src/HOL/Finite_Set.thy

added finite_set_choice

(*  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 Nat Product_Type Power
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_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 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_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_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_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)

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_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])


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
  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)

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
  by default (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_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 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_insert fold_fun_comm)

lemma fold_rec:
assumes "finite A" and "x \<in> A"
shows "fold f z A = f x (fold f z (A - {x}))"
proof -
  have A: "A = insert x (A - {x})" using `x \<in> A` by blast
  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
  also have "\<dots> = f x (fold f z (A - {x}))"
    by (rule fold_insert) (simp add: `finite A`)+
  finally show ?thesis .
qed

lemma fold_insert_remove:
  assumes "finite A"
  shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
proof -
  from `finite A` have "finite (insert x A)" by auto
  moreover have "x \<in> insert x A" by auto
  ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
    by (rule fold_rec)
  then show ?thesis by simp
qed

end

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

locale fun_left_comm_idem = fun_left_comm +
  assumes fun_left_idem: "f x (f x z) = f x z"
begin

text{* The nice version: *}
lemma fun_comp_idem : "f x o f x = f x"
by (simp add: fun_left_idem expand_fun_eq)

lemma fold_insert_idem:
  assumes fin: "finite A"
  shows "fold f z (insert x A) = f x (fold f z A)"
proof cases
  assume "x \<in> A"
  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
  then show ?thesis using assms by (simp add:fun_left_idem)
next
  assume "x \<notin> A" then show ?thesis using assms by simp
qed

declare fold_insert[simp del] fold_insert_idem[simp]

lemma fold_insert_idem2:
  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
by(simp add:fold_fun_comm)

end

subsubsection{* 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 I.fold_insert)
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 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:
  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)

end


subsection {* Generalized summation over a set *}

interpretation comm_monoid_add!: comm_monoid_mult "0::'a::comm_monoid_add" "op +"
  proof qed (auto intro: add_assoc add_commute)

definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
where "setsum f A == if finite A then fold_image (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" == "CONST setsum (%i. b) A"
  "\<Sum>i\<in>A. b" == "CONST 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" => "CONST setsum (%x. t) {x. P}"
  "\<Sum>x|P. t" => "CONST 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_image_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_reindex_nonzero: 
  assumes fS: "finite S"
  and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
  shows "setsum h (f ` S) = setsum (h o f) S"
using nz
proof(induct rule: finite_induct[OF fS])
  case 1 thus ?case by simp
next
  case (2 x F) 
  {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
    then obtain y where y: "y \<in> F" "f x = f y" by auto 
    from "2.hyps" y have xy: "x \<noteq> y" by auto
    
    from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
    also have "\<dots> = setsum (h o f) (insert x F)" 
      unfolding setsum_insert[OF `finite F` `x\<notin>F`]
      using h0 
      apply simp
      apply (rule "2.hyps"(3))
      apply (rule_tac y="y" in  "2.prems")
      apply simp_all
      done
    finally have ?case .}
  moreover
  {assume fxF: "f x \<notin> f ` F"
    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
      using fxF "2.hyps" by simp 
    also have "\<dots> = setsum (h o f) (insert x F)"
      unfolding setsum_insert[OF `finite F` `x\<notin>F`]
      apply simp
      apply (rule cong[OF refl[of "op + (h (f x))"]])
      apply (rule "2.hyps"(3))
      apply (rule_tac y="y" in  "2.prems")
      apply simp_all
      done
    finally have ?case .}
  ultimately show ?case by blast
qed

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_image_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_image_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_image_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)

lemma setsum_mono_zero_left: 
  assumes fT: "finite T" and ST: "S \<subseteq> T"
  and z: "\<forall>i \<in> T - S. f i = 0"
  shows "setsum f S = setsum f T"
proof-
  have eq: "T = S \<union> (T - S)" using ST by blast
  have d: "S \<inter> (T - S) = {}" using ST by blast
  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
  show ?thesis 
  by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
qed

lemma setsum_mono_zero_right: 
  assumes fT: "finite T" and ST: "S \<subseteq> T"
  and z: "\<forall>i \<in> T - S. f i = 0"
  shows "setsum f T = setsum f S"
using setsum_mono_zero_left[OF fT ST z] by simp

lemma setsum_mono_zero_cong_left: 
  assumes fT: "finite T" and ST: "S \<subseteq> T"
  and z: "\<forall>i \<in> T - S. g i = 0"
  and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
  shows "setsum f S = setsum g T"
proof-
  have eq: "T = S \<union> (T - S)" using ST by blast
  have d: "S \<inter> (T - S) = {}" using ST by blast
  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
  show ?thesis 
    using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
qed

lemma setsum_mono_zero_cong_right: 
  assumes fT: "finite T" and ST: "S \<subseteq> T"
  and z: "\<forall>i \<in> T - S. f i = 0"
  and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
  shows "setsum f T = setsum g S"
using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto 

lemma setsum_delta: 
  assumes fS: "finite S"
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
proof-
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
  {assume a: "a \<notin> S"
    hence "\<forall> k\<in> S. ?f k = 0" by simp
    hence ?thesis  using a by simp}
  moreover 
  {assume a: "a \<in> S"
    let ?A = "S - {a}"
    let ?B = "{a}"
    have eq: "S = ?A \<union> ?B" using a by blast 
    have dj: "?A \<inter> ?B = {}" by simp
    from fS have fAB: "finite ?A" "finite ?B" by auto  
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
      by simp
    then have ?thesis  using a by simp}
  ultimately show ?thesis by blast
qed
lemma setsum_delta': 
  assumes fS: "finite S" shows 
  "setsum (\<lambda>k. if a = k then b k else 0) S = 
     (if a\<in> S then b a else 0)"
  using setsum_delta[OF fS, of a b, symmetric] 
  by (auto intro: setsum_cong)


(*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_image_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_image_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_image_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: algebra_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: algebra_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 assms
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