converse obviously fails. *} 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}" -- {* The converse obviously fails. *} by(simp add:Collect_conj_eq) lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}" by(simp add: le_eq_less_or_eq) lemma finite_insert [simp]: "finite (insert a A) = finite A" apply (subst insert_is_Un) apply (simp only: finite_Un, blast) done lemma finite_Union[simp, intro]: "\ finite A; !!M. M \ A \ finite M \ \ finite(\A)" by (induct rule:finite_induct) simp_all lemma finite_Inter[intro]: "EX A:M. finite(A) \ finite(Inter M)" by (blast intro: Inter_lower finite_subset) lemma finite_INT[intro]: "EX x:I. finite(A x) \ finite(INT x:I. A x)" by (blast intro: INT_lower finite_subset) lemma finite_empty_induct: assumes "finite A" and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" shows "P {}" proof - have "P (A - A)" proof - { fix c b :: "'a set" assume c: "finite c" and b: "finite b" and P1: "P b" and P2: "!!x y. finite y ==> x \ y ==> P y ==> P (y - {x})" have "c \ b ==> P (b - c)" using c proof induct case empty from P1 show ?case by simp next case (insert x F) have "P (b - F - {x})" proof (rule P2) from _ b show "finite (b - F)" by (rule finite_subset) blast from insert show "x \ b - F" by simp from insert show "P (b - F)" by simp qed also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) finally show ?case . qed } then show ?thesis by this (simp_all add: assms) qed then show ?thesis by simp qed lemma finite_Diff [simp]: "finite A ==> finite (A - B)" by (rule Diff_subset [THEN finite_subset]) lemma finite_Diff2 [simp]: assumes "finite B" shows "finite (A - B) = finite A" proof - have "finite A \ finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int) also have "\ \ finite(A-B)" using `finite B` by(simp) finally show ?thesis .. 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_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" apply (subst Diff_insert) apply (case_tac "a : A - B") apply (rule finite_insert [symmetric, THEN trans]) apply (subst insert_Diff, simp_all) done text {* Image and Inverse Image over Finite Sets *} lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" -- {* The image of a finite set is finite. *} by (induct set: finite) 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_surj: "finite A ==> B <= f ` A ==> finite B" apply (frule finite_imageI) apply (erule finite_subset, assumption) done lemma finite_range_imageI: "finite (range g) ==> finite (range (%x. f (g x)))" apply (drule finite_imageI, simp add: range_composition) done lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" proof - have aux: "!!A. finite (A - {}) = finite A" by simp fix B :: "'a set" assume "finite B" thus "!!A. f`A = B ==> inj_on f A ==> finite A" apply induct apply simp apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") apply clarify apply (simp (no_asm_use) add: inj_on_def) apply (blast dest!: aux [THEN iffD1], atomize) apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) apply (frule subsetD [OF equalityD2 insertI1], clarify) apply (rule_tac x = xa in bexI) apply (simp_all add: inj_on_image_set_diff) done qed (rule refl) lemma inj_vimage_singleton: "inj f ==> f-`{a} \ {THE x. f x = a}" -- {* The inverse image of a singleton under an injective function is included in a singleton. *} apply (auto simp add: inj_on_def) apply (blast intro: the_equality [symmetric]) done lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)" -- {* The inverse image of a finite set under an injective function is finite. *} apply (induct set: finite) apply simp_all apply (subst vimage_insert) apply (simp add: finite_subset [OF inj_vimage_singleton]) done lemma finite_vimageD: assumes fin: "finite (h -` F)" and surj: "surj h" shows "finite F" proof - have "finite (h ` (h -` F))" using fin by (rule finite_imageI) also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq) finally show "finite F" . qed lemma finite_vimage_iff: "bij h \ finite (h -` F) \ finite F" unfolding bij_def by (auto elim: finite_vimageD finite_vimageI) text {* The finite UNION of finite sets *} lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" by (induct set: finite) simp_all text {* Strengthen RHS to @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \ {}})"}? We'd need to prove @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \ {}}"} by induction. *} lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" by (blast intro: finite_UN_I finite_subset) lemma finite_Collect_bex[simp]: "finite A \ finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})" apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})") apply auto done lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \ finite{x. EX y. P y & Q x y} = (ALL y. P y \ finite{x. Q x y})" apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})") apply auto done 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 hence "finite (Inl ` A :: ('a + 'b) set)" using fin by(rule finite_subset) thus "finite A" by(rule finite_imageD)(auto intro: inj_onI) next have "Inr ` B \ A <+> B" by auto hence "finite (Inr ` B :: ('a + 'b) set)" using fin by(rule finite_subset) thus "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) text {* Sigma of finite sets *} lemma finite_SigmaI [simp]: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" by (unfold Sigma_def) (blast intro!: finite_UN_I) 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)" apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") apply (erule ssubst) apply (erule finite_SigmaI, auto) done lemma finite_cartesian_productD1: "[| finite (A <*> B); B \ {} |] ==> finite A" apply (auto simp add: finite_conv_nat_seg_image) apply (drule_tac x=n in spec) apply (drule_tac x="fst o f" in spec) apply (auto simp add: o_def) prefer 2 apply (force dest!: equalityD2) apply (drule equalityD1) apply (rename_tac y x) apply (subgoal_tac "\k. k B); A \ {} |] ==> finite B" apply (auto simp add: finite_conv_nat_seg_image) apply (drule_tac x=n in spec) apply (drule_tac x="snd o f" in spec) apply (auto simp add: o_def) prefer 2 apply (force dest!: equalityD2) apply (drule equalityD1) apply (rename_tac x y) apply (subgoal_tac "\k. k finite{B. B \ A}" by(simp add: Pow_def[symmetric]) lemma finite_UnionD: "finite(\A) \ finite A" by(blast intro: finite_subset[OF subset_Pow_Union]) lemma finite_subset_image: assumes "finite B" shows "B \ f ` A \ \C\A. finite C \ B = f ` C" using assms proof(induct) case empty thus ?case by simp next case insert thus ?case by (clarsimp simp del: image_insert simp add: image_insert[symmetric]) blast qed subsection {* Class @{text 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)+ end lemma UNIV_unit [no_atp]: "UNIV = {()}" by auto instance unit :: finite proof qed (simp add: UNIV_unit) lemma UNIV_bool [no_atp]: "UNIV = {False, True}" by auto instance bool :: finite proof qed (simp add: UNIV_bool) instance * :: (finite, finite) finite proof qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite) lemma finite_option_UNIV [simp]: "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)" by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some) instance option :: (finite) finite proof qed (simp add: UNIV_option_conv) lemma inj_graph: "inj (%f. {(x, y). y = f x})" by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq) instance "fun" :: (finite, finite) finite proof show "finite (UNIV :: ('a => 'b) set)" proof (rule finite_imageD) let ?graph = "%f::'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 "+" :: (finite, finite) finite proof qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite) subsection {* A basic fold functional for finite sets *} text {* The intended behaviour is @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\ (f x\<^isub>n z)\)"} if @{text f} is ``left-commutative'': *} locale fun_left_comm = fixes f :: "'a \ 'b \ 'b" assumes fun_left_comm: "f x (f y z) = f y (f x z)" begin text{* On a functional level it looks much nicer: *} lemma fun_comp_comm: "f x \ f y = f y \ f x" by (simp add: fun_left_comm expand_fun_eq) 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" definition fold :: "('a \ 'b \ 'b) \ 'b \ 'a set \ 'b" where [code del]: "fold f z A = (THE y. fold_graph f z A y)" 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 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}. The proofs become ugly. It is not worth the effort. (???) *} lemma Diff1_fold_graph: "fold_graph f z (A - {x}) y \ x \ A \ fold_graph f z A (f x y)" by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto) lemma fold_graph_imp_finite: "fold_graph f z A x \ finite A" by (induct set: fold_graph) auto lemma finite_imp_fold_graph: "finite A \ \x. fold_graph f z A x" by (induct set: finite) auto subsubsection{*From @{const fold_graph} to @{term fold}*} context fun_left_comm begin 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 (insertI x A y) show ?case proof (cases "x = a") assume "x = a" with insertI show ?case by auto next assume "x \ a" then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'" using insertI by auto have 1: "f x y = f a (f x y')" unfolding y by (rule fun_left_comm) have 2: "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) from 1 2 show ?case by fast qed qed simp 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 (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 fast lemma fold_equality: "fold_graph f z A y \ fold f z A = y" by (unfold fold_def) (blast intro: fold_graph_determ) lemma fold_graph_fold: "finite A \ fold_graph f z A (fold f z A)" unfolding fold_def apply (rule theI') apply (rule ex_ex1I) apply (erule finite_imp_fold_graph) apply (erule (1) fold_graph_determ) done text{* The base case for @{text fold}: *} lemma (in -) fold_empty [simp]: "fold f z {} = z" by (unfold fold_def) blast text{* The various recursion equations for @{const fold}: *} lemma fold_insert [simp]: "finite A ==> x \ A ==> fold f z (insert x A) = f x (fold f z A)" apply (rule fold_equality) apply (erule fold_graph.insertI) apply (erule fold_graph_fold) done lemma fold_fun_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 y A) 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_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 end text{* A simplified version for idempotent functions: *} locale fun_left_comm_idem = fun_left_comm + assumes fun_left_idem: "f x (f x z) = f x z" begin text{* The nice version: *} lemma fun_comp_idem : "f x o f x = f x" by (simp add: fun_left_idem expand_fun_eq) 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: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_comm) end subsubsection {* Expressing set operations via @{const fold} *} lemma (in fun_left_comm) fun_left_comm_apply: "fun_left_comm (\x. f (g x))" proof qed (simp_all add: fun_left_comm) lemma (in fun_left_comm_idem) fun_left_comm_idem_apply: "fun_left_comm_idem (\x. f (g x))" by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales) (simp_all add: fun_left_idem) lemma fun_left_comm_idem_insert: "fun_left_comm_idem insert" proof qed auto lemma fun_left_comm_idem_remove: "fun_left_comm_idem (\x A. A - {x})" proof qed auto lemma (in semilattice_inf) fun_left_comm_idem_inf: "fun_left_comm_idem inf" proof qed (auto simp add: inf_left_commute) lemma (in semilattice_sup) fun_left_comm_idem_sup: "fun_left_comm_idem sup" proof qed (auto simp add: sup_left_commute) lemma union_fold_insert: assumes "finite A" shows "A \ B = fold insert B A" proof - interpret fun_left_comm_idem insert by (fact fun_left_comm_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 (\x A. A - {x}) B A" proof - interpret fun_left_comm_idem "\x A. A - {x}" by (fact fun_left_comm_idem_remove) from `finite A` show ?thesis by (induct A arbitrary: B) auto qed context complete_lattice begin lemma inf_Inf_fold_inf: assumes "finite A" shows "inf B (Inf A) = fold inf B A" proof - interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf) from `finite A` show ?thesis by (induct A arbitrary: B) (simp_all add: Inf_empty Inf_insert inf_commute fold_fun_comm) qed lemma sup_Sup_fold_sup: assumes "finite A" shows "sup B (Sup A) = fold sup B A" proof - interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup) from `finite A` show ?thesis by (induct A arbitrary: B) (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm) qed lemma Inf_fold_inf: assumes "finite A" shows "Inf A = fold inf top A" using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2) lemma Sup_fold_sup: assumes "finite A" shows "Sup A = fold sup bot A" using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2) lemma inf_INFI_fold_inf: assumes "finite A" shows "inf B (INFI A f) = fold (\A. inf (f A)) B A" (is "?inf = ?fold") proof (rule sym) interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf) interpret fun_left_comm_idem "\A. inf (f A)" by (fact fun_left_comm_idem_apply) from `finite A` show "?fold = ?inf" by (induct A arbitrary: B) (simp_all add: INFI_def Inf_empty Inf_insert inf_left_commute) qed lemma sup_SUPR_fold_sup: assumes "finite A" shows "sup B (SUPR A f) = fold (\A. sup (f A)) B A" (is "?sup = ?fold") proof (rule sym) interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup) interpret fun_left_comm_idem "\A. sup (f A)" by (fact fun_left_comm_idem_apply) from `finite A` show "?fold = ?sup" by (induct A arbitrary: B) (simp_all add: SUPR_def Sup_empty Sup_insert sup_left_commute) qed lemma INFI_fold_inf: assumes "finite A" shows "INFI A f = fold (\A. inf (f A)) top A" using assms inf_INFI_fold_inf [of A top] by simp lemma SUPR_fold_sup: assumes "finite A" shows "SUPR A f = fold (\A. sup (f A)) bot A" using assms sup_SUPR_fold_sup [of A bot] by simp end subsection {* The derived combinator @{text fold_image} *} definition fold_image :: "('b \ 'b \ 'b) \ ('a \ 'b) \ 'b \ 'a set \ 'b" where "fold_image f g = fold (%x y. f (g x) y)" lemma fold_image_empty[simp]: "fold_image f g z {} = z" by(simp add:fold_image_def) context ab_semigroup_mult begin lemma fold_image_insert[simp]: assumes "finite A" and "a \ A" shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)" proof - interpret I: fun_left_comm "%x y. (g x) * y" by unfold_locales (simp add: mult_ac) show ?thesis using assms by(simp add:fold_image_def) qed (* lemma fold_commute: "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)" apply (induct set: finite) apply simp apply (simp add: mult_left_commute [of x]) done lemma fold_nest_Un_Int: "finite A ==> finite B ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)" apply (induct set: finite) apply simp apply (simp add: fold_commute Int_insert_left insert_absorb) done lemma fold_nest_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> fold times g z (A Un B) = fold times g (fold times g z B) A" by (simp add: fold_nest_Un_Int) *) lemma fold_image_reindex: assumes fin: "finite A" shows "inj_on h A \ fold_image times g z (h`A) = fold_image times (g\h) z A" using fin by induct auto (* text{* Fusion theorem, as described in Graham Hutton's paper, A Tutorial on the Universality and Expressiveness of Fold, JFP 9:4 (355-372), 1999. *} lemma fold_fusion: assumes "ab_semigroup_mult g" assumes fin: "finite A" and hyp: "\x y. h (g x y) = times x (h y)" shows "h (fold g j w A) = fold times j (h w) A" proof - class_interpret ab_semigroup_mult [g] by fact show ?thesis using fin hyp by (induct set: finite) simp_all qed *) lemma fold_image_cong: "finite A \ (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A" apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C") apply simp apply (erule finite_induct, simp) apply (simp add: subset_insert_iff, clarify) apply (subgoal_tac "finite C") prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) apply (subgoal_tac "C = insert x (C - {x})") prefer 2 apply blast apply (erule ssubst) apply (drule spec) apply (erule (1) notE impE) apply (simp add: Ball_def del: insert_Diff_single) done end context comm_monoid_mult begin lemma fold_image_1: "finite S \ (\x\S. f x = 1) \ fold_image op * f 1 S = 1" apply (induct set: finite) apply simp by auto lemma fold_image_Un_Int: "finite A ==> finite B ==> fold_image times g 1 A * fold_image times g 1 B = fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)" by (induct set: finite) (auto simp add: mult_ac insert_absorb Int_insert_left) lemma fold_image_Un_one: assumes fS: "finite S" and fT: "finite T" and I0: "\x \ S\T. f x = 1" shows "fold_image (op *) f 1 (S \ T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T" proof- have "fold_image op * f 1 (S \ T) = 1" apply (rule fold_image_1) using fS fT I0 by auto with fold_image_Un_Int[OF fS fT] show ?thesis by simp qed corollary fold_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> fold_image times g 1 (A Un B) = fold_image times g 1 A * fold_image times g 1 B" by (simp add: fold_image_Un_Int) lemma fold_image_UN_disjoint: "\ finite I; ALL i:I. finite (A i); ALL i:I. ALL j:I. i \ j --> A i Int A j = {} \ \ fold_image times g 1 (UNION I A) = fold_image times (%i. fold_image times g 1 (A i)) 1 I" apply (induct set: finite, simp, atomize) apply (subgoal_tac "ALL i:F. x \ i") prefer 2 apply blast apply (subgoal_tac "A x Int UNION F A = {}") prefer 2 apply blast apply (simp add: fold_Un_disjoint) done lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==> fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A = fold_image times (split g) 1 (SIGMA x:A. B x)" apply (subst Sigma_def) apply (subst fold_image_UN_disjoint, assumption, simp) apply blast apply (erule fold_image_cong) apply (subst fold_image_UN_disjoint, simp, simp) apply blast apply simp done lemma fold_image_distrib: "finite A \ fold_image times (%x. g x * h x) 1 A = fold_image times g 1 A * fold_image times h 1 A" by (erule finite_induct) (simp_all add: mult_ac) lemma fold_image_related: assumes Re: "R e e" and Rop: "\x1 y1 x2 y2. R x1 x2 \ R y1 y2 \ R (x1 * y1) (x2 * y2)" and fS: "finite S" and Rfg: "\x\S. R (h x) (g x)" shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)" using fS by (rule finite_subset_induct) (insert assms, auto) lemma fold_image_eq_general: assumes fS: "finite S" and h: "\y\S'. \!x. x\ S \ h(x) = y" and f12: "\x\S. h x \ S' \ f2(h x) = f1 x" shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'" proof- from h f12 have hS: "h ` S = S'" by auto {fix x y assume H: "x \ S" "y \ S" "h x = h y" from f12 h H have "x = y" by auto } hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast from f12 have th: "\x. x \ S \ (f2 \ h) x = f1 x" by auto from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp also have "\ = fold_image (op *) (f2 o h) e S" using fold_image_reindex[OF fS hinj, of f2 e] . also have "\ = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e] by blast finally show ?thesis .. qed lemma fold_image_eq_general_inverses: assumes fS: "finite S" and kh: "\y. y \ T \ k y \ S \ h (k y) = y" and hk: "\x. x \ S \ h x \ T \ k (h x) = x \ g (h x) = f x" shows "fold_image (op *) f e S = fold_image (op *) g e T" (* metis solves it, but not yet available here *) apply (rule fold_image_eq_general[OF fS, of T h g f e]) apply (rule ballI) apply (frule kh) apply (rule ex1I[]) apply blast apply clarsimp apply (drule hk) apply simp apply (rule sym) apply (erule conjunct1[OF conjunct2[OF hk]]) apply (rule ballI) apply (drule hk) apply blast done end subsection {* A fold functional for non-empty sets *} text{* Does not require start value. *} inductive fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool" for f :: "'a => 'a => 'a" where fold1Set_insertI [intro]: "\ fold_graph f a A x; a \ A \ \ fold1Set f (insert a A) x" definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where "fold1 f A == THE x. fold1Set f A x" lemma fold1Set_nonempty: "fold1Set f A x \ A \ {}" by(erule fold1Set.cases, simp_all) inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x" inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x" lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)" by (blast elim: fold_graph.cases) lemma fold1_singleton [simp]: "fold1 f {a} = a" by (unfold fold1_def) blast lemma finite_nonempty_imp_fold1Set: "\ finite A; A \ {} \ \ EX x. fold1Set f A x" apply (induct A rule: finite_induct) apply (auto dest: finite_imp_fold_graph [of _ f]) done text{*First, some lemmas about @{const fold_graph}.*} context ab_semigroup_mult begin lemma fun_left_comm: "fun_left_comm(op *)" by unfold_locales (simp add: mult_ac) lemma fold_graph_insert_swap: assumes fold: "fold_graph times (b::'a) A y" and "b \ A" shows "fold_graph times z (insert b A) (z * y)" proof - interpret fun_left_comm "op *::'a \ 'a \ 'a" by (rule fun_left_comm) from assms show ?thesis proof (induct rule: fold_graph.induct) case emptyI show ?case by (subst mult_commute [of z b], fast) next case (insertI x A y) have "fold_graph times z (insert x (insert b A)) (x * (z * y))" using insertI by force --{*how does @{term id} get unfolded?*} thus ?case by (simp add: insert_commute mult_ac) qed qed lemma fold_graph_permute_diff: assumes fold: "fold_graph times b A x" shows "!!a. \a \ A; b \ A\ \ fold_graph times a (insert b (A-{a})) x" using fold proof (induct rule: fold_graph.induct) case emptyI thus ?case by simp next case (insertI x A y) have "a = x \ a \ A" using insertI by simp thus ?case proof assume "a = x" with insertI show ?thesis by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap) next assume ainA: "a \ A" hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)" using insertI by force moreover have "insert x (insert b (A - {a})) = insert b (insert x A - {a})" using ainA insertI by blast ultimately show ?thesis by simp qed qed lemma fold1_eq_fold: assumes "finite A" "a \ A" shows "fold1 times (insert a A) = fold times a A" proof - interpret fun_left_comm "op *::'a \ 'a \ 'a" by (rule fun_left_comm) from assms show ?thesis apply (simp add: fold1_def fold_def) apply (rule the_equality) apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times]) apply (rule sym, clarify) apply (case_tac "Aa=A") apply (best intro: fold_graph_determ) apply (subgoal_tac "fold_graph times a A x") apply (best intro: fold_graph_determ) apply (subgoal_tac "insert aa (Aa - {a}) = A") prefer 2 apply (blast elim: equalityE) apply (auto dest: fold_graph_permute_diff [where a=a]) done qed lemma nonempty_iff: "(A \ {}) = (\x B. A = insert x B & x \ B)" apply safe apply simp apply (drule_tac x=x in spec) apply (drule_tac x="A-{x}" in spec, auto) done lemma fold1_insert: assumes nonempty: "A \ {}" and A: "finite A" "x \ A" shows "fold1 times (insert x A) = x * fold1 times A" proof - interpret fun_left_comm "op *::'a \ 'a \ 'a" by (rule fun_left_comm) from nonempty obtain a A' where "A = insert a A' & a ~: A'" by (auto simp add: nonempty_iff) with A show ?thesis by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) qed end context ab_semigroup_idem_mult begin lemma fun_left_comm_idem: "fun_left_comm_idem(op *)" apply unfold_locales apply (rule mult_left_commute) apply (rule mult_left_idem) done lemma fold1_insert_idem [simp]: assumes nonempty: "A \ {}" and A: "finite A" shows "fold1 times (insert x A) = x * fold1 times A" proof - interpret fun_left_comm_idem "op *::'a \ 'a \ 'a" by (rule fun_left_comm_idem) from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" by (auto simp add: nonempty_iff) show ?thesis proof cases assume "a = x" thus ?thesis proof cases assume "A' = {}" with prems show ?thesis by simp next assume "A' \ {}" with prems show ?thesis by (simp add: fold1_insert mult_assoc [symmetric]) qed next assume "a \ x" with prems show ?thesis by (simp add: insert_commute fold1_eq_fold) qed qed lemma hom_fold1_commute: assumes hom: "!!x y. h (x * y) = h x * h y" and N: "finite N" "N \ {}" shows "h (fold1 times N) = fold1 times (h ` N)" using N proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next case (insert n N) then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp also have "\ = h n * h (fold1 times N)" by(rule hom) also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert) also have "times (h n) \ = fold1 times (insert (h n) (h ` N))" using insert by(simp) also have "insert (h n) (h ` N) = h ` insert n N" by simp finally show ?case . qed lemma fold1_eq_fold_idem: assumes "finite A" shows "fold1 times (insert a A) = fold times a A" proof (cases "a \ A") case False with assms show ?thesis by (simp add: fold1_eq_fold) next interpret fun_left_comm_idem times by (fact fun_left_comm_idem) case True then obtain b B where A: "A = insert a B" and "a \ B" by (rule set_insert) with assms have "finite B" by auto then have "fold times a (insert a B) = fold times (a * a) B" using `a \ B` by (rule fold_insert2) then show ?thesis using `a \ B` `finite B` by (simp add: fold1_eq_fold A) qed end text{* Now the recursion rules for definitions: *} lemma fold1_singleton_def: "g = fold1 f \ g {a} = a" by simp lemma (in ab_semigroup_mult) fold1_insert_def: "\ g = fold1 times; finite A; x \ A; A \ {} \ \ g (insert x A) = x * g A" by (simp add:fold1_insert) lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def: "\ g = fold1 times; finite A; A \ {} \ \ g (insert x A) = x * g A" by simp subsubsection{* Determinacy for @{term fold1Set} *} (*Not actually used!!*) (* context ab_semigroup_mult begin lemma fold_graph_permute: "[|fold_graph times id b (insert a A) x; a \ A; b \ A|] ==> fold_graph times id a (insert b A) x" apply (cases "a=b") apply (auto dest: fold_graph_permute_diff) done lemma fold1Set_determ: "fold1Set times A x ==> fold1Set times A y ==> y = x" proof (clarify elim!: fold1Set.cases) fix A x B y a b assume Ax: "fold_graph times id a A x" assume By: "fold_graph times id b B y" assume anotA: "a \ A" assume bnotB: "b \ B" assume eq: "insert a A = insert b B" show "y=x" proof cases assume same: "a=b" hence "A=B" using anotA bnotB eq by (blast elim!: equalityE) thus ?thesis using Ax By same by (blast intro: fold_graph_determ) next assume diff: "a\b" let ?D = "B - {a}" have B: "B = insert a ?D" and A: "A = insert b ?D" and aB: "a \ B" and bA: "b \ A" using eq anotA bnotB diff by (blast elim!:equalityE)+ with aB bnotB By have "fold_graph times id a (insert b ?D) y" by (auto intro: fold_graph_permute simp add: insert_absorb) moreover have "fold_graph times id a (insert b ?D) x" by (simp add: A [symmetric] Ax) ultimately show ?thesis by (blast intro: fold_graph_determ) qed qed lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y" by (unfold fold1_def) (blast intro: fold1Set_determ) end *) declare empty_fold_graphE [rule del] fold_graph.intros [rule del] empty_fold1SetE [rule del] insert_fold1SetE [rule del] -- {* No more proofs involve these relations. *} subsubsection {* Lemmas about @{text fold1} *} context ab_semigroup_mult begin lemma fold1_Un: assumes A: "finite A" "A \ {}" shows "finite B \ B \ {} \ A Int B = {} \ fold1 times (A Un B) = fold1 times A * fold1 times B" using A by (induct rule: finite_ne_induct) (simp_all add: fold1_insert mult_assoc) lemma fold1_in: assumes A: "finite (A)" "A \ {}" and elem: "\x y. x * y \ {x,y}" shows "fold1 times A \ A" using A proof (induct rule:finite_ne_induct) case singleton thus ?case by simp next case insert thus ?case using elem by (force simp add:fold1_insert) qed end lemma (in ab_semigroup_idem_mult) fold1_Un2: assumes A: "finite A" "A \ {}" shows "finite B \ B \ {} \ fold1 times (A Un B) = fold1 times A * fold1 times B" using A proof(induct rule:finite_ne_induct) case singleton thus ?case by simp next case insert thus ?case by (simp add: mult_assoc) qed subsection {* Locales as mini-packages for fold operations *} subsubsection {* The natural case *} locale folding = fixes f :: "'a \ 'b \ 'b" fixes F :: "'a set \ 'b \ 'b" assumes commute_comp: "f y \ f x = f x \ f y" assumes eq_fold: "finite A \ F A s = fold f s A" begin lemma empty [simp]: "F {} = id" by (simp add: eq_fold expand_fun_eq) lemma insert [simp]: assumes "finite A" and "x \ A" shows "F (insert x A) = F A \ f x" proof - interpret fun_left_comm f proof qed (insert commute_comp, simp add: expand_fun_eq) from fold_insert2 assms have "\s. fold f s (insert x A) = fold f (f x s) A" . with `finite A` show ?thesis by (simp add: eq_fold expand_fun_eq) qed lemma remove: assumes "finite A" and "x \ A" shows "F A = F (A - {x}) \ f 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` this have "finite B" by simp ultimately show ?thesis by simp qed lemma insert_remove: assumes "finite A" shows "F (insert x A) = F (A - {x}) \ f x" using assms by (cases "x \ A") (simp_all add: remove insert_absorb) lemma commute_left_comp: "f y \ (f x \ g) = f x \ (f y \ g)" by (simp add: o_assoc commute_comp) lemma commute_comp': assumes "finite A" shows "f x \ F A = F A \ f x" using assms by (induct A) (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: o_assoc [symmetric] commute_comp) lemma commute_left_comp': assumes "finite A" shows "f x \ (F A \ g) = F A \ (f x \ g)" using assms by (simp add: o_assoc commute_comp') lemma commute_comp'': assumes "finite A" and "finite B" shows "F B \ F A = F A \ F B" using assms by (induct A) (simp_all add: o_assoc, simp add: o_assoc [symmetric] commute_comp') lemma commute_left_comp'': assumes "finite A" and "finite B" shows "F B \ (F A \ g) = F A \ (F B \ g)" using assms by (simp add: o_assoc commute_comp'') lemmas commute_comps = o_assoc [symmetric] commute_comp commute_left_comp commute_comp' commute_left_comp' commute_comp'' commute_left_comp'' lemma union_inter: assumes "finite A" and "finite B" shows "F (A \ B) \ F (A \ B) = F A \ F B" using assms by (induct A) (simp_all del: o_apply add: insert_absorb Int_insert_left commute_comps, simp add: o_assoc) lemma union: assumes "finite A" and "finite B" and "A \ B = {}" shows "F (A \ B) = F A \ F B" proof - from union_inter `finite A` `finite B` have "F (A \ B) \ F (A \ B) = F A \ F B" . with `A \ B = {}` show ?thesis by simp qed end subsubsection {* The natural case with idempotency *} locale folding_idem = folding + assumes idem_comp: "f x \ f x = f x" begin lemma idem_left_comp: "f x \ (f x \ g) = f x \ g" by (simp add: o_assoc idem_comp) lemma in_comp_idem: assumes "finite A" and "x \ A" shows "F A \ f x = F A" using assms by (induct A) (auto simp add: commute_comps idem_comp, simp add: commute_left_comp' [symmetric] commute_comp') lemma subset_comp_idem: assumes "finite A" and "B \ A" shows "F A \ F B = F A" proof - from assms have "finite B" by (blast dest: finite_subset) then show ?thesis using `B \ A` by (induct B) (simp_all add: o_assoc