(*  Title:      HOL/Finite_Set.thy

    ID:         $Id$

    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

                with contributions by Jeremy Avigad

*)

header {* Finite sets *}

theory Finite_Set

imports Divides

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 prems 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{* 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)

subsubsection{* Finiteness and set theoretic constructions *}

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

  -- {* The union of two finite sets is finite. *}

  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 finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"

using finite_subset[of "{x \<in> A. P x}" "A"] by blast

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

  -- {* The converse obviously fails. *}

  by (blast intro: finite_subset)

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_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 B ==> finite (B - Ba)"

  by (rule Diff_subset [THEN finite_subset])

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

lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"

  by simp

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_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)

  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_Un finite_subset [OF inj_vimage_singleton])

  done

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

by (simp add: Plus_def)

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: finite_UnI finite_imageI Pow_insert)

qed

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

by(blast intro: finite_subset[OF subset_Pow_Union])

lemma finite_converse [iff]: "finite (r^-1) = finite r"

  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")

   apply simp

   apply (rule iffI)

    apply (erule finite_imageD [unfolded inj_on_def])

    apply (simp split add: split_split)

   apply (erule finite_imageI)

  apply (simp add: converse_def image_def, auto)

  apply (rule bexI)

   prefer 2 apply assumption

  apply simp

  done

text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi

Ehmety) *}

lemma finite_Field: "finite r ==> finite (Field r)"

  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}

  apply (induct set: finite)

   apply (auto simp add: Field_def Domain_insert Range_insert)

  done

lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"

  apply clarify

  apply (erule trancl_induct)

   apply (auto simp add: Field_def)

  done

lemma finite_trancl: "finite (r^+) = finite r"

  apply auto

   prefer 2

   apply (rule trancl_subset_Field2 [THEN finite_subset])

   apply (rule finite_SigmaI)

    prefer 3

    apply (blast intro: r_into_trancl' finite_subset)

   apply (auto simp add: finite_Field)

  done

subsection {* Class @{text finite} and code generation *}

lemma finite_code [code func]:

  "finite {} \<longleftrightarrow> True"

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

  by auto

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

class finite (attach UNIV) = type +

  fixes itself :: "'a itself"

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

setup {* Sign.parent_path *}

hide const finite

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

  by (rule finite_subset [OF subset_UNIV finite_UNIV])

lemma univ_unit [noatp]:

  "UNIV = {()}" by auto

instantiation unit :: finite

begin

definition

  "itself = TYPE(unit)"

instance proof

  have "finite {()}" by simp

  also note univ_unit [symmetric]

  finally show "finite (UNIV :: unit set)" .

qed

end

lemmas [code func] = univ_unit

lemma univ_bool [noatp]:

  "UNIV = {False, True}" by auto

instantiation bool :: finite

begin

definition

  "itself = TYPE(bool)"

instance proof

  have "finite {False, True}" by simp

  also note univ_bool [symmetric]

  finally show "finite (UNIV :: bool set)" .

qed

end

lemmas [code func] = univ_bool

instantiation * :: (finite, finite) finite

begin

definition

  "itself = TYPE('a \<times> 'b)"

instance proof

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

  proof (rule finite_Prod_UNIV)

    show "finite (UNIV :: 'a set)" by (rule finite)

    show "finite (UNIV :: 'b set)" by (rule finite)

  qed

qed

end

lemma univ_prod [noatp, code func]:

  "UNIV = (UNIV \<Colon> 'a\<Colon>finite set) \<times> (UNIV \<Colon> 'b\<Colon>finite set)"

  unfolding UNIV_Times_UNIV ..

instantiation "+" :: (finite, finite) finite

begin

definition

  "itself = TYPE('a + 'b)"

instance proof

  have a: "finite (UNIV :: 'a set)" by (rule finite)

  have b: "finite (UNIV :: 'b set)" by (rule finite)

  from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"

    by (rule finite_Plus)

  thus "finite (UNIV :: ('a + 'b) set)" by simp

qed

end

lemma univ_sum [noatp, code func]:

  "UNIV = (UNIV \<Colon> 'a\<Colon>finite set) <+> (UNIV \<Colon> 'b\<Colon>finite set)"

  unfolding UNIV_Plus_UNIV ..

instantiation set :: (finite) finite

begin

definition

  "itself = TYPE('a set)"

instance proof

  have "finite (UNIV :: 'a set)" by (rule finite)

  hence "finite (Pow (UNIV :: 'a set))"

    by (rule finite_Pow_iff [THEN iffD2])

  thus "finite (UNIV :: 'a set set)" by simp

qed

end

lemma univ_set [noatp, code func]:

  "UNIV = Pow (UNIV \<Colon> 'a\<Colon>finite set)" unfolding Pow_UNIV ..

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

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

instantiation "fun" :: (finite, finite) finite

begin

definition

  "itself \<equiv> TYPE('a \<Rightarrow> 'b)"

instance proof

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

  proof (rule finite_imageD)

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

    show "finite (range ?graph)" by (rule finite)

    show "inj ?graph" by (rule inj_graph)

  qed

qed

end

hide (open) const itself

subsection {* Equality and order on functions *}

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

lemma eq_fun [code func]:

  fixes f g :: "'a\<Colon>finite \<Rightarrow> 'b\<Colon>eq"

  shows "f = g \<longleftrightarrow> (\<forall>x\<in>UNIV. f x = g x)"

  unfolding expand_fun_eq by auto

lemma order_fun [code func]:

  fixes f g :: "'a\<Colon>finite \<Rightarrow> 'b\<Colon>order"

  shows "f \<le> g \<longleftrightarrow> (\<forall>x\<in>UNIV. f x \<le> g x)"

    and "f < g \<longleftrightarrow> f \<le> g \<and> (\<exists>x\<in>UNIV. f x \<noteq> g x)"

  by (auto simp add: expand_fun_eq le_fun_def less_fun_def order_less_le)

subsection {* A fold functional for finite sets *}

text {* The intended behaviour is

@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}

if @{text f} is associative-commutative. For an application of @{text fold}

se the definitions of sums and products over finite sets.

*}

inductive

  foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool"

  for f ::  "'a => 'a => 'a"

  and g :: "'b => 'a"

  and z :: 'a

where

  emptyI [intro]: "foldSet f g z {} z"

| insertI [intro]:

     "\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk>

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

inductive_cases empty_foldSetE [elim!]: "foldSet f g z {} x"

constdefs

  fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"

  "fold f g z A == THE x. foldSet f g z A x"

text{*A tempting alternative for the definiens is

@{term "if finite A then THE x. foldSet f g e A x else e"}.

It allows the removal of finiteness assumptions from the theorems

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

The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}

lemma Diff1_foldSet:

  "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)"

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

lemma foldSet_imp_finite: "foldSet f g z A x==> finite A"

  by (induct set: foldSet) auto

lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x"

  by (induct set: finite) auto

subsubsection{*From @{term foldSet} 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 = "swap k m h"

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

    by (simp add: inj_on_swap_iff 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 (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 ab_semigroup_mult

begin

lemma foldSet_determ_aux:

  "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 

                foldSet times g z A x; foldSet times g z A x' \<rbrakk>

   \<Longrightarrow> x' = x"

proof (induct n rule: less_induct)

  case (less n)

    have IH: "!!m h A x x'. 

               \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 

                foldSet times g z A x; foldSet times g z A x'\<rbrakk> \<Longrightarrow> x' = x" by fact

    have Afoldx: "foldSet times g z A x" and Afoldx': "foldSet times g z A x'"

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

    show ?case

    proof (rule foldSet.cases [OF Afoldx])

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

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

    next

      fix B b u

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

         and notinB: "b \<notin> B" and Bu: "foldSet times g z B u"

      show "x'=x" 

      proof (rule foldSet.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' = g c * v"

           and notinC: "c \<notin> C" and Cv: "foldSet times g 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 foldSet_imp_finite[OF Afoldx])

	  with AbB have "finite ?D" by simp

	  then obtain d where Dfoldd: "foldSet times g z ?D d"

	    using finite_imp_foldSet by iprover

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

	  ultimately have "foldSet times g z B (g c * d)"

	    by(rule Diff1_foldSet)

	  then have "g c * d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 

          then have "u = g c * d" ..

          moreover have "v = g b * d"

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

	    show "foldSet times g z C (g b * d)" using C notinB Dfoldd

	      by fastsimp

	  qed

	  ultimately show ?thesis using x x'

	    by (simp add: mult_left_commute)

	qed

      qed

    qed

  qed

lemma foldSet_determ:

  "foldSet times g z A x ==> foldSet times g z A y ==> y = x"

apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 

apply (blast intro: foldSet_determ_aux [rule_format])

done

lemma fold_equality: "foldSet times g z A y ==> fold times g z A = y"

  by (unfold fold_def) (blast intro: foldSet_determ)

text{* The base case for @{text fold}: *}

lemma (in -) fold_empty [simp]: "fold f g z {} = z"

  by (unfold fold_def) blast

lemma fold_insert_aux: "x \<notin> A ==>

    (foldSet times g z (insert x A) v) =

    (EX y. foldSet times g z A y & v = g x * y)"

  apply auto

  apply (rule_tac A1 = A and f1 = times in finite_imp_foldSet [THEN exE])

   apply (fastsimp dest: foldSet_imp_finite)

  apply (blast intro: foldSet_determ)

  done

text{* The recursion equation for @{text fold}: *}

lemma fold_insert [simp]:

    "finite A ==> x \<notin> A ==> fold times g z (insert x A) = g x * fold times g z A"

  apply (unfold fold_def)

  apply (simp add: fold_insert_aux)

  apply (rule the_equality)

  apply (auto intro: finite_imp_foldSet

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

  done

lemma fold_rec:

assumes fin: "finite A" and a: "a:A"

shows "fold times g z A = g a * fold times g z (A - {a})"

proof-

  have A: "A = insert a (A - {a})" using a by blast

  hence "fold times g z A = fold times g z (insert a (A - {a}))" by simp

  also have "\<dots> = g a * fold times g z (A - {a})"

    by(rule fold_insert) (simp add:fin)+

  finally show ?thesis .

qed

end

text{* A simplified version for idempotent functions: *}

context ab_semigroup_idem_mult

begin

lemma fold_insert_idem:

assumes finA: "finite A"

shows "fold times g z (insert a A) = g a * fold times g z A"

proof cases

  assume "a \<in> A"

  then obtain B where A: "A = insert a B" and disj: "a \<notin> B"

    by(blast dest: mk_disjoint_insert)

  show ?thesis

  proof -

    from finA A have finB: "finite B" by(blast intro: finite_subset)

    have "fold times g z (insert a A) = fold times g z (insert a B)" using A by simp

    also have "\<dots> = g a * fold times g z B"

      using finB disj by simp

    also have "\<dots> = g a * fold times g z A"

      using A finB disj

	by (simp add: mult_idem mult_assoc [symmetric])

    finally show ?thesis .

  qed

next

  assume "a \<notin> A"

  with finA show ?thesis by simp

qed

lemma foldI_conv_id:

  "finite A \<Longrightarrow> fold times g z A = fold times id z (g ` A)"

by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)

end

subsubsection{*Lemmas about @{text fold}*}

context ab_semigroup_mult

begin

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_reindex:

assumes fin: "finite A"

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

using fin apply induct

 apply simp

apply simp

done

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:

  includes 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"

  using fin hyp by (induct set: finite) simp_all

lemma fold_cong:

  "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold times g z A = fold times h z A"

  apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold times g z C = fold 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_Un_Int:

  "finite A ==> finite B ==>

    fold times g 1 A * fold times g 1 B =

    fold times g 1 (A Un B) * fold 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 times g 1 (A Un B) = fold times g 1 A * fold times g 1 B"

  by (simp add: fold_Un_Int)

lemma fold_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 times g 1 (UNION I A) =

       fold times (%i. fold 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_Sigma: "finite A ==> ALL x:A. finite (B x) ==>

  fold times (%x. fold times (g x) 1 (B x)) 1 A =

  fold times (split g) 1 (SIGMA x:A. B x)"

apply (subst Sigma_def)

apply (subst fold_UN_disjoint, assumption, simp)

 apply blast

apply (erule fold_cong)

apply (subst fold_UN_disjoint, simp, simp)

 apply blast

apply simp

done

lemma fold_distrib: "finite A \<Longrightarrow>

   fold times (%x. g x * h x) 1 A = fold times g 1 A *  fold times h 1 A"

  by (erule finite_induct) (simp_all add: mult_ac)

end

subsection {* Generalized summation over a set *}

interpretation comm_monoid_add: comm_monoid_mult ["0::'a::comm_monoid_add" "op +"]

  by unfold_locales (auto intro: add_assoc add_commute)

constdefs

  setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"

  "setsum f A == if finite A then fold (op +) f 0 A else 0"

abbreviation

  Setsum  ("\<Sum>_" [1000] 999) where

  "\<Sum>A == setsum (%x. x) A"

text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is

written @{text"\<Sum>x\<in>A. e"}. *}

syntax

  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)

syntax (xsymbols)

  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)

syntax (HTML output)

  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)

translations -- {* Beware of argument permutation! *}

  "SUM i:A. b" == "setsum (%i. b) A"

  "\<Sum>i\<in>A. b" == "setsum (%i. b) A"

text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter

 @{text"\<Sum>x|P. e"}. *}

syntax

  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)

syntax (xsymbols)

  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)

syntax (HTML output)

  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)

translations

  "SUM x|P. t" => "setsum (%x. t) {x. P}"

  "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"

print_translation {*

let

  fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 

    if x<>y then raise Match

    else let val x' = Syntax.mark_bound x

             val t' = subst_bound(x',t)

             val P' = subst_bound(x',P)

         in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end

in [("setsum", setsum_tr')] end

*}

lemma setsum_empty [simp]: "setsum f {} = 0"

  by (simp add: setsum_def)

lemma setsum_insert [simp]:

    "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"

  by (simp add: setsum_def)

lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"

  by (simp add: setsum_def)

lemma setsum_reindex:

     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"

by(auto simp add: setsum_def comm_monoid_add.fold_reindex dest!:finite_imageD)

lemma setsum_reindex_id:

     "inj_on f B ==> setsum f B = setsum id (f ` B)"

by (auto simp add: setsum_reindex)