(*  Title:      HOL/Finite_Set.thy

    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

                with contributions by Jeremy Avigad

*)

header {* Finite sets *}

theory Finite_Set

imports Power Option

begin

subsection {* Predicate for finite sets *}

inductive finite :: "'a set => bool"

  where

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

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

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

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

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

proof -

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

  thus ?thesis by blast

qed

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

  "finite F ==>

    P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"

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

proof -

  assume "P {}" and

    insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"

  assume "finite F"

  thus "P F"

  proof induct

    show "P {}" by fact

    fix x F assume F: "finite F" and P: "P F"

    show "P (insert x F)"

    proof cases

      assume "x \<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

qed

lemma finite_ne_induct[case_names singleton insert, consumes 2]:

assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>

 \<lbrakk> \<And>x. P{x};

   \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>

 \<Longrightarrow> P F"

using fin

proof induct

  case empty thus ?case by simp

next

  case (insert x F)

  show ?case

  proof cases

    assume "F = {}"

    thus ?thesis using `P {x}` by simp

  next

    assume "F \<noteq> {}"

    thus ?thesis using insert by blast

  qed

qed

lemma finite_subset_induct [consumes 2, case_names empty insert]:

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

    and empty: "P {}"

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

  shows "P F"

proof -

  from `finite F` and `F \<subseteq> A`

  show ?thesis

  proof induct

    show "P {}" by fact

  next

    fix x F

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

      P: "F \<subseteq> A ==> 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

qed

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

lemma finite_set_choice:

  "finite A \<Longrightarrow> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)"

proof (induct set: finite)

  case empty thus ?case by simp

next

  case (insert a A)

  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto

  show ?case (is "EX f. ?P f")

  proof

    show "?P(%x. if x = a then b else f x)" using f ab by auto

  qed

qed

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

lemma finite_imp_nat_seg_image_inj_on:

  assumes fin: "finite A" 

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

using fin

proof induct

  case empty

  show ?case  

  proof show "\<exists>f. {} = f ` {i::nat. i < 0} & 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:

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

proof (induct n)

  case 0 thus ?case by simp

next

  case (Suc n)

  let ?B = "f ` {i. i < n}"

  have finB: "finite ?B" by(rule Suc.hyps[OF refl])

  show ?case

  proof cases

    assume "\<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 = (\<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 "EX f n::nat. f`A = {i. i<n} & inj_on f A"

proof -

  from finite_imp_nat_seg_image_inj_on[OF `finite A`]

  obtain f and n::nat where bij: "bij_betw f {i. i<n} A"

    by (auto simp:bij_betw_def)

  let ?f = "the_inv_into {i. i<n} f"

  have "inj_on ?f A & ?f ` A = {i. i<n}"

    by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])

  thus ?thesis by blast

qed

lemma finite_Collect_less_nat[iff]: "finite{n::nat. n<k}"

by(fastsimp simp: finite_conv_nat_seg_image)

text {* Finiteness and set theoretic constructions *}

lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"

by (induct set: finite) simp_all

lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"

  -- {* Every subset of a finite set is finite. *}

proof -

  assume "finite B"

  thus "!!A. A \<subseteq> B ==> finite A"

  proof induct

    case empty

    thus ?case by simp

  next

    case (insert x F A)

    have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> 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

qed

lemma rev_finite_subset: "finite B ==> A \<subseteq> B ==> finite A"

by (rule finite_subset)

lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"

by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)

lemma finite_Collect_disjI[simp]:

  "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"

by(simp add:Collect_disj_eq)

lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"

  -- {* The converse obviously fails. *}

by (blast intro: finite_subset)

lemma finite_Collect_conjI [simp, intro]:

  "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"

  -- {* The converse obviously fails. *}

by(simp add:Collect_conj_eq)

lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}"

by(simp add: le_eq_less_or_eq)

lemma finite_insert [simp]: "finite (insert a A) = finite A"

  apply (subst insert_is_Un)

  apply (simp only: finite_Un, blast)

  done

lemma finite_Union[simp, intro]:

 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"

by (induct rule:finite_induct) simp_all

lemma finite_Inter[intro]: "EX A:M. finite(A) \<Longrightarrow> finite(Inter M)"

by (blast intro: Inter_lower finite_subset)

lemma finite_INT[intro]: "EX x:I. finite(A x) \<Longrightarrow> finite(INT x:I. A x)"

by (blast intro: INT_lower finite_subset)

lemma finite_empty_induct:

  assumes "finite A"

    and "P A"

    and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"

  shows "P {}"

proof -

  have "P (A - A)"

  proof -

    {

      fix c b :: "'a set"

      assume c: "finite c" and b: "finite b"

        and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"

      have "c \<subseteq> b ==> P (b - c)"

        using c

      proof induct

        case empty

        from P1 show ?case by simp

      next

        case (insert x F)

        have "P (b - F - {x})"

        proof (rule P2)

          from _ b show "finite (b - F)" by (rule finite_subset) blast

          from insert show "x \<in> b - F" by simp

          from insert show "P (b - F)" by simp

        qed

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

        finally show ?case .

      qed

    }

    then show ?thesis by this (simp_all add: assms)

  qed

  then show ?thesis by simp

qed

lemma finite_Diff [simp]: "finite A ==> finite (A - B)"

by (rule Diff_subset [THEN finite_subset])

lemma finite_Diff2 [simp]:

  assumes "finite B" shows "finite (A - B) = finite A"

proof -

  have "finite A \<longleftrightarrow> finite((A-B) Un (A Int 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_compl[simp]:

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

by(simp add:Compl_eq_Diff_UNIV)

lemma finite_Collect_not[simp]:

  "finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)"

by(simp add:Collect_neg_eq)

lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"

  apply (subst Diff_insert)

  apply (case_tac "a : A - B")

   apply (rule finite_insert [symmetric, THEN trans])

   apply (subst insert_Diff, simp_all)

  done

text {* Image and Inverse Image over Finite Sets *}

lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"

  -- {* The image of a finite set is finite. *}

  by (induct set: finite) simp_all

lemma finite_image_set [simp]:

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

  by (simp add: image_Collect [symmetric])

lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"

  apply (frule finite_imageI)

  apply (erule finite_subset, assumption)

  done

lemma finite_range_imageI:

    "finite (range g) ==> finite (range (%x. f (g x)))"

  apply (drule finite_imageI, simp add: range_composition)

  done

lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"

proof -

  have aux: "!!A. finite (A - {}) = finite A" by simp

  fix B :: "'a set"

  assume "finite B"

  thus "!!A. f`A = B ==> inj_on f A ==> finite A"

    apply induct

     apply simp

    apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")

     apply clarify

     apply (simp (no_asm_use) add: inj_on_def)

     apply (blast dest!: aux [THEN iffD1], atomize)

    apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)

    apply (frule subsetD [OF equalityD2 insertI1], clarify)

    apply (rule_tac x = xa in bexI)

     apply (simp_all add: inj_on_image_set_diff)

    done

qed (rule refl)

lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"

  -- {* The inverse image of a singleton under an injective function

         is included in a singleton. *}

  apply (auto simp add: inj_on_def)

  apply (blast intro: the_equality [symmetric])

  done

lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"

  -- {* The inverse image of a finite set under an injective function

         is finite. *}

  apply (induct set: finite)

   apply simp_all

  apply (subst vimage_insert)

  apply (simp add: finite_subset [OF inj_vimage_singleton])

  done

lemma finite_vimageD:

  assumes fin: "finite (h -` F)" and surj: "surj h"

  shows "finite F"

proof -

  have "finite (h ` (h -` F))" using fin by (rule finite_imageI)

  also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)

  finally show "finite F" .

qed

lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"

  unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)

text {* The finite UNION of finite sets *}

lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"

  by (induct set: finite) simp_all

text {*

  Strengthen RHS to

  @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?

  We'd need to prove

  @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}

  by induction. *}

lemma finite_UN [simp]:

  "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"

by (blast intro: finite_UN_I finite_subset)

lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow>

  finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})"

apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})")

 apply auto

done

lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow>

  finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})"

apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})")

 apply auto

done

lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"

by (simp add: Plus_def)

lemma finite_PlusD: 

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

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

  shows "finite A" "finite B"

proof -

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

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

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

next

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

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

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

qed

lemma finite_Plus_iff[simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"

by(auto intro: finite_PlusD finite_Plus)

lemma finite_Plus_UNIV_iff[simp]:

  "finite (UNIV :: ('a + 'b) set) =

  (finite (UNIV :: 'a set) & finite (UNIV :: 'b set))"

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

text {* Sigma of finite sets *}

lemma finite_SigmaI [simp]:

    "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"

  by (unfold Sigma_def) (blast intro!: finite_UN_I)

lemma finite_cartesian_product: "[| finite A; finite B |] ==>

    finite (A <*> B)"

  by (rule finite_SigmaI)

lemma finite_Prod_UNIV:

    "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"

  apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")

   apply (erule ssubst)

   apply (erule finite_SigmaI, auto)

  done

lemma finite_cartesian_productD1:

     "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"

apply (auto simp add: finite_conv_nat_seg_image) 

apply (drule_tac x=n in spec) 

apply (drule_tac x="fst o f" in spec) 

apply (auto simp add: o_def) 

 prefer 2 apply (force dest!: equalityD2) 

apply (drule equalityD1) 

apply (rename_tac y x)

apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 

 prefer 2 apply force

apply clarify

apply (rule_tac x=k in image_eqI, auto)

done

lemma finite_cartesian_productD2:

     "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"

apply (auto simp add: finite_conv_nat_seg_image) 

apply (drule_tac x=n in spec) 

apply (drule_tac x="snd o f" in spec) 

apply (auto simp add: o_def) 

 prefer 2 apply (force dest!: equalityD2) 

apply (drule equalityD1)

apply (rename_tac x y)

apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 

 prefer 2 apply force

apply clarify

apply (rule_tac x=k in image_eqI, auto)

done

text {* The powerset of a finite set *}

lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"

proof

  assume "finite (Pow A)"

  with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast

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

next

  assume "finite A"

  thus "finite (Pow A)"

    by induct (simp_all add: Pow_insert)

qed

lemma 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])

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 thus ?case by simp

next

  case insert thus ?case

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

       blast

qed

subsection {* Class @{text finite}  *}

setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}

class finite =

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

setup {* Sign.parent_path *}

hide const finite

context finite

begin

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

  by (rule subset_UNIV finite_UNIV finite_subset)+

end

lemma UNIV_unit [no_atp]:

  "UNIV = {()}" by auto

instance unit :: finite proof

qed (simp add: UNIV_unit)

lemma UNIV_bool [no_atp]:

  "UNIV = {False, True}" by auto

instance bool :: finite proof

qed (simp add: UNIV_bool)

instance * :: (finite, finite) finite proof

qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)

lemma finite_option_UNIV [simp]:

  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"

  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)

instance option :: (finite) finite proof

qed (simp add: UNIV_option_conv)

lemma inj_graph: "inj (%f. {(x, y). y = f x})"

  by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)

instance "fun" :: (finite, finite) finite

proof

  show "finite (UNIV :: ('a => 'b) set)"

  proof (rule finite_imageD)

    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"

    have "range ?graph \<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 "+" :: (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 fun_left_comm: "f x (f y z) = f y (f x z)"

begin

text{* On a functional level it looks much nicer: *}

lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"

by (simp add: fun_left_comm expand_fun_eq)

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

[code del]: "fold f z A = (THE y. fold_graph f z A y)"

text{*A tempting alternative for the definiens is

@{term "if finite A then THE y. fold_graph f z A y else e"}.

It allows the removal of finiteness assumptions from the theorems

@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.

The proofs become ugly. It is not worth the effort. (???) *}

lemma Diff1_fold_graph:

  "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"

by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)

lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"

by (induct set: fold_graph) auto

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

by (induct set: finite) auto

subsubsection{*From @{const fold_graph} to @{term fold}*}

lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"

  by (auto simp add: less_Suc_eq) 

lemma insert_image_inj_on_eq:

     "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 

        inj_on h {i. i < Suc m}|] 

      ==> A = h ` {i. i < m}"

apply (auto simp add: image_less_Suc inj_on_def)

apply (blast intro: less_trans) 

done

lemma insert_inj_onE:

  assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 

      and inj_on: "inj_on h {i::nat. i<n}"

  shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"

proof (cases n)

  case 0 thus ?thesis using aA by auto

next

  case (Suc m)

  have nSuc: "n = Suc m" by fact

  have mlessn: "m<n" by (simp add: nSuc)

  from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)

  let ?hm = "Fun.swap k m h"

  have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 

    by (simp add: inj_on)

  show ?thesis

  proof (intro exI conjI)

    show "inj_on ?hm {i. i < m}" using inj_hm

      by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)

    show "m<n" by (rule mlessn)

    show "A = ?hm ` {i. i < m}" 

    proof (rule insert_image_inj_on_eq)

      show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp

      show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 

      show "insert (?hm m) A = ?hm ` {i. i < Suc m}"

        using aA hkeq nSuc klessn

        by (auto simp add: swap_def image_less_Suc fun_upd_image 

                           less_Suc_eq inj_on_image_set_diff [OF inj_on])

    qed

  qed

qed

context fun_left_comm

begin

lemma fold_graph_determ_aux:

  "A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n}

   \<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x'

   \<Longrightarrow> x' = x"

proof (induct n arbitrary: A x x' h rule: less_induct)

  case (less n)

  have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m}

      \<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x

      \<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact

  have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'"

    and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+

  show ?case

  proof (rule fold_graph.cases [OF Afoldx])

    assume "A = {}" and "x = z"

    with Afoldx' show "x' = x" by auto

  next

    fix B b u

    assume AbB: "A = insert b B" and x: "x = f b u"

      and notinB: "b \<notin> B" and Bu: "fold_graph f z B u"

    show "x'=x" 

    proof (rule fold_graph.cases [OF Afoldx'])

      assume "A = {}" and "x' = z"

      with AbB show "x' = x" by blast

    next

      fix C c v

      assume AcC: "A = insert c C" and x': "x' = f c v"

        and notinC: "c \<notin> C" and Cv: "fold_graph f z C v"

      from A AbB have Beq: "insert b B = h`{i. i<n}" by simp

      from insert_inj_onE [OF Beq notinB injh]

      obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 

        and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto 

      from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp

      from insert_inj_onE [OF Ceq notinC injh]

      obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"

        and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto 

      show "x'=x"

      proof cases

        assume "b=c"

        then moreover have "B = C" using AbB AcC notinB notinC by auto

        ultimately show ?thesis  using Bu Cv x x' IH [OF lessC Ceq inj_onC]

          by auto

      next

        assume diff: "b \<noteq> c"

        let ?D = "B - {c}"

        have B: "B = insert c ?D" and C: "C = insert b ?D"

          using AbB AcC notinB notinC diff by(blast elim!:equalityE)+

        have "finite A" by(rule fold_graph_imp_finite [OF Afoldx])

        with AbB have "finite ?D" by simp

        then obtain d where Dfoldd: "fold_graph f z ?D d"

          using finite_imp_fold_graph by iprover

        moreover have cinB: "c \<in> B" using B by auto

        ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph)

        hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 

        moreover have "f b d = v"

        proof (rule IH[OF lessC Ceq inj_onC Cv])

          show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp

        qed

        ultimately show ?thesis

          using fun_left_comm [of c b] x x' by (auto simp add: o_def)

      qed

    qed

  qed

qed

lemma fold_graph_determ:

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

apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 

apply (blast intro: fold_graph_determ_aux [rule_format])

done

lemma fold_equality:

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

by (unfold fold_def) (blast intro: fold_graph_determ)

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_aux: "x \<notin> A

  \<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow>

      (\<exists>y. fold_graph f z A y \<and> v = f x y)"

apply auto

apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE])

 apply (fastsimp dest: fold_graph_imp_finite)

apply (blast intro: fold_graph_determ)

done

lemma fold_insert [simp]:

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

apply (simp add: fold_def fold_insert_aux)

apply (rule the_equality)

 apply (auto intro: finite_imp_fold_graph

        cong add: conj_cong simp add: fold_def[symmetric] fold_equality)

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 expand_fun_eq)

lemma fold_insert_idem:

  assumes fin: "finite A"

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

proof cases

  assume "x \<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: fun_left_comm)

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_empty Inf_insert inf_commute fold_fun_comm)

qed

lemma sup_Sup_fold_sup:

  assumes "finite A"

  shows "sup B (Sup A) = fold sup B A"

proof -

  interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)

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

    (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm)

qed

lemma Inf_fold_inf:

  assumes "finite A"

  shows "Inf A = fold inf top A"

  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)

lemma Sup_fold_sup:

  assumes "finite A"

  shows "Sup A = fold sup bot A"

  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)

lemma inf_INFI_fold_inf:

  assumes "finite A"

  shows "inf B (INFI A f) = fold (\<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_empty 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_empty 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"

    by unfold_locales (simp add: mult_ac)

  show ?thesis using assms by(simp add:fold_image_def)

qed

(*

lemma fold_commute:

  "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"

  apply (induct set: finite)

   apply simp

  apply (simp add: mult_left_commute [of x])

  done

lemma fold_nest_Un_Int:

  "finite A ==> finite B

    ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"

  apply (induct set: finite)

   apply simp

  apply (simp add: fold_commute Int_insert_left insert_absorb)

  done

lemma fold_nest_Un_disjoint:

  "finite A ==> finite B ==> A Int B = {}

    ==> fold times g z (A Un B) = fold times g (fold times g z B) A"

  by (simp add: fold_nest_Un_Int)

*)

lemma fold_image_reindex:

assumes fin: "finite A"

shows "inj_on h A \<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,