(*  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 {* Definition and basic properties *}

 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)

 subsubsection{* 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 [noatp]:

   "UNIV = {()}" by auto

 instance unit :: finite proof

 qed (simp add: UNIV_unit)

 lemma UNIV_bool [noatp]:

   "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 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

 context ab_semigroup_idem_mult

 begin

 lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"

 apply unfold_locales

  apply (rule mult_left_commute)

 apply (rule mult_left_idem)

 done

 end

 context semilattice_inf

 begin

 lemma ab_semigroup_idem_mult_inf: "ab_semigroup_idem_mult inf"

 proof qed (rule inf_assoc inf_commute inf_idem)+

 lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> fold inf b (insert a A) = inf a (fold inf b A)"

 by(rule fun_left_comm_idem.fold_insert_idem[OF ab_semigroup_idem_mult.fun_left_comm_idem[OF ab_semigroup_idem_mult_inf]])

 lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> fold inf c A"

 by (induct pred: finite) (auto intro: le_infI1)

 lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> fold inf b A \<le> inf a b"

 proof(induct arbitrary: a pred:finite)

   case empty thus ?case by simp

 next

   case (insert x A)

   show ?case

   proof cases

     assume "A = {}" thus ?thesis using insert by simp

   next

     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)

   qed

 qed

 end

 context semilattice_sup

 begin

 lemma ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup"

 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)

 lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"

 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)

 lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"

 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)

 lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"

 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)

 end

 subsubsection{* 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,

   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_Un_Int:

   "finite A ==> finite B ==>

     fold_image times g 1 A * fold_image times g 1 B =

     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"

 by (induct set: finite) 

    (auto simp add: mult_ac insert_absorb Int_insert_left)

 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 set: finite, simp, 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 *)"

 by unfold_locales (simp add: 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 thus ?case by (force simp add: fold_insert_aux mult_commute)

 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 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 = x"

     thus ?thesis

     proof cases

       assume "A' = {}"

       with prems show ?thesis by simp

     next

       assume "A' \<noteq> {}"

       with prems show ?thesis

         by (simp add: fold1_insert mult_assoc [symmetric])

     qed

   next

     assume "a \<noteq> x"

     with prems show ?thesis

       by (simp add: insert_commute fold1_eq_fold)

   qed

 qed

 lemma hom_fold1_commute:

 assumes hom: "!!x y. h (x * y) = h x * h y"

 and N: "finite N" "N \<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 {* 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 {* Locales as mini-packages *}

 locale folding =

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

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

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

   assumes eq_fold: "F A s = Finite_Set.fold f s A"

 begin

 lemma fun_left_commute:

   "f x (f y s) = f y (f x s)"

   using commute_comp [of x y] by (simp add: expand_fun_eq)

 lemma fun_left_comm:

   "fun_left_comm f"

 proof

 qed (fact fun_left_commute)

 lemma empty [simp]:

   "F {} = id"

   by (simp add: eq_fold expand_fun_eq)

 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 by (fact fun_left_comm)

   from fold_insert2 assms

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

   then show ?thesis by (simp add: eq_fold expand_fun_eq)

 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"

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

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

 next

   case False with assms show ?thesis by simp

 qed

 lemma commute_comp':

   assumes "finite A"

   shows "f x \<circ> F A = F A \<circ> f x"

 proof (rule ext)

   fix s

   from assms show "(f x \<circ> F A) s = (F A \<circ> f x) s"

     by (induct A arbitrary: s) (simp_all add: fun_left_commute)

 qed

 lemma fun_left_commute':

   assumes "finite A"

   shows "f x (F A s) = F A (f x s)"

   using commute_comp' assms by (simp add: expand_fun_eq)

 lemma union:

   assumes "finite A" and "finite B"

   and "A \<inter> B = {}"

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

 using `finite A` `A \<inter> B = {}` proof (induct A)

   case empty show ?case by simp

 next

   case (insert x A)

   then have "A \<inter> B = {}" by auto

   with insert(3) have "F (A \<union> B) = F A \<circ> F B" .

   moreover from insert have "x \<notin> B" by simp

   moreover from `finite A` `finite B` have fin: "finite (A \<union> B)" by simp

   moreover from `x \<notin> A` `x \<notin> B` have "x \<notin> A \<union> B" by simp

   ultimately show ?case by (simp add: fun_left_commute')

 qed

 end

 locale folding_idem = folding +

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

 begin

 declare insert [simp del]

 lemma fun_idem:

   "f x (f x s) = f x s"

   using idem_comp [of x] by (simp add: expand_fun_eq)

 lemma fun_left_comm_idem:

   "fun_left_comm_idem f"

 proof

 qed (fact fun_left_commute fun_idem)+

 lemma insert_idem [simp]:

   assumes "finite A"

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

 proof -

   interpret fun_left_comm_idem f by (fact fun_left_comm_idem)

   from fold_insert_idem2 assms

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

   then show ?thesis by (simp add: eq_fold expand_fun_eq)

 qed

 lemma union_idem:

   assumes "finite A" and "finite B"

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

 using `finite A` proof (induct A)

   case empty show ?case by simp

 next

   case (insert x A)

   from insert(3) have "F (A \<union> B) = F A \<circ> F B" .

   moreover from `finite A` `finite B` have fin: "finite (A \<union> B)" by simp

   ultimately show ?case by (simp add: fun_left_commute')

 qed

 end

 end