in_comp_idem `finite A`) qed declare insert [simp del] lemma insert_idem [simp]: assumes "finite A" shows "F (insert x A) = F A \ f x" using assms by (cases "x \ A") (simp_all add: insert in_comp_idem insert_absorb) lemma union_idem: assumes "finite A" and "finite B" shows "F (A \ B) = F A \ F B" proof - from assms have "finite (A \ B)" and "A \ B \ A \ B" by auto then have "F (A \ B) \ F (A \ B) = F (A \ B)" by (rule subset_comp_idem) with assms show ?thesis by (simp add: union_inter) qed end subsubsection {* The image case with fixed function *} no_notation times (infixl "*" 70) no_notation Groups.one ("1") locale folding_image_simple = comm_monoid + fixes g :: "('b \ 'a)" fixes F :: "'b set \ 'a" assumes eq_fold_g: "finite A \ F A = fold_image f g 1 A" begin lemma empty [simp]: "F {} = 1" by (simp add: eq_fold_g) lemma insert [simp]: assumes "finite A" and "x \ A" shows "F (insert x A) = g x * F A" proof - interpret fun_left_comm "%x y. (g x) * y" proof qed (simp add: ac_simps) with assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A" by (simp add: fold_image_def) with `finite A` show ?thesis by (simp add: eq_fold_g) qed lemma remove: assumes "finite A" and "x \ A" shows "F A = g 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` this have "finite B" by simp ultimately show ?thesis by simp qed lemma insert_remove: assumes "finite A" shows "F (insert x A) = g x * F (A - {x})" using assms by (cases "x \ A") (simp_all add: remove insert_absorb) lemma neutral: assumes "finite A" and "\x\A. g x = 1" shows "F A = 1" using assms by (induct A) simp_all lemma union_inter: assumes "finite A" and "finite B" shows "F (A \ B) * F (A \ B) = F A * F B" using assms proof (induct A) case empty then show ?case by simp next case (insert x A) then show ?case by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute) qed corollary union_inter_neutral: assumes "finite A" and "finite B" and I0: "\x \ A\B. g x = 1" shows "F (A \ B) = F A * F B" using assms by (simp add: union_inter [symmetric] neutral) corollary union_disjoint: assumes "finite A" and "finite B" assumes "A \ B = {}" shows "F (A \ B) = F A * F B" using assms by (simp add: union_inter_neutral) end subsubsection {* The image case with flexible function *} locale folding_image = comm_monoid + fixes F :: "('b \ 'a) \ 'b set \ 'a" assumes eq_fold: "\g. finite A \ F g A = fold_image f g 1 A" sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof qed (fact eq_fold) context folding_image begin lemma reindex: (* FIXME polymorhism *) assumes "finite A" and "inj_on h A" shows "F g (h ` A) = F (g \ h) A" using assms by (induct A) auto lemma cong: assumes "finite A" and "\x. x \ A \ g x = h x" shows "F g A = F h A" proof - from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C" apply - apply (erule finite_induct) apply simp apply (simp add: subset_insert_iff, clarify) apply (subgoal_tac "finite C") prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) apply (subgoal_tac "C = insert x (C - {x})") prefer 2 apply blast apply (erule ssubst) apply (drule spec) apply (erule (1) notE impE) apply (simp add: Ball_def del: insert_Diff_single) done with assms show ?thesis by simp qed lemma UNION_disjoint: assumes "finite I" and "\i\I. finite (A i)" and "\i\I. \j\I. i \ j \ A i \ A j = {}" shows "F g (UNION I A) = F (F g \ A) I" apply (insert assms) apply (induct set: finite, simp, atomize) apply (subgoal_tac "\i\Fa. x \ i") prefer 2 apply blast apply (subgoal_tac "A x Int UNION Fa A = {}") prefer 2 apply blast apply (simp add: union_disjoint) done lemma distrib: assumes "finite A" shows "F (\x. g x * h x) A = F g A * F h A" using assms by (rule finite_induct) (simp_all add: assoc commute left_commute) lemma related: assumes Re: "R 1 1" and Rop: "\x1 y1 x2 y2. R x1 x2 \ R y1 y2 \ R (x1 * y1) (x2 * y2)" and fS: "finite S" and Rfg: "\x\S. R (h x) (g x)" shows "R (F h S) (F g S)" using fS by (rule finite_subset_induct) (insert assms, auto) lemma eq_general: assumes fS: "finite S" and h: "\y\S'. \!x. x \ S \ h x = y" and f12: "\x\S. h x \ S' \ f2 (h x) = f1 x" shows "F f1 S = F f2 S'" proof- from h f12 have hS: "h ` S = S'" by blast {fix x y assume H: "x \ S" "y \ S" "h x = h y" from f12 h H have "x = y" by auto } hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast from f12 have th: "\x. x \ S \ (f2 \ h) x = f1 x" by auto from hS have "F f2 S' = F f2 (h ` S)" by simp also have "\ = F (f2 o h) S" using reindex [OF fS hinj, of f2] . also have "\ = F f1 S " using th cong [OF fS, of "f2 o h" f1] by blast finally show ?thesis .. qed lemma eq_general_inverses: assumes fS: "finite S" and kh: "\y. y \ T \ k y \ S \ h (k y) = y" and hk: "\x. x \ S \ h x \ T \ k (h x) = x \ g (h x) = j x" shows "F j S = F g T" (* metis solves it, but not yet available here *) apply (rule eq_general [OF fS, of T h g j]) apply (rule ballI) apply (frule kh) apply (rule ex1I[]) apply blast apply clarsimp apply (drule hk) apply simp apply (rule sym) apply (erule conjunct1[OF conjunct2[OF hk]]) apply (rule ballI) apply (drule hk) apply blast done end subsubsection {* The image case with fixed function and idempotency *} locale folding_image_simple_idem = folding_image_simple + assumes idem: "x * x = x" sublocale folding_image_simple_idem < semilattice proof qed (fact idem) context folding_image_simple_idem begin lemma in_idem: assumes "finite A" and "x \ A" shows "g x * F A = F A" using assms by (induct A) (auto simp add: left_commute) lemma subset_idem: assumes "finite A" and "B \ A" shows "F B * F A = F A" proof - from assms have "finite B" by (blast dest: finite_subset) then show ?thesis using `B \ A` by (induct B) (auto simp add: assoc in_idem `finite A`) qed declare insert [simp del] lemma insert_idem [simp]: assumes "finite A" shows "F (insert x A) = g x * F A" using assms by (cases "x \ A") (simp_all add: insert in_idem insert_absorb) lemma union_idem: assumes "finite A" and "finite B" shows "F (A \ B) = F A * F B" proof - from assms have "finite (A \ B)" and "A \ B \ A \ B" by auto then have "F (A \ B) * F (A \ B) = F (A \ B)" by (rule