(*  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 comp_fun_commute =

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

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

begin

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

  using comp_fun_commute 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 comp_fun_commute

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 "f x y = f a (f x y')"

      unfolding y by (rule fun_left_comm)

    moreover have "fold_graph f z (insert x A - {a}) (f x y')"

      using y' and `x \<noteq> a` and `x \<notin> A`

      by (simp add: insert_Diff_if fold_graph.insertI)

    ultimately 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]:

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

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

proof (rule fold_equality)

  from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)

  with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)

qed

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 comp_fun_idem = comp_fun_commute +

  assumes comp_fun_idem: "f x o f x = f x"

begin

lemma fun_left_idem: "f x (f x z) = f x z"

  using comp_fun_idem by (simp add: fun_eq_iff)

lemma fold_insert_idem:

  assumes fin: "finite A"

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

proof cases

  assume "x \<in> A"

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

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

  "comp_fun_commute (f \<circ> g)"

proof

qed (simp_all add: comp_fun_commute)

lemma (in comp_fun_idem) comp_comp_fun_idem:

  "comp_fun_idem (f \<circ> g)"

  by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)

    (simp_all add: comp_fun_idem)

lemma comp_fun_idem_insert:

  "comp_fun_idem insert"

proof

qed auto

lemma comp_fun_idem_remove:

  "comp_fun_idem (\<lambda>x A. A - {x})"

proof

qed auto

lemma (in semilattice_inf) comp_fun_idem_inf:

  "comp_fun_idem inf"

proof

qed (auto simp add: inf_left_commute)

lemma (in semilattice_sup) comp_fun_idem_sup:

  "comp_fun_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 comp_fun_idem insert by (fact comp_fun_idem_insert)

  from `finite A` show ?thesis by (induct A arbitrary: B) simp_all

qed

lemma minus_fold_remove:

  assumes "finite A"

  shows "B - A = fold (\<lambda>x A. A - {x}) B A"

proof -

  interpret comp_fun_idem "\<lambda>x A. A - {x}" by (fact comp_fun_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 comp_fun_idem inf by (fact comp_fun_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 comp_fun_idem sup by (fact comp_fun_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 (inf \<circ> f) B A" (is "?inf = ?fold") 

proof (rule sym)

  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)

  interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)

  from `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 (sup \<circ> f) B A" (is "?sup = ?fold") 

proof (rule sym)

  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)

  interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)

  from `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 (inf \<circ> f) 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 (sup \<circ> f) 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 (\<lambda>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 comp_fun_commute "%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_image_reindex:

  assumes "finite A"

  shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"

  using assms by induct auto

lemma fold_image_cong:

  assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"

  shows "fold_image times g z A = fold_image times h z A"

proof -

  from `finite A`

  have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"

  proof (induct arbitrary: C)

    case empty then show ?case by simp

  next

    case (insert x F) then show ?case apply -

    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 (simp add: Ball_def del: insert_Diff_single)

    done

  qed

  with g_h show ?thesis by simp

qed

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: