src/HOL/Finite_Set.thy

prove lemma finite in context of finite class

(*  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 Transitive_Closure
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 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_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}  *}

setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
class finite = itself +
  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
  by default (simp add: UNIV_unit)

lemma UNIV_bool [noatp]:
  "UNIV = {False, True}" by auto

instance bool :: finite
  by default (simp add: UNIV_bool)

instance * :: (finite, finite) finite
  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)

instance "+" :: (finite, finite) finite
  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)

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


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 = "Fun.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 (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 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)

lemma setsum_cong:
  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_cong)

lemma strong_setsum_cong[cong]:
  "A = B ==> (!!x. x:B =simp=> f x = g x)
   ==> setsum (%x. f x) A = setsum (%x. g x) B"
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_cong)

lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
  by (rule setsum_cong[OF refl], auto);

lemma setsum_reindex_cong:
     "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
      ==> setsum h B = setsum g A"
  by (simp add: setsum_reindex cong: setsum_cong)

lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
apply (clarsimp simp: setsum_def)
apply (erule finite_induct, auto)
done

lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
by(simp add:setsum_cong)

lemma setsum_Un_Int: "finite A ==> finite B ==>
  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
by(simp add: setsum_def comm_monoid_add.fold_Un_Int [symmetric])

lemma setsum_Un_disjoint: "finite A ==> finite B
  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
by (subst setsum_Un_Int [symmetric], auto)

(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
  the lhs need not be, since UNION I A could still be finite.*)
lemma setsum_UN_disjoint:
    "finite I ==> (ALL i:I. finite (A i)) ==>
        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
by(simp add: setsum_def comm_monoid_add.fold_UN_disjoint cong: setsum_cong)

text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
lemma setsum_Union_disjoint:
  "[| (ALL A:C. finite A);
      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
   ==> setsum f (Union C) = setsum (setsum f) C"
apply (cases "finite C") 
 prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
  apply (frule setsum_UN_disjoint [of C id f])
 apply (unfold Union_def id_def, assumption+)
done

(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
  the rhs need not be, since SIGMA A B could still be finite.*)
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
by(simp add:setsum_def comm_monoid_add.fold_Sigma split_def cong:setsum_cong)

text{*Here we can eliminate the finiteness assumptions, by cases.*}
lemma setsum_cartesian_product: 
   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
apply (cases "finite A") 
 apply (cases "finite B") 
  apply (simp add: setsum_Sigma)
 apply (cases "A={}", simp)
 apply (simp) 
apply (auto simp add: setsum_def
            dest: finite_cartesian_productD1 finite_cartesian_productD2) 
done

lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
by(simp add:setsum_def comm_monoid_add.fold_distrib)


subsubsection {* Properties in more restricted classes of structures *}

lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
  apply (case_tac "finite A")
   prefer 2 apply (simp add: setsum_def)
  apply (erule rev_mp)
  apply (erule finite_induct, auto)
  done

lemma setsum_eq_0_iff [simp]:
    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
  by (induct set: finite) auto

lemma setsum_Un_nat: "finite A ==> finite B ==>
    (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
  -- {* For the natural numbers, we have subtraction. *}
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)

lemma setsum_Un: "finite A ==> finite B ==>
    (setsum f (A Un B) :: 'a :: ab_group_add) =
      setsum f A + setsum f B - setsum f (A Int B)"
  by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)

lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
    (if a:A then setsum f A - f a else setsum f A)"
  apply (case_tac "finite A")
   prefer 2 apply (simp add: setsum_def)
  apply (erule finite_induct)
   apply (auto simp add: insert_Diff_if)
  apply (drule_tac a = a in mk_disjoint_insert, auto)
  done

lemma setsum_diff1: "finite A \<Longrightarrow>
  (setsum f (A - {a}) :: ('a::ab_group_add)) =
  (if a:A then setsum f A - f a else setsum f A)"
  by (erule finite_induct) (auto simp add: insert_Diff_if)

lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
  apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
  apply (auto simp add: insert_Diff_if add_ac)
  done

