(*  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 Option Power

 begin

 subsection {* Predicate for finite sets *}

 inductive finite :: "'a set \<Rightarrow> bool"

   where

     emptyI [simp, intro!]: "finite {}"

   | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"

 lemma finite_induct [case_names empty insert, induct set: finite]:

   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}

   assumes "finite F"

   assumes "P {}"

     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"

   shows "P F"

 using `finite 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 \<in> F"

     hence "insert x F = F" by (rule insert_absorb)

     with P show ?thesis by (simp only:)

   next

     assume "x \<notin> F"

     from F this P show ?thesis by (rule insert)

   qed

 qed

 subsubsection {* Choice principles *}

 lemma ex_new_if_finite: -- "does not depend on def of finite at all"

   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"

   shows "\<exists>a::'a. a \<notin> A"

 proof -

   from assms have "A \<noteq> UNIV" by blast

   then show ?thesis by blast

 qed

 text {* A finite choice principle. Does not need the SOME choice operator. *}

 lemma finite_set_choice:

   "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"

 proof (induct rule: finite_induct)

   case empty then show ?case by simp

 next

   case (insert a A)

   then obtain f b where f: "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

 subsubsection {* Finite sets are the images of initial segments of natural numbers *}

 lemma finite_imp_nat_seg_image_inj_on:

   assumes "finite A" 

   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"

 using assms proof induct

   case empty

   show ?case

   proof

     show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 

   qed

 next

   case (insert a A)

   have notinA: "a \<notin> A" by fact

   from insert.hyps obtain n f

     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast

   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:

   "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"

 proof (induct n arbitrary: A)

   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 "\<exists>k<n. f n = f k"

     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)

     thus ?thesis using finB by simp

   next

     assume "\<not>(\<exists> k<n. f n = f k)"

     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)

     thus ?thesis using finB by simp

   qed

 qed

 lemma finite_conv_nat_seg_image:

   "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"

   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)

 lemma finite_imp_inj_to_nat_seg:

   assumes "finite A"

   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> 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])

   thus ?thesis by blast

 qed

 lemma finite_Collect_less_nat [iff]:

   "finite {n::nat. n < k}"

   by (fastsimp simp: finite_conv_nat_seg_image)

 lemma finite_Collect_le_nat [iff]:

   "finite {n::nat. n \<le> k}"

   by (simp add: le_eq_less_or_eq Collect_disj_eq)

 subsubsection {* Finiteness and common set operations *}

 lemma rev_finite_subset:

   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"

 proof (induct arbitrary: A rule: finite_induct)

   case empty

   then show ?case by simp

 next

   case (insert x F A)

   have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+

   show "finite A"

   proof cases

     assume x: "x \<in> A"

     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)

     with r have "finite (A - {x})" .

     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 \<subseteq> F ==> ?thesis" by fact

     assume "x \<notin> A"

     with A show "A \<subseteq> F" by (simp add: subset_insert_iff)

   qed

 qed

 lemma finite_subset:

   "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"

   by (rule rev_finite_subset)

 lemma finite_UnI:

   assumes "finite F" and "finite G"

   shows "finite (F \<union> G)"

   using assms by induct simp_all

 lemma finite_Un [iff]:

   "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"

   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])

 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"

 proof -

   have "finite {a} \<and> finite A \<longleftrightarrow> finite A" by simp

   then have "finite ({a} \<union> A) \<longleftrightarrow> finite A" by (simp only: finite_Un)

   then show ?thesis by simp

 qed

 lemma finite_Int [simp, intro]:

   "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"

   by (blast intro: finite_subset)

 lemma finite_Collect_conjI [simp, intro]:

   "finite {x. P x} \<or> finite {x. Q x} \<Longrightarrow> finite {x. P x \<and> Q x}"

   by (simp add: Collect_conj_eq)

 lemma finite_Collect_disjI [simp]:

   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"

   by (simp add: Collect_disj_eq)

 lemma finite_Diff [simp, intro]:

   "finite A \<Longrightarrow> finite (A - B)"

   by (rule finite_subset, rule Diff_subset)

 lemma finite_Diff2 [simp]:

   assumes "finite B"

   shows "finite (A - B) \<longleftrightarrow> finite A"

 proof -

   have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)

   also have "\<dots> \<longleftrightarrow> finite (A - B)" using `finite B` by simp

   finally show ?thesis ..

 qed

 lemma finite_Diff_insert [iff]:

   "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"

 proof -

   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp

   moreover have "A - insert a B = A - B - {a}" by auto

   ultimately show ?thesis by simp

 qed

 lemma finite_compl[simp]:

   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"

   by (simp add: Compl_eq_Diff_UNIV)

 lemma finite_Collect_not[simp]:

   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"

   by (simp add: Collect_neg_eq)

 lemma finite_Union [simp, intro]:

   "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"

   by (induct rule: finite_induct) simp_all

 lemma finite_UN_I [intro]:

   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"

   by (induct rule: finite_induct) simp_all

 lemma finite_UN [simp]:

   "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"

   by (blast intro: finite_subset)

 lemma finite_Inter [intro]:

   "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"

   by (blast intro: Inter_lower finite_subset)

 lemma finite_INT [intro]:

   "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"

   by (blast intro: INT_lower finite_subset)

 lemma finite_imageI [simp, intro]:

   "finite F \<Longrightarrow> finite (h ` F)"

   by (induct rule: finite_induct) simp_all

 lemma finite_image_set [simp]:

   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"

   by (simp add: image_Collect [symmetric])

 lemma finite_imageD:

   assumes "finite (f ` A)" and "inj_on f A"

   shows "finite A"

 using assms proof (induct "f ` A" arbitrary: A)

   case empty then show ?case by simp

 next

   case (insert x B)

   then have B_A: "insert x B = f ` A" by simp

   then obtain y where "x = f y" and "y \<in> A" by blast

   from B_A `x \<notin> B` have "B = f ` A - {x}" by blast

   with B_A `x \<notin> B` `x = f y` `inj_on f A` `y \<in> 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_surj:

   "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"

   by (erule finite_subset) (rule finite_imageI)

 lemma finite_range_imageI:

   "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"

   by (drule finite_imageI) (simp add: range_composition)

 lemma finite_subset_image:

   assumes "finite B"

   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"

 using assms proof induct

   case empty then show ?case by simp

 next

   case insert then show ?case

     by (clarsimp simp del: image_insert simp add: image_insert [symmetric])

        blast

 qed

 lemma finite_vimageI:

   "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"

   apply (induct rule: finite_induct)

    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 \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"

   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)

 lemma finite_Collect_bex [simp]:

   assumes "finite A"

   shows "finite {x. \<exists>y\<in>A. Q x y} \<longleftrightarrow> (\<forall>y\<in>A. finite {x. Q x y})"

 proof -

   have "{x. \<exists>y\<in>A. Q x y} = (\<Union>y\<in>A. {x. Q x y})" by auto

   with assms show ?thesis by simp

 qed

 lemma finite_Collect_bounded_ex [simp]:

   assumes "finite {y. P y}"

   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"

 proof -

   have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto

   with assms show ?thesis by simp

 qed

 lemma finite_Plus:

   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"

   by (simp add: Plus_def)

 lemma finite_PlusD: 

   fixes A :: "'a set" and B :: "'b set"

   assumes fin: "finite (A <+> B)"

   shows "finite A" "finite B"

 proof -

   have "Inl ` A \<subseteq> A <+> B" by auto

   then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)

   then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)

 next

   have "Inr ` B \<subseteq> A <+> B" by auto

   then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)

   then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)

 qed

 lemma finite_Plus_iff [simp]:

   "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"

   by (auto intro: finite_PlusD finite_Plus)

 lemma finite_Plus_UNIV_iff [simp]:

   "finite (UNIV :: ('a + 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"

   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)

 lemma finite_SigmaI [simp, intro]:

   "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"

   by (unfold Sigma_def) blast

 lemma finite_cartesian_product:

   "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"

   by (rule finite_SigmaI)

 lemma finite_Prod_UNIV:

   "finite (UNIV :: 'a set) \<Longrightarrow> finite (UNIV :: 'b set) \<Longrightarrow> finite (UNIV :: ('a \<times> 'b) set)"

   by (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product)

 lemma finite_cartesian_productD1:

   assumes "finite (A \<times> B)" and "B \<noteq> {}"

   shows "finite A"

 proof -

   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"

     by (auto simp add: finite_conv_nat_seg_image)

   then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp

   with `B \<noteq> {}` have "A = (fst \<circ> f) ` {i::nat. i < n}"

     by (simp add: image_compose)

   then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast

   then show ?thesis

     by (auto simp add: finite_conv_nat_seg_image)

 qed

 lemma finite_cartesian_productD2:

   assumes "finite (A \<times> B)" and "A \<noteq> {}"

   shows "finite B"

 proof -

   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"

     by (auto simp add: finite_conv_nat_seg_image)

   then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp

   with `A \<noteq> {}` have "B = (snd \<circ> f) ` {i::nat. i < n}"

     by (simp add: image_compose)

   then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast

   then show ?thesis

     by (auto simp add: finite_conv_nat_seg_image)

 qed

 lemma finite_Pow_iff [iff]:

   "finite (Pow A) \<longleftrightarrow> finite A"

 proof

   assume "finite (Pow A)"

   then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)

   then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp

 next

   assume "finite A"

   then show "finite (Pow A)"

     by induct (simp_all add: Pow_insert)

 qed

 corollary finite_Collect_subsets [simp, intro]:

   "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"

   by (simp add: Pow_def [symmetric])

 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"

   by (blast intro: finite_subset [OF subset_Pow_Union])

 subsubsection {* Further induction rules on finite sets *}

 lemma finite_ne_induct [case_names singleton insert, consumes 2]:

   assumes "finite F" and "F \<noteq> {}"

   assumes "\<And>x. P {x}"

     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"

   shows "P F"

 using assms proof induct

   case empty then show ?case by simp

 next

   case (insert x F) then show ?case by cases auto

 qed

 lemma finite_subset_induct [consumes 2, case_names empty insert]:

   assumes "finite F" and "F \<subseteq> A"

   assumes empty: "P {}"

     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"

   shows "P F"

 using `finite F` `F \<subseteq> A` proof induct

   show "P {}" by fact

 next

   fix x F

   assume "finite F" and "x \<notin> F" and

     P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"

   show "P (insert x F)"

   proof (rule insert)

     from i show "x \<in> A" by blast

     from i have "F \<subseteq> A" by blast

     with P show "P F" .

     show "finite F" by fact

     show "x \<notin> F" by fact

   qed

 qed

 lemma finite_empty_induct:

   assumes "finite A"

   assumes "P A"

     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"

   shows "P {}"

 proof -

   have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"

   proof -

     fix B :: "'a set"

     assume "B \<subseteq> A"

     with `finite A` have "finite B" by (rule rev_finite_subset)

     from this `B \<subseteq> 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 \<in> A - B" by simp

         from insert show "P (A - B)" by simp

       qed

       also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])

       finally show ?case .

     qed

   qed

   then have "P (A - A)" by blast

   then show ?thesis by simp

 qed

 subsection {* Class @{text finite}  *}

 class finite =

   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"

 begin

 lemma finite [simp]: "finite (A \<Colon> 'a set)"

   by (rule subset_UNIV finite_UNIV finite_subset)+

 lemma finite_code [code]: "finite (A \<Colon> 'a set) = True"

   by simp

 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 prod :: (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: set_eq_iff fun_eq_iff)

 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 \<subseteq> Pow UNIV" by simp

     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"

       by (simp only: finite_Pow_iff finite)

     ultimately show "finite (range ?graph)"

       by (rule finite_subset)

     show "inj ?graph" by (rule inj_graph)

   qed

 qed

 instance sum :: (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 (\<dots> (f x\<^isub>n z)\<dots>)"}

 if @{text f} is ``left-commutative'':

 *}

 locale fun_left_comm =

   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"

   assumes commute_comp: "f y \<circ> f x = f x \<circ> f y"

 begin

 lemma fun_left_comm: "f x (f y z) = f y (f x z)"

   using commute_comp by (simp add: fun_eq_iff)

 end

 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"

 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where

   emptyI [intro]: "fold_graph f z {} z" |

   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y

       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"

 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"

 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where

   "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 finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"

 by (induct rule: finite_induct) 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 \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"

 proof (induct set: fold_graph)

   case (insertI x A y) show ?case

   proof (cases "x = a")

     assume "x = a" with insertI show ?case by auto

   next

     assume "x \<noteq> 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 \<noteq> a` and `x \<notin> 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 \<notin> A"

   obtains y where "v = f x y" and "fold_graph f z A y"

 using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])

 lemma fold_graph_determ:

   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"

 proof (induct arbitrary: y set: fold_graph)

   case (insertI x A y v)

   from `fold_graph f z (insert x A) v` and `x \<notin> 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 \<Longrightarrow> fold f z A = y"

 by (unfold fold_def) (blast intro: fold_graph_determ)

 lemma fold_graph_fold:

   assumes "finite A"

   shows "fold_graph f z A (fold f z A)"

 proof -

   from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)

   moreover note fold_graph_determ

   ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)

   then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')

   then show ?thesis by (unfold fold_def)

 qed

 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 \<notin> 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 \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"

 proof (induct rule: finite_induct)

   case empty then show ?case by simp

 next

   case (insert y A) then show ?case

     by (simp add: fun_left_comm[of x])

 qed

 lemma fold_insert2:

   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"

 by (simp add: fold_fun_comm)

 lemma fold_rec:

 assumes "finite A" and "x \<in> A"

 shows "fold f z A = f x (fold f z (A - {x}))"

 proof -

   have A: "A = insert x (A - {x})" using `x \<in> A` by blast

   then have "fold f z A = fold f z (insert x (A - {x}))" by simp

   also have "\<dots> = f x (fold f z (A - {x}))"

     by (rule fold_insert) (simp add: `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 \<in> insert x A" by auto

   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"

     by (rule fold_rec)

   then show ?thesis by simp

 qed

 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 fun_eq_iff)

 lemma fold_insert_idem:

   assumes fin: "finite A"

   shows "fold f z (insert x A) = f x (fold f z A)"

 proof cases

   assume "x \<in> A"

   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)

   then show ?thesis using assms by (simp add:fun_left_idem)

 next

   assume "x \<notin> A" then show ?thesis using assms by simp

 qed

 declare fold_insert[simp del] fold_insert_idem[simp]

 lemma fold_insert_idem2:

   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"

 by(simp add:fold_fun_comm)

 end

 subsubsection {* Expressing set operations via @{const fold} *}

 lemma (in fun_left_comm) fun_left_comm_apply:

   "fun_left_comm (\<lambda>x. f (g x))"

 proof

 qed (simp_all add: commute_comp)

 lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:

   "fun_left_comm_idem (\<lambda>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 (\<lambda>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 \<union> 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 (\<lambda>x A. A - {x}) B A"

 proof -

   interpret fun_left_comm_idem "\<lambda>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_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_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 (\<lambda>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 "\<lambda>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_insert inf_left_commute)

 qed

 lemma sup_SUPR_fold_sup:

   assumes "finite A"

   shows "sup B (SUPR A f) = fold (\<lambda>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 "\<lambda>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_insert sup_left_commute)

 qed

 lemma INFI_fold_inf:

   assumes "finite A"

   shows "INFI A f = fold (\<lambda>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 (\<lambda>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 \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> '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 \<notin> 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" proof

   qed (simp add: fun_eq_iff 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 \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>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: "\<And>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 \<Longrightarrow>

   (!!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 \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"

   apply (induct rule: finite_induct)

   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)"

   apply (induct rule: finite_induct)

 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: "\<forall>x \<in> S\<inter>T. f x = 1"

   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"

 proof-

   have "fold_image op * f 1 (S \<inter> 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:

   "\<lbrakk> finite I; ALL i:I. finite (A i);

      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>

    \<Longrightarrow> fold_image times g 1 (UNION I A) =

        fold_image times (%i. fold_image times g 1 (A i)) 1 I"

 apply (induct rule: finite_induct)

 apply simp

 apply atomize

 apply (subgoal_tac "ALL i:F. x \<noteq> 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 \<Longrightarrow>

    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: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 

   and fS: "finite S" and Rfg: "\<forall>x\<in>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: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" 