subset_idem) with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute) qed end subsubsection {* The image case with flexible function and idempotency *} locale folding_image_idem = folding_image + assumes idem: "x * x = x" sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof qed (fact idem) subsubsection {* The neutral-less case *} locale folding_one = abel_semigroup + fixes F :: "'a set \ 'a" assumes eq_fold: "finite A \ F A = fold1 f A" begin lemma singleton [simp]: "F {x} = x" by (simp add: eq_fold) lemma eq_fold': assumes "finite A" and "x \ A" shows "F (insert x A) = fold (op *) x A" proof - interpret ab_semigroup_mult "op *" proof qed (simp_all add: ac_simps) with assms show ?thesis by (simp add: eq_fold fold1_eq_fold) qed lemma insert [simp]: assumes "finite A" and "x \ A" and "A \ {}" shows "F (insert x A) = x * F A" proof - from `A \ {}` obtain b where "b \ A" by blast then obtain B where *: "A = insert b B" "b \ B" by (blast dest: mk_disjoint_insert) with `finite A` have "finite B" by simp interpret fold: folding "op *" "\a b. fold (op *) b a" proof qed (simp_all add: expand_fun_eq ac_simps) thm fold.commute_comp' [of B b, simplified expand_fun_eq, simplified] from `finite B` fold.commute_comp' [of B x] have "op * x \ (\b. fold op * b B) = (\b. fold op * b B) \ op * x" by simp then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: expand_fun_eq commute) from `finite B` * fold.insert [of B b] have "(\x. fold op * x (insert b B)) = (\x. fold op * x B) \ op * b" by simp then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: expand_fun_eq) from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert) qed lemma remove: assumes "finite A" and "x \ A" shows "F A = (if A - {x} = {} then x else x * F (A - {x}))" proof - from assms obtain B where "A = insert x B" and "x \ B" by (blast dest: mk_disjoint_insert) with assms show ?thesis by simp qed lemma insert_remove: assumes "finite A" shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))" using assms by (cases "x \ A") (simp_all add: insert_absorb remove) lemma union_disjoint: assumes "finite A" "A \ {}" and "finite B" "B \ {}" and "A \ B = {}" shows "F (A \ B) = F A * F B" using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps) lemma union_inter: assumes "finite A" and "finite B" and "A \ B \ {}" shows "F (A \ B) * F (A \ B) = F A * F B" proof - from assms have "A \ {}" and "B \ {}" by auto from `finite A` `A \ {}` `A \ B \ {}` show ?thesis proof (induct A rule: finite_ne_induct) case (singleton x) then show ?case by (simp add: insert_absorb ac_simps) next case (insert x A) show ?case proof (cases "x \ B") case True then have "B \ {}" by auto with insert True `finite B` show ?thesis by (cases "A \ B = {}") (simp_all add: insert_absorb ac_simps union_disjoint) next case False with insert have "F (A \ B) * F (A \ B) = F A * F B" by simp moreover from False `finite B` insert have "finite (A \ B)" "x \ A \ B" "A \ B \ {}" by auto ultimately show ?thesis using False `finite A` `x \ A` `A \ {}` by (simp add: assoc) qed qed qed lemma closed: assumes "finite A" "A \ {}" and elem: "\x y. x * y \ {x, y}" shows "F A \ A" using `finite A` `A \ {}` proof (induct rule: finite_ne_induct) case singleton then show ?case by simp next case insert with elem show ?case by force qed end subsubsection {* The neutral-less case with idempotency *} locale folding_one_idem = folding_one + assumes idem: "x * x = x" sublocale folding_one_idem < semilattice proof qed (fact idem) context folding_one_idem begin lemma in_idem: assumes "finite A" and "x \ A" shows "x * F A = F A" proof - from assms have "A \ {}" by auto with `finite A` show ?thesis using `x \ A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps) qed lemma subset_idem: assumes "finite A" "B \ {}" and "B \ A" shows "F B * F A = F A" proof - from assms have "finite B" by (blast dest: finite_subset) then show ?thesis using `B \ {}` `B \ A` by (induct B rule: finite_ne_induct) (simp_all add: assoc in_idem `finite A`) qed lemma eq_fold_idem': assumes "finite A" shows "F (insert a A) = fold (op *) a A" proof - interpret ab_semigroup_idem_mult "op *" proof qed (simp_all add: ac_simps) with assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem) qed lemma insert_idem [simp]: assumes "finite A" and "A \ {}" shows "F (insert x A) = x * F A" proof (cases "x \ A") case False from `finite A` `x \ A` `A \ {}` show ?thesis by (rule insert) next case True from `finite A` `A \ {}` show ?thesis by (simp add: in_idem insert_absorb True) qed lemma union_idem: assumes "finite A" "A \ {}" and "finite B" "B \ {}" shows "F (A \ B) = F A * F B" proof (cases "A \ B = {}") case True with assms show ?thesis by (simp add: union_disjoint) next case False from assms have "finite (A \ B)" and "A \ B \ A \ B" by auto with False have "F (A \ B) * F (A \ B) = F (A \ B)" by (auto intro: subset_idem) with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute) qed lemma hom_commute: assumes hom: "\x y. h (x * y) = h x * h y" and N: "finite N" "N \ {}" shows "h (F N) = F (h ` N)" using N proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next case (insert n N) then have "h (F (insert n N)) = h (n * F N)" by simp also have "\ = h n * h (F N)" by (rule hom) also have "h (F N) = F (h ` N)" by(rule insert) also have "h n * \ = F (insert (h n) (h ` N))" using insert by(simp) also have "insert (h n) (h ` N) = h ` insert n N" by simp finally show ?case . qed end notation times (infixl "*" 70) notation Groups.one ("1") subsection {* Finite cardinality *} text {* This definition, although traditional, is ugly to work with: @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}. But now that we have @{text fold_image} things are easy: *} definition card :: "'a set \ nat" where "card A = (if finite A then fold_image (op +) (\x. 1) 0 A else 0)" interpretation card!: folding_image_simple "op +" 0 "\x. 1" card proof qed (simp add: card_def) lemma card_infinite [simp]: "\ finite A \ card A = 0" by (simp add: card_def) 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 simp 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, no_atp]: "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_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_tac t = A in insert_Diff [THEN subst], assumption) apply(simp del:insert_Diff_single) 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 (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert) lemma card_insert_le: "finite A ==> card A <= card (insert x A)" by (simp add: card_insert_if) 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 set: finite, simp, 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 add: split_if_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: "finite A ==> finite B ==> card A + card B = card (A Un B) + card (A Int B)" by (fact card.union_inter [symmetric]) lemma card_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> card (A Un B) = card A + card B" by (fact card.union_disjoint) lemma card_Diff_subset: assumes "finite B" and "B \ A" shows "card (A - B) = card A - card B" 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 AB: "finite (A \ B)" shows "card (A - B) = card A - card (A \ B)" proof - have "A - B = A - A \ B" by auto thus ?thesis by (simp add: card_Diff_subset AB) qed lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" apply (rule Suc_less_SucD) apply (simp add: card_Suc_Diff1 del:card_Diff_insert) done lemma card_Diff2_less: "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" apply (case_tac "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" apply (case_tac "x : A") apply (simp_all add: card_Diff1_less less_imp_le) done lemma card_psubset: "finite B ==> A \ B ==> card A < card B ==> A < B" by (erule psubsetI, blast) lemma insert_partition: "\ x \ F; \c1 \ insert x F. \c2 \ insert x F. c1 \ c2 \ c1 \ c2 = {} \ \ x \ \ F = {}" by auto lemma finite_psubset_induct[consumes 1, case_names psubset]: assumes fin: "finite A" and major: "\A. finite A \ (\B. B \ A \ P B) \ P A" shows "P A" using fin proof (induct A taking: card rule: measure_induct_rule) case (less A) have fin: "finite A" by fact have ih: "\B. \card B < card A; finite B\ \ P B" by fact { fix B assume asm: "B \ A" from asm have "card B < card A" using psubset_card_mono fin by blast moreover from asm have "B \ A" by auto then have "finite B" using fin finite_subset by blast ultimately have "P B" using ih by simp } with fin show "P A" using major by blast qed 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)" apply (erule finite_induct, simp) apply (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\ (insert x F)"]) done 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 fix x show "x \ A" 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(fastsimp 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(rule iffI) apply(erule card_eq_SucD) apply(auto) apply(subst card_insert) apply(auto intro:ccontr) done lemma finite_fun_UNIVD2: assumes fin: "finite (UNIV :: ('a \ 'b) set)" shows "finite (UNIV :: 'b set)" proof - from fin have "finite (range (\f :: 'a \ 'b. f arbitrary))" by(rule finite_imageI) moreover have "UNIV = range (\f :: 'a \ 'b. f arbitrary)" by(rule UNIV_eq_I) auto ultimately show "finite (UNIV :: 'b set)" by simp qed lemma card_UNIV_unit: "card (UNIV :: unit set) = 1" unfolding UNIV_unit by simp subsubsection {* Cardinality of image *} lemma card_image_le: "finite A ==> card (f ` A) <= card A" apply (induct set: finite) apply simp apply (simp add: le_SucI card_insert_if) done lemma card_image: assumes "inj_on f A" shows "card (f ` A) = card A" proof (cases "finite A") case True then show ?thesis using assms by (induct A) simp_all next case False then have "\ finite (f ` A)" using assms by (auto dest: finite_imageD) with False show ?thesis by simp qed 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: "[| finite A; card(f ` A) = card A |] ==> inj_on f A" apply (induct rule:finite_induct) apply simp apply(frule card_image_le[where f = f]) apply(simp add:card_insert_if split:if_splits) done 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: "[|inj_on f A; f ` A \ B; finite B |] ==> card A \ card B" apply (subgoal_tac "finite A") apply (force intro: card_mono simp add: card_image [symmetric]) apply (blast intro: finite_imageD dest: finite_subset) done 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) subsubsection {* Cardinality of sums *} lemma card_Plus: assumes "finite A" and "finite B" shows "card (A <+> B) = card A + card B" proof - have "Inl`A \ Inr`B = {}" by fast with assms show ?thesis unfolding Plus_def by (simp add: 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) subsubsection {* Cardinality of the Powerset *} lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) apply (induct set: finite) apply (simp_all add: Pow_insert) apply (subst card_Un_disjoint, blast) apply (blast intro: finite_imageI, blast) apply (subgoal_tac "inj_on (insert x) (Pow F)") apply (simp add: card_image Pow_insert) apply (unfold inj_on_def) apply (blast elim!: equalityE) done text {* Relates to equivalence classes. Based on a theorem of F. Kammüller. *} lemma dvd_partition: "finite (Union C) ==> ALL c : C. k dvd card c ==> (ALL c1: C. ALL c2: C. c1 \ c2 --> c1 Int c2 = {}) ==> k dvd card (Union C)" apply(frule finite_UnionD) apply(rotate_tac -1) apply (induct set: finite, simp_all, clarify) apply (subst card_Un_disjoint) apply (auto simp add: disjoint_eq_subset_Compl) done subsubsection {* Relating injectivity and surjectivity *} lemma finite_surj_inj: "finite(A) \ A <= f`A \ inj_on f A" apply(rule eq_card_imp_inj_on, assumption) apply(frule finite_imageI) apply(drule (1) card_seteq) apply(erule card_image_le) apply simp done lemma finite_UNIV_surj_inj: fixes f :: "'a \ 'a" shows "finite(UNIV:: 'a set) \ surj f \ inj f" by (blast intro: finite_surj_inj subset_UNIV dest:surj_range) lemma finite_UNIV_inj_surj: fixes f :: "'a \ 'a" shows "finite(UNIV:: 'a set) \ inj f \ surj f" by(fastsimp 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 (* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *) lemma infinite_UNIV_char_0[no_atp]: "\ 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 end