(* By Jeremy Siek: *)

lemma setsum_diff_nat: 
  assumes "finite B"
    and "B \<subseteq> A"
  shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
  using prems
proof induct
  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
next
  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
    and xFinA: "insert x F \<subseteq> A"
    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
    by (simp add: setsum_diff1_nat)
  from xFinA have "F \<subseteq> A" by simp
  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
    by simp
  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
    by simp
  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
    by simp
  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
qed

lemma setsum_diff:
  assumes le: "finite A" "B \<subseteq> A"
  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
proof -
  from le have finiteB: "finite B" using finite_subset by auto
  show ?thesis using finiteB le
  proof induct
    case empty
    thus ?case by auto
  next
    case (insert x F)
    thus ?case using le finiteB 
      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
  qed
qed

lemma setsum_mono:
  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
proof (cases "finite K")
  case True
  thus ?thesis using le
  proof induct
    case empty
    thus ?case by simp
  next
    case insert
    thus ?case using add_mono by fastsimp
  qed
next
  case False
  thus ?thesis
    by (simp add: setsum_def)
qed

lemma setsum_strict_mono:
  fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
  assumes "finite A"  "A \<noteq> {}"
    and "!!x. x:A \<Longrightarrow> f x < g x"
  shows "setsum f A < setsum g A"
  using prems
proof (induct rule: finite_ne_induct)
  case singleton thus ?case by simp
next
  case insert thus ?case by (auto simp: add_strict_mono)
qed

lemma setsum_negf:
  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
proof (cases "finite A")
  case True thus ?thesis by (induct set: finite) auto
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_subtractf:
  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
    setsum f A - setsum g A"
proof (cases "finite A")
  case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_nonneg:
  assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
  shows "0 \<le> setsum f A"
proof (cases "finite A")
  case True thus ?thesis using nn
  proof induct
    case empty then show ?case by simp
  next
    case (insert x F)
    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
    with insert show ?case by simp
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_nonpos:
  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
  shows "setsum f A \<le> 0"
proof (cases "finite A")
  case True thus ?thesis using np
  proof induct
    case empty then show ?case by simp
  next
    case (insert x F)
    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
    with insert show ?case by simp
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_mono2:
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
shows "setsum f A \<le> setsum f B"
proof -
  have "setsum f A \<le> setsum f A + setsum f (B-A)"
    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
  also have "A \<union> (B-A) = B" using sub by blast
  finally show ?thesis .
qed

lemma setsum_mono3: "finite B ==> A <= B ==> 
    ALL x: B - A. 
      0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
        setsum f A <= setsum f B"
  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
  apply (erule ssubst)
  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
  apply simp
  apply (rule add_left_mono)
  apply (erule setsum_nonneg)
  apply (subst setsum_Un_disjoint [THEN sym])
  apply (erule finite_subset, assumption)
  apply (rule finite_subset)
  prefer 2
  apply assumption
  apply auto
  apply (rule setsum_cong)
  apply auto
done

lemma setsum_right_distrib: 
  fixes f :: "'a => ('b::semiring_0)"
  shows "r * setsum f A = setsum (%n. r * f n) A"
proof (cases "finite A")
  case True
  thus ?thesis
  proof induct
    case empty thus ?case by simp
  next
    case (insert x A) thus ?case by (simp add: right_distrib)
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_left_distrib:
  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
proof (cases "finite A")
  case True
  then show ?thesis
  proof induct
    case empty thus ?case by simp
  next
    case (insert x A) thus ?case by (simp add: left_distrib)
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_divide_distrib:
  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
proof (cases "finite A")
  case True
  then show ?thesis
  proof induct
    case empty thus ?case by simp
  next
    case (insert x A) thus ?case by (simp add: add_divide_distrib)
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_abs[iff]: 
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
proof (cases "finite A")
  case True
  thus ?thesis
  proof induct
    case empty thus ?case by simp
  next
    case (insert x A)
    thus ?case by (auto intro: abs_triangle_ineq order_trans)
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma setsum_abs_ge_zero[iff]: 
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
  shows "0 \<le> setsum (%i. abs(f i)) A"
proof (cases "finite A")
  case True
  thus ?thesis
  proof induct
    case empty thus ?case by simp
  next
    case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed

lemma abs_setsum_abs[simp]: 
  fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
proof (cases "finite A")
  case True
  thus ?thesis
  proof induct
    case empty thus ?case by simp
  next
    case (insert a A)
    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
      by (simp del: abs_of_nonneg)
    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
    finally show ?case .
  qed
next
  case False thus ?thesis by (simp add: setsum_def)
qed


text {* Commuting outer and inner summation *}

lemma swap_inj_on:
  "inj_on (%(i, j). (j, i)) (A \<times> B)"
  by (unfold inj_on_def) fast

lemma swap_product:
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
  by (simp add: split_def image_def) blast

lemma setsum_commute:
  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
proof (simp add: setsum_cartesian_product)
  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
    (is "?s = _")
    apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
    apply (simp add: split_def)
    done
  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
    (is "_ = ?t")
    apply (simp add: swap_product)
    done
  finally show "?s = ?t" .
qed

lemma setsum_product:
  fixes f :: "'a => ('b::semiring_0)"
  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)