   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> 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 \<in> S" "y \<in> 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: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> 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 "\<dots> = fold_image (op *) (f2 o h) e S" 

     using fold_image_reindex[OF fS hinj, of f2 e] .

   also have "\<dots> = 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: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> 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]:

    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> A \<noteq> {}"

 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:

   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> 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 *)" proof

 qed (simp add: fun_eq_iff mult_ac)

 lemma fold_graph_insert_swap:

 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"

 shows "fold_graph times z (insert b A) (z * y)"

 proof -

   interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> '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. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> 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 \<or> a \<in> 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 \<in> 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 \<notin> A" shows "fold1 times (insert a A) = fold times a A"

 proof -

   interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> '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 \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> 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 \<noteq> {}" and A: "finite A" "x \<notin> A"

   shows "fold1 times (insert x A) = x * fold1 times A"

 proof -

   interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> '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 *)" proof

 qed (simp add: fun_eq_iff mult_left_commute, rule mult_left_idem)

 lemma fold1_insert_idem [simp]:

   assumes nonempty: "A \<noteq> {}" and A: "finite A" 

   shows "fold1 times (insert x A) = x * fold1 times A"

 proof -

   interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> '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: "a = x"

     show ?thesis

     proof cases

       assume "A' = {}"

       with A' a show ?thesis by simp

     next

       assume "A' \<noteq> {}"

       with A A' a show ?thesis

         by (simp add: fold1_insert mult_assoc [symmetric])

     qed

   next

     assume "a \<noteq> x"

     with A A' 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 \<noteq> {}" 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 "\<dots> = 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) \<dots> = 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 \<in> 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 \<notin> 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 \<notin> B` by (rule fold_insert2)

   then show ?thesis

     using `a \<notin> 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 \<Longrightarrow> g {a} = a"

 by simp

 lemma (in ab_semigroup_mult) fold1_insert_def:

   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"

 by (simp add:fold1_insert)

 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:

   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> 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 \<notin> A; b \<notin> 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 \<notin> A"

   assume bnotB:  "b \<notin> 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\<noteq>b"

     let ?D = "B - {a}"

     have B: "B = insert a ?D" and A: "A = insert b ?D"

      and aB: "a \<in> B" and bA: "b \<in> 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 \<noteq> {}"

 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>

        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 \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"

   shows "fold1 times A \<in> 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 \<noteq> {}"

 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>

        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 \<Rightarrow> 'b \<Rightarrow> 'b"

   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"

   assumes commute_comp: "f y \<circ> f x = f x \<circ> f y"

   assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"

 begin

 lemma empty [simp]:

   "F {} = id"

   by (simp add: eq_fold fun_eq_iff)

 lemma insert [simp]:

   assumes "finite A" and "x \<notin> A"

   shows "F (insert x A) = F A \<circ> f x"

 proof -

   interpret fun_left_comm f proof

   qed (insert commute_comp, simp add: fun_eq_iff)

   from fold_insert2 assms

   have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .

   with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)

 qed

 lemma remove:

   assumes "finite A" and "x \<in> A"

   shows "F A = F (A - {x}) \<circ> f x"

 proof -

   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> 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}) \<circ> f x"

   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)

 lemma commute_left_comp:

   "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"

   by (simp add: o_assoc commute_comp)

 lemma commute_comp':

   assumes "finite A"

   shows "f x \<circ> F A = F A \<circ> 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 \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"

   using assms by (simp add: o_assoc commute_comp')

 lemma commute_comp'':

   assumes "finite A" and "finite B"

   shows "F B \<circ> F A = F A \<circ> 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 \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> 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 \<union> B) \<circ> F (A \<inter> B) = F A \<circ> 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 \<inter> B = {}"

   shows "F (A \<union> B) = F A \<circ> F B"

 proof -

   from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .

   with `A \<inter> B = {}` show ?thesis by simp

 qed

 end

 subsubsection {* The natural case with idempotency *}

 locale folding_idem = folding +

   assumes idem_comp: "f x \<circ> f x = f x"

 begin

 lemma idem_left_comp:

   "f x \<circ> (f x \<circ> g) = f x \<circ> g"

   by (simp add: o_assoc idem_comp)

 lemma in_comp_idem:

   assumes "finite A" and "x \<in> A"

   shows "F A \<circ> 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 \<subseteq> A"

   shows "F A \<circ> F B = F A"

 proof -

   from assms have "finite B" by (blast dest: finite_subset)

   then show ?thesis using `B \<subseteq> 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 \<circ> f x"

   using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)

 lemma union_idem:

   assumes "finite A" and "finite B"

   shows "F (A \<union> B) = F A \<circ> F B"

 proof -

   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto

   then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> 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 \<Rightarrow> 'a)"

   fixes F :: "'b set \<Rightarrow> 'a"

   assumes eq_fold_g: "finite A \<Longrightarrow> 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 \<notin> 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 fun_eq_iff)

   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 \<in> A"

   shows "F A = g x * F (A - {x})"

 proof -

   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> 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 \<in> A") (simp_all add: remove insert_absorb)

 lemma neutral:

   assumes "finite A" and "\<forall>x\<in>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 \<union> B) * F (A \<inter> 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: "\<forall>x \<in> A\<inter>B. g x = 1"

   shows "F (A \<union> 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 \<inter> B = {}"

   shows "F (A \<union> 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 \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"

   assumes eq_fold: "\<And>g. finite A \<Longrightarrow> 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 \<circ> h) A"

   using assms by (induct A) auto

 lemma cong:

   assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> 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 "\<forall>i\<in>I. finite (A i)"

   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"

   shows "F g (UNION I A) = F (F g \<circ> A) I"

 apply (insert assms)

 apply (induct rule: finite_induct)

 apply simp

 apply atomize

 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> 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 (\<lambda>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: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 

   and fS: "finite S" and Rfg: "\<forall>x\<in>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: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 

   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> 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 \<in> S" "y \<in> 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: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 

   from hS have "F f2 S' = F f2 (h ` S)" by simp

   also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .

   also have "\<dots> = 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: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> 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 \<in> 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 \<subseteq> A"

   shows "F B * F A = F A"

 proof -

   from assms have "finite B" by (blast dest: finite_subset)

   then show ?thesis using `B \<subseteq> 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 \<in> A") (simp_all add: insert in_idem insert_absorb)

 lemma union_idem:

   assumes "finite A" and "finite B"

   shows "F (A \<union> B) = F A * F B"

 proof -

   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto

   then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> 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 \<Rightarrow> 'a"

   assumes eq_fold: "finite A \<Longrightarrow> 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 \<notin> 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 \<notin> A" and "A \<noteq> {}"

   shows "F (insert x A) = x * F A"

 proof -

   from `A \<noteq> {}` obtain b where "b \<in> A" by blast

   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)

   with `finite A` have "finite B" by simp

   interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof

   qed (simp_all add: fun_eq_iff ac_simps)

   thm fold.commute_comp' [of B b, simplified fun_eq_iff, simplified]

   from `finite B` fold.commute_comp' [of B x]

     have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp

   then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)

   from `finite B` * fold.insert [of B b]

     have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp

   then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)

   from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)

 qed

 lemma remove:

   assumes "finite A" and "x \<in> 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 \<notin> 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 \<in> A") (simp_all add: insert_absorb remove)

 lemma union_disjoint:

   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"

   shows "F (A \<union> 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 \<inter> B \<noteq> {}"

   shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"

 proof -

   from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto

   from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` 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 \<in> B")

       case True then have "B \<noteq> {}" by auto

       with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")

         (simp_all add: insert_absorb ac_simps union_disjoint)

     next

       case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp

       moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"

         by auto

       ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)

     qed

   qed

 qed

 lemma closed:

   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"

   shows "F A \<in> A"

 using `finite A` `A \<noteq> {}` 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 \<in> A"

   shows "x * F A = F A"

 proof -

   from assms have "A \<noteq> {}" by auto

   with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)

 qed

 lemma subset_idem:

   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"

   shows "F B * F A = F A"

 proof -

   from assms have "finite B" by (blast dest: finite_subset)

   then show ?thesis using `B \<noteq> {}` `B \<subseteq> 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 \<noteq> {}"

   shows "F (insert x A) = x * F A"

 proof (cases "x \<in> A")

   case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)

 next

   case True

   from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)

 qed

 lemma union_idem:

   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"

   shows "F (A \<union> B) = F A * F B"

 proof (cases "A \<inter> B = {}")

   case True with assms show ?thesis by (simp add: union_disjoint)

 next

   case False

   from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto

   with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> 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: "\<And>x y. h (x * y) = h x * h y"

   and N: "finite N" "N \<noteq> {}" 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 "\<dots> = h n * h (F N)" by (rule hom)

   also have "h (F N) = F (h ` N)" by(rule insert)

   also have "h n * \<dots> = 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 \<Rightarrow> nat" where

   "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"

 interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof

 qed (simp add: card_def)

 lemma card_infinite [simp]:

   "\<not> finite A \<Longrightarrow> card A = 0"

   by (simp add: card_def)

 lemma card_empty:

   "card {} = 0"

   by (fact card.empty)

 lemma card_insert_disjoint:

   "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"

   by simp

 lemma card_insert_if:

   "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"

   by auto (simp add: card.insert_remove card.remove)

 lemma card_ge_0_finite:

   "card A > 0 \<Longrightarrow> finite A"

   by (rule ccontr) simp

 lemma card_0_eq [simp, no_atp]:

   "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"

   by (auto dest: mk_disjoint_insert)

 lemma finite_UNIV_card_ge_0:

   "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"

   by (rule ccontr) simp

 lemma card_eq_0_iff:

   "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"

   by auto

 lemma card_gt_0_iff:

   "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"

   by (simp add: neq0_conv [symmetric] card_eq_0_iff) 

 lemma card_Suc_Diff1: "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_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 \<subseteq> B"

   shows "card A \<le> card B"

 proof -

   from assms have "finite A" by (auto intro: finite_subset)

   then show ?thesis using assms proof (induct A arbitrary: B)

     case empty then show ?case by simp

   next

     case (insert x A)

     then have "x \<in> B" by simp

     from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto

     with insert.hyps have "card A \<le> card (B - {x})" by auto

     with `finite A` `x \<notin> A` `finite B` `x \<in> B` show ?case by simp (simp only: card.remove)

   qed

 qed

 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"

 apply (induct rule: finite_induct)

 apply simp

 apply clarify

 apply (subgoal_tac "finite A & A - {x} <= F")

  prefer 2 apply (blast intro: finite_subset, atomize)

 apply (drule_tac x = "A - {x}" in spec)

 apply (simp add: card_Diff_singleton_if split 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 \<subseteq> 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 \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"

 proof -

   have "A - B = A - A \<inter> B" by auto

   thus ?thesis

     by (simp add: card_Diff_subset AB) 

 qed

 lemma diff_card_le_card_Diff:

 assumes "finite B" shows "card A - card B \<le> card(A - B)"

 proof-

   have "card A - card B \<le> card A - card (A \<inter> B)"

     using card_mono[OF assms Int_lower2, of A] by arith

   also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)

   finally show ?thesis .

 qed

 lemma card_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 \<subseteq> B ==> card A < card B ==> A < B"

 by (erule psubsetI, blast)

 lemma insert_partition:

   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>

   \<Longrightarrow> x \<inter> \<Union> F = {}"

 by auto

 lemma finite_psubset_induct[consumes 1, case_names psubset]:

   assumes fin: "finite A" 

   and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> 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: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact

   { fix B 

     assume asm: "B \<subset> A"

     from asm have "card B < card A" using psubset_card_mono fin by blast

     moreover

     from asm have "B \<subseteq> 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 (\<Union> C) -->

      (\<forall>c\<in>C. card c = k) -->

      (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->

      k * card(C) = card (\<Union> C)"

 apply (erule finite_induct, simp)

 apply (simp add: card_Un_disjoint insert_partition 

        finite_subset [of _ "\<Union> (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 \<subseteq> UNIV" by simp

   show "UNIV \<subseteq> A"

   proof

     fix x

     show "x \<in> A"

     proof (rule ccontr)

       assume "x \<notin> A"

       then have "A \<subset> UNIV" by auto

       with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)

       with card show False by simp

     qed

   qed

 qed

 text{*The form of a finite set of given cardinality*}

 lemma card_eq_SucD:

 assumes "card A = Suc k"

 shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"

 proof -

   have fin: "finite A" using assms by (auto intro: ccontr)

   moreover have "card A \<noteq> 0" using assms by auto

   ultimately obtain b where b: "b \<in> A" by auto

   show ?thesis

   proof (intro exI conjI)

     show "A = insert b (A-{b})" using b by blast

     show "b \<notin> A - {b}" by blast

     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"

       using assms b fin by(fastsimp dest:mk_disjoint_insert)+

   qed

 qed

 lemma card_Suc_eq:

   "(card A = Suc k) =

    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> 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 \<Rightarrow> 'b) set)"

   shows "finite (UNIV :: 'b set)"

 proof -

   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"

     by(rule finite_imageI)

   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> '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 rule: finite_induct)

  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 "\<not> 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 \<Longrightarrow> card A = card B"

 by(auto simp: card_image bij_betw_def)

 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> 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 \<subseteq> B; finite B |] ==> card A \<le> 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 \<subseteq> B; inj_on g B; g ` B \<subseteq> A;

      finite A; finite B |] ==> card A = card B"

 by (auto intro: le_antisym card_inj_on_le)

 lemma bij_betw_finite:

   assumes "bij_betw f A B"

   shows "finite A \<longleftrightarrow> finite B"

 using assms unfolding bij_betw_def

 using finite_imageD[of f A] by auto

 subsubsection {* Pigeonhole Principles *}

 lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "

 by (auto dest: card_image less_irrefl_nat)

 lemma pigeonhole_infinite:

 assumes  "~ finite A" and "finite(f`A)"

 shows "EX a0:A. ~finite{a:A. f a = f a0}"

 proof -

   have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"

   proof(induct "f`A" arbitrary: A rule: finite_induct)

     case empty thus ?case by simp

   next

     case (insert b F)

     show ?case

     proof cases

       assume "finite{a:A. f a = b}"

       hence "~ finite(A - {a:A. f a = b})" using `\<not> finite A` by simp

       also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast

       finally have "~ finite({a:A. f a \<noteq> b})" .

       from insert(3)[OF _ this]

       show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)

     next

       assume 1: "~finite{a:A. f a = b}"

       hence "{a \<in> A. f a = b} \<noteq> {}" by force

       thus ?thesis using 1 by blast

     qed

   qed

   from this[OF assms(2,1)] show ?thesis .

 qed

 lemma pigeonhole_infinite_rel:

 assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"

 shows "EX b:B. ~finite{a:A. R a b}"

 proof -

    let ?F = "%a. {b:B. R a b}"

    from finite_Pow_iff[THEN iffD2, OF `finite B`]

    have "finite(?F ` A)" by(blast intro: rev_finite_subset)

    from pigeonhole_infinite[where f = ?F, OF assms(1) this]

    obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..

    obtain b0 where "b0 : B" and "R a0 b0" using `a0:A` assms(3) by blast

    { assume "finite{a:A. R a b0}"

      then have "finite {a\<in>A. ?F a = ?F a0}"

        using `b0 : B` `R a0 b0` by(blast intro: rev_finite_subset)

    }

    with 1 `b0 : B` show ?thesis by blast

 qed

 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 \<inter> 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 \<and> 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 rule: finite_induct)

  apply (simp_all add: Pow_insert)

 apply (subst card_Un_disjoint, blast)

   apply (blast, 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. Kamm\"uller.  *}

 lemma dvd_partition:

   "finite (Union C) ==>

     ALL c : C. k dvd card c ==>

     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>

   k dvd card (Union C)"

 apply (frule finite_UnionD)

 apply (rotate_tac -1)

 apply (induct rule: finite_induct)

 apply simp_all

 apply 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 \<Longrightarrow> A \<subseteq> f ` A \<Longrightarrow> 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 \<Rightarrow> 'a"

 shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"

 by (blast intro: finite_surj_inj subset_UNIV)

 lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"

 shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> 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]:

   "\<not> 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 \<Rightarrow> 'a)"

     by (simp add: inj_on_def)

   ultimately have "finite (UNIV::nat set)"

     by (rule finite_imageD)

   then show "False"

     by simp

 qed

 end