(*  Title:      HOL/Big_Operators.thy

     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

                 with contributions by Jeremy Avigad

 *)

 header {* Big operators and finite (non-empty) sets *}

 theory Big_Operators

 imports Plain

 begin

 subsection {* Generic monoid operation over a set *}

 no_notation times (infixl "*" 70)

 no_notation Groups.one ("1")

 locale comm_monoid_big = comm_monoid +

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

   assumes F_eq: "F g A = (if finite A then fold_image (op *) g 1 A else 1)"

 sublocale comm_monoid_big < folding_image proof

 qed (simp add: F_eq)

 context comm_monoid_big

 begin

 lemma infinite [simp]:

   "\<not> finite A \<Longrightarrow> F g A = 1"

   by (simp add: F_eq)

 lemma F_cong:

   assumes "A = B" "\<And>x. x \<in> B \<Longrightarrow> h x = g x"

   shows "F h A = F g B"

 proof cases

   assume "finite A"

   with assms show ?thesis unfolding `A = B` by (simp cong: cong)

 next

   assume "\<not> finite A"

   then show ?thesis unfolding `A = B` by simp

 qed

 lemma If_cases:

   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"

   assumes fA: "finite A"

   shows "F (\<lambda>x. if P x then h x else g x) A =

          F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"

 proof-

   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 

           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 

     by blast+

   from fA 

   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto

   let ?g = "\<lambda>x. if P x then h x else g x"

   from union_disjoint[OF f a(2), of ?g] a(1)

   show ?thesis

     by (subst (1 2) F_cong) simp_all

 qed

 end

 text {* for ad-hoc proofs for @{const fold_image} *}

 lemma (in comm_monoid_add) comm_monoid_mult:

   "class.comm_monoid_mult (op +) 0"

 proof qed (auto intro: add_assoc add_commute)

 notation times (infixl "*" 70)

 notation Groups.one ("1")

 subsection {* Generalized summation over a set *}

 definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) => 'b set => 'a" where

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

 sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof

 qed (fact setsum_def)

 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 (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =

         if x <> y then raise Match

         else

           let

             val x' = Syntax_Trans.mark_bound x;

             val t' = subst_bound (x', t);

             val P' = subst_bound (x', P);

           in Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound x $ P' $ t' end

     | setsum_tr' _ = raise Match;

 in [(@{const_syntax setsum}, setsum_tr')] end

 *}

 lemma setsum_empty:

   "setsum f {} = 0"

   by (fact setsum.empty)

 lemma setsum_insert:

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

   by (fact setsum.insert)

 lemma setsum_infinite:

   "~ finite A ==> setsum f A = 0"

   by (fact setsum.infinite)

 lemma (in comm_monoid_add) setsum_reindex:

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

 proof -

   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)

   from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex dest!:finite_imageD)

 qed

 lemma (in comm_monoid_add) setsum_reindex_id:

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

   by (simp add: setsum_reindex)

 lemma (in comm_monoid_add) 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 (in comm_monoid_add) setsum_cong:

   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"

   by (cases "finite A") (auto intro: setsum.cong)

 lemma (in comm_monoid_add) 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 (rule setsum_cong) (simp_all add: simp_implies_def)

 lemma (in comm_monoid_add) setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"

   by (auto intro: setsum_cong)

 lemma (in comm_monoid_add) 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)

 lemma (in comm_monoid_add) setsum_0[simp]: "setsum (%i. 0) A = 0"

   by (cases "finite A") (erule finite_induct, auto)

 lemma (in comm_monoid_add) setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"

   by (simp add:setsum_cong)

 lemma (in comm_monoid_add) 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 (fact setsum.union_inter)

 lemma (in comm_monoid_add) setsum_Un_disjoint: "finite A ==> finite B

   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"

   by (fact setsum.union_disjoint)

 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: 

   "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. f i = 0 \<Longrightarrow> setsum f T = setsum f S"

 by(blast intro!: setsum_mono_zero_left[symmetric])

 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)

 lemma setsum_restrict_set:

   assumes fA: "finite A"

   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"

 proof-

   from fA have fab: "finite (A \<inter> B)" by auto

   have aba: "A \<inter> B \<subseteq> A" by blast

   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"

   from setsum_mono_zero_left[OF fA aba, of ?g]

   show ?thesis by simp

 qed

 lemma setsum_cases:

   assumes fA: "finite A"

   shows "setsum (\<lambda>x. if P x then f x else g x) A =

          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"

   using setsum.If_cases[OF fA] .

 (*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 (in comm_monoid_add) setsum_UN_disjoint:

   assumes "finite I" and "ALL i:I. finite (A i)"

     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"

   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"

 proof -

   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)

   from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint)

 qed

 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:

   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"

   shows "setsum f (Union C) = setsum (setsum f) C"

 proof cases

   assume "finite C"

   from setsum_UN_disjoint[OF this assms]

   show ?thesis

     by (simp add: SUP_def)

 qed (force dest: finite_UnionD simp add: setsum_def)

 (*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 (in comm_monoid_add) setsum_Sigma:

   assumes "finite A" and  "ALL x:A. finite (B x)"

   shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"

 proof -

   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)

   from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def)

 qed

 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 (in comm_monoid_add) setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"

   by (cases "finite A") (simp_all add: setsum.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_eq_Suc0_iff: "finite A \<Longrightarrow>

   (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"

 apply(erule finite_induct)

 apply (auto simp add:add_is_1)

 done

 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]

 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 (in comm_monoid_add) setsum_eq_general_reverses:

   assumes fS: "finite S" and fT: "finite T"

   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"

   shows "setsum f S = setsum g T"

 proof -

   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)

   show ?thesis

   apply (simp add: setsum_def fS fT)

   apply (rule fold_image_eq_general_inverses)

   apply (rule fS)

   apply (erule kh)

   apply (erule hk)

   done

 qed

 lemma (in comm_monoid_add) setsum_Un_zero:  

   assumes fS: "finite S" and fT: "finite T"

   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"

   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"

 proof -

   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)

   show ?thesis

   using fS fT

   apply (simp add: setsum_def)

   apply (rule fold_image_Un_one)

   using I0 by auto

 qed

 lemma setsum_UNION_zero: 

   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"

   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"

   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"

   using fSS f0

 proof(induct rule: finite_induct[OF fS])

   case 1 thus ?case by simp

 next

   case (2 T F)

   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 

     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto

   from fTF have fUF: "finite (\<Union>F)" by auto

   from "2.prems" TF fTF

   show ?case 

     by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])

 qed

 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

 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"

   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"

 unfolding setsum_diff1'[OF assms] by auto

 (* 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, ordered_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 fastforce

   qed

 next

   case False

   thus ?thesis

     by (simp add: setsum_def)

 qed

 lemma setsum_strict_mono:

   fixes f :: "'a \<Rightarrow> 'b::{ordered_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 assms

 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_strict_mono_ex1:

 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"

 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"

 shows "setsum f A < setsum g A"

 proof-

   from assms(3) obtain a where a: "a:A" "f a < g a" by blast

   have "setsum f A = setsum f ((A-{a}) \<union> {a})"

     by(simp add:insert_absorb[OF `a:A`])

   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"

     using `finite A` by(subst setsum_Un_disjoint) auto

   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"

     by(rule setsum_mono)(simp add: assms(2))

   also have "setsum f {a} < setsum g {a}" using a by simp

   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"

     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto

   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])

   finally show ?thesis by (metis add_right_mono add_strict_left_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::{ordered_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::{ordered_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_nonneg_leq_bound:

   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"

   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"

   shows "f i \<le> B"

 proof -

   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"

     using assms by (auto intro!: setsum_nonneg)

   moreover

   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"

     using assms by (simp add: setsum_diff1)

   ultimately show ?thesis by auto

 qed

 lemma setsum_nonneg_0:

   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"

   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"

   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"

   shows "f i = 0"

   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto

 lemma setsum_mono2:

 fixes f :: "'a \<Rightarrow> 'b :: ordered_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,ordered_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 simp add: sup_absorb2)

 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::ordered_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::ordered_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

   qed

 next

   case False thus ?thesis by (simp add: setsum_def)

 qed

 lemma abs_setsum_abs[simp]: 

   fixes f :: "'a => ('b::ordered_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

 lemma setsum_Plus:

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

   assumes fin: "finite A" "finite B"

   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"

 proof -

   have "A <+> B = Inl ` A \<union> Inr ` B" by auto

   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"

     by auto

   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto

   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)

   ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)

 qed

 text {* Commuting outer and inner summation *}

 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)

 lemma setsum_mult_setsum_if_inj:

 fixes f :: "'a => ('b::semiring_0)"

 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>

   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"

 by(auto simp: setsum_product setsum_cartesian_product

         intro!:  setsum_reindex_cong[symmetric])

 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"

 apply (cases "finite A")

 apply (erule finite_induct)

 apply (auto simp add: algebra_simps)

 done

 lemma setsum_bounded:

   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"

   shows "setsum f A \<le> of_nat(card A) * K"

 proof (cases "finite A")

   case True

   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp

 next

   case False thus ?thesis by (simp add: setsum_def)

 qed

 subsubsection {* Cardinality as special case of @{const setsum} *}

 lemma card_eq_setsum:

   "card A = setsum (\<lambda>x. 1) A"

   by (simp only: card_def setsum_def)

 lemma card_UN_disjoint:

   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"

     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"

   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"

 proof -

   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp

   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)

 qed

 lemma card_Union_disjoint:

   "finite C ==> (ALL A:C. finite A) ==>

    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})

    ==> card (Union C) = setsum card C"

 apply (frule card_UN_disjoint [of C id])

 apply (simp_all add: SUP_def id_def)

 done

 text{*The image of a finite set can be expressed using @{term fold_image}.*}

 lemma image_eq_fold_image:

   "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"

 proof (induct rule: finite_induct)

   case empty then show ?case by simp

 next

   interpret ab_semigroup_mult "op Un"

     proof qed auto

   case insert 

   then show ?case by simp

 qed

 subsubsection {* Cardinality of products *}

 lemma card_SigmaI [simp]:

   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>

   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"

 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)

 (*

 lemma SigmaI_insert: "y \<notin> A ==>

   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"

   by auto

 *)

 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"

   by (cases "finite A \<and> finite B")

     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)

 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"

 by (simp add: card_cartesian_product)

 subsection {* Generalized product over a set *}

 definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where

   "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"

 sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof

 qed (fact setprod_def)

 abbreviation

   Setprod  ("\<Prod>_" [1000] 999) where

   "\<Prod>A == setprod (%x. x) A"

 syntax

   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)

 syntax (xsymbols)

   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)

 syntax (HTML output)

   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)

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

   "PROD i:A. b" == "CONST setprod (%i. b) A" 

   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 

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

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

 syntax

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

 syntax (xsymbols)

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

 syntax (HTML output)

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

 translations

   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"

   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"

 lemma setprod_empty: "setprod f {} = 1"

   by (fact setprod.empty)

 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>

     setprod f (insert a A) = f a * setprod f A"

   by (fact setprod.insert)

 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"

   by (fact setprod.infinite)

 lemma setprod_reindex:

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

 by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD)

 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"

 by (auto simp add: setprod_reindex)

 lemma setprod_cong:

   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"

 by(fastforce simp: setprod_def intro: fold_image_cong)

 lemma strong_setprod_cong[cong]:

   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"

 by(fastforce simp: simp_implies_def setprod_def intro: fold_image_cong)

 lemma setprod_reindex_cong: "inj_on f A ==>

     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"

 by (frule setprod_reindex, simp)

 lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"

   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"

   shows "setprod h B = setprod g A"

 proof-

     have "setprod h B = setprod (h o f) A"

       by (simp add: B setprod_reindex[OF i, of h])

     then show ?thesis apply simp

       apply (rule setprod_cong)

       apply simp

       by (simp add: eq)

 qed

 lemma setprod_Un_one:  

   assumes fS: "finite S" and fT: "finite T"

   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"

   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"

   using fS fT

   apply (simp add: setprod_def)

   apply (rule fold_image_Un_one)

   using I0 by auto

 lemma setprod_1: "setprod (%i. 1) A = 1"

 apply (case_tac "finite A")

 apply (erule finite_induct, auto simp add: mult_ac)

 done

 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"

 apply (subgoal_tac "setprod f F = setprod (%x. 1) F")

 apply (erule ssubst, rule setprod_1)

 apply (rule setprod_cong, auto)

 done

 lemma setprod_Un_Int: "finite A ==> finite B

     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"

 by(simp add: setprod_def fold_image_Un_Int[symmetric])

 lemma setprod_Un_disjoint: "finite A ==> finite B

   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"

 by (subst setprod_Un_Int [symmetric], auto)

 lemma setprod_mono_one_left: 

   assumes fT: "finite T" and ST: "S \<subseteq> T"

   and z: "\<forall>i \<in> T - S. f i = 1"

   shows "setprod f S = setprod 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: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z])

 qed

 lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym]

 lemma setprod_delta: 

   assumes fS: "finite S"

   shows "setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"

 proof-

   let ?f = "(\<lambda>k. if k=a then b k else 1)"

   {assume a: "a \<notin> S"

     hence "\<forall> k\<in> S. ?f k = 1" by simp

     hence ?thesis  using a by (simp add: setprod_1) }

   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 fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto

     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"

       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]

       by simp

     then have ?thesis using a by (simp add: fA1 cong: setprod_cong cong del: if_weak_cong)}

   ultimately show ?thesis by blast

 qed

 lemma setprod_delta': 

   assumes fS: "finite S" shows 

   "setprod (\<lambda>k. if a = k then b k else 1) S = 

      (if a\<in> S then b a else 1)"

   using setprod_delta[OF fS, of a b, symmetric] 

   by (auto intro: setprod_cong)

 lemma setprod_UN_disjoint:

     "finite I ==> (ALL i:I. finite (A i)) ==>

         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>

       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"

   by (simp add: setprod_def fold_image_UN_disjoint)

 lemma setprod_Union_disjoint:

   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 

   shows "setprod f (Union C) = setprod (setprod f) C"

 proof cases

   assume "finite C"

   from setprod_UN_disjoint[OF this assms]

   show ?thesis

     by (simp add: SUP_def)

 qed (force dest: finite_UnionD simp add: setprod_def)

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

     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =

     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"

 by(simp add:setprod_def fold_image_Sigma split_def)

 text{*Here we can eliminate the finiteness assumptions, by cases.*}

 lemma setprod_cartesian_product: 

      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"

 apply (cases "finite A") 

  apply (cases "finite B") 

   apply (simp add: setprod_Sigma)

  apply (cases "A={}", simp)

  apply (simp add: setprod_1) 

 apply (auto simp add: setprod_def

             dest: finite_cartesian_productD1 finite_cartesian_productD2) 

 done

 lemma setprod_timesf:

      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"

 by(simp add:setprod_def fold_image_distrib)

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

 lemma setprod_eq_1_iff [simp]:

   "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"

 by (induct set: finite) auto

 lemma setprod_zero:

      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"

 apply (induct set: finite, force, clarsimp)

 apply (erule disjE, auto)

 done

 lemma setprod_nonneg [rule_format]:

    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"

 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)

 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)

   --> 0 < setprod f A"

 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)

 lemma setprod_zero_iff[simp]: "finite A ==> 

   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =

   (EX x: A. f x = 0)"

 by (erule finite_induct, auto simp:no_zero_divisors)

 lemma setprod_pos_nat:

   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"

 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])

 lemma setprod_pos_nat_iff[simp]:

   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"

 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])

 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>

   (setprod f (A Un B) :: 'a ::{field})

    = setprod f A * setprod f B / setprod f (A Int B)"

 by (subst setprod_Un_Int [symmetric], auto)

 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>

   (setprod f (A - {a}) :: 'a :: {field}) =

   (if a:A then setprod f A / f a else setprod f A)"

   by (erule finite_induct) (auto simp add: insert_Diff_if)

 lemma setprod_inversef: 

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

   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"

 by (erule finite_induct) auto

 lemma setprod_dividef:

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

   shows "finite A

     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"

 apply (subgoal_tac

          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")

 apply (erule ssubst)

 apply (subst divide_inverse)

 apply (subst setprod_timesf)

 apply (subst setprod_inversef, assumption+, rule refl)

 apply (rule setprod_cong, rule refl)

 apply (subst divide_inverse, auto)

 done

 lemma setprod_dvd_setprod [rule_format]: 

     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"

   apply (cases "finite A")

   apply (induct set: finite)

   apply (auto simp add: dvd_def)

   apply (rule_tac x = "k * ka" in exI)

   apply (simp add: algebra_simps)

 done

 lemma setprod_dvd_setprod_subset:

   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"

   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")

   apply (unfold dvd_def, blast)

   apply (subst setprod_Un_disjoint [symmetric])

   apply (auto elim: finite_subset intro: setprod_cong)

 done

 lemma setprod_dvd_setprod_subset2:

   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 

       setprod f A dvd setprod g B"

   apply (rule dvd_trans)

   apply (rule setprod_dvd_setprod, erule (1) bspec)

   apply (erule (1) setprod_dvd_setprod_subset)

 done

 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 

     (f i ::'a::comm_semiring_1) dvd setprod f A"

 by (induct set: finite) (auto intro: dvd_mult)

 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 

     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"

   apply (cases "finite A")

   apply (induct set: finite)

   apply auto

 done

 lemma setprod_mono:

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

   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"

   shows "setprod f A \<le> setprod g A"

 proof (cases "finite A")

   case True

   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]

   proof (induct A rule: finite_subset_induct)

     case (insert a F)

     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"

       unfolding setprod_insert[OF insert(1,3)]

       using assms[rule_format,OF insert(2)] insert

       by (auto intro: mult_mono mult_nonneg_nonneg)

   qed auto

   thus ?thesis by simp

 qed auto

 lemma abs_setprod:

   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"

   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"

 proof (cases "finite A")

   case True thus ?thesis

     by induct (auto simp add: field_simps abs_mult)

 qed auto

 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"

 apply (erule finite_induct)

 apply auto

 done

 lemma setprod_gen_delta:

   assumes fS: "finite S"

   shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::{comm_monoid_mult}) * c^ (card S - 1) else c^ card S)"

 proof-

   let ?f = "(\<lambda>k. if k=a then b k else c)"

   {assume a: "a \<notin> S"

     hence "\<forall> k\<in> S. ?f k = c" by simp

     hence ?thesis  using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) }

   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 fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"

       apply (rule setprod_cong) by auto

     have cA: "card ?A = card S - 1" using fS a by auto

     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto

     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"

       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]

       by simp

     then have ?thesis using a cA

       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}

   ultimately show ?thesis by blast

 qed

 subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}

 no_notation times (infixl "*" 70)

 no_notation Groups.one ("1")

 locale semilattice_big = semilattice +

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

   assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"

 sublocale semilattice_big < folding_one_idem proof

 qed (simp_all add: F_eq)

 notation times (infixl "*" 70)

 notation Groups.one ("1")

 context lattice

 begin

 definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900) where

   "Inf_fin = fold1 inf"

 definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900) where

   "Sup_fin = fold1 sup"

 end

 sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof

 qed (simp add: Inf_fin_def)

 sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof

 qed (simp add: Sup_fin_def)

 context semilattice_inf

 begin

 lemma ab_semigroup_idem_mult_inf:

   "class.ab_semigroup_idem_mult inf"

 proof qed (rule inf_assoc inf_commute inf_idem)+

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

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

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

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

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

 proof(induct arbitrary: a pred:finite)

   case empty thus ?case by simp

 next

   case (insert x A)

   show ?case

   proof cases

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

   next

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

   qed

 qed

 lemma below_fold1_iff:

   assumes "finite A" "A \<noteq> {}"

   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"

 proof -

   interpret ab_semigroup_idem_mult inf

     by (rule ab_semigroup_idem_mult_inf)

   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all

 qed

 lemma fold1_belowI:

   assumes "finite A"

     and "a \<in> A"

   shows "fold1 inf A \<le> a"

 proof -

   from assms have "A \<noteq> {}" by auto

   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis

   proof (induct rule: finite_ne_induct)

     case singleton thus ?case by simp

   next

     interpret ab_semigroup_idem_mult inf

       by (rule ab_semigroup_idem_mult_inf)

     case (insert x F)

     from insert(5) have "a = x \<or> a \<in> F" by simp

     thus ?case

     proof

       assume "a = x" thus ?thesis using insert

         by (simp add: mult_ac)

     next

       assume "a \<in> F"

       hence bel: "fold1 inf F \<le> a" by (rule insert)

       have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"

         using insert by (simp add: mult_ac)

       also have "inf (fold1 inf F) a = fold1 inf F"

         using bel by (auto intro: antisym)

       also have "inf x \<dots> = fold1 inf (insert x F)"

         using insert by (simp add: mult_ac)

       finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .

       moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp

       ultimately show ?thesis by simp

     qed

   qed

 qed

 end

 context semilattice_sup

 begin

 lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"

 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)

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

 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)

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

 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)

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

 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)

 end

 context lattice

 begin

 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"

 apply(unfold Sup_fin_def Inf_fin_def)

 apply(subgoal_tac "EX a. a:A")

 prefer 2 apply blast

 apply(erule exE)

 apply(rule order_trans)

 apply(erule (1) fold1_belowI)

 apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])

 done

 lemma sup_Inf_absorb [simp]:

   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"

 apply(subst sup_commute)

 apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)

 done

 lemma inf_Sup_absorb [simp]:

   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"

 by (simp add: Sup_fin_def inf_absorb1

   semilattice_inf.fold1_belowI [OF dual_semilattice])

 end

 context distrib_lattice

 begin

 lemma sup_Inf1_distrib:

   assumes "finite A"

     and "A \<noteq> {}"

   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"

 proof -

   interpret ab_semigroup_idem_mult inf

     by (rule ab_semigroup_idem_mult_inf)

   from assms show ?thesis

     by (simp add: Inf_fin_def image_def

       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])

         (rule arg_cong [where f="fold1 inf"], blast)

 qed

 lemma sup_Inf2_distrib:

   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"

   shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"

 using A proof (induct rule: finite_ne_induct)

   case singleton thus ?case

     by (simp add: sup_Inf1_distrib [OF B])

 next

   interpret ab_semigroup_idem_mult inf

     by (rule ab_semigroup_idem_mult_inf)

   case (insert x A)

   have finB: "finite {sup x b |b. b \<in> B}"

     by(rule finite_surj[where f = "sup x", OF B(1)], auto)

   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"

   proof -

     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"

       by blast

     thus ?thesis by(simp add: insert(1) B(1))

   qed

   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast

   have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"

     using insert by simp

   also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)

   also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"

     using insert by(simp add:sup_Inf1_distrib[OF B])

   also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"

     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")

     using B insert

     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])

   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"

     by blast

   finally show ?case .

 qed

 lemma inf_Sup1_distrib:

   assumes "finite A" and "A \<noteq> {}"

   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"

 proof -

   interpret ab_semigroup_idem_mult sup

     by (rule ab_semigroup_idem_mult_sup)

   from assms show ?thesis

     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])

       (rule arg_cong [where f="fold1 sup"], blast)

 qed

 lemma inf_Sup2_distrib:

   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"

   shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"

 using A proof (induct rule: finite_ne_induct)

   case singleton thus ?case

     by(simp add: inf_Sup1_distrib [OF B])

 next

   case (insert x A)

   have finB: "finite {inf x b |b. b \<in> B}"

     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)

   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"

   proof -

     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"

       by blast

     thus ?thesis by(simp add: insert(1) B(1))

   qed

   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast

   interpret ab_semigroup_idem_mult sup

     by (rule ab_semigroup_idem_mult_sup)

   have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"

     using insert by simp

   also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)

   also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"

     using insert by(simp add:inf_Sup1_distrib[OF B])

   also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"

     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")

     using B insert

     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])

   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"

     by blast

   finally show ?case .

 qed

 end

 context complete_lattice

 begin

 lemma Inf_fin_Inf:

   assumes "finite A" and "A \<noteq> {}"

   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"

 proof -

   interpret ab_semigroup_idem_mult inf

     by (rule ab_semigroup_idem_mult_inf)

   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto

   moreover with `finite A` have "finite B" by simp

   ultimately show ?thesis

     by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])

 qed

 lemma Sup_fin_Sup:

   assumes "finite A" and "A \<noteq> {}"

   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"

 proof -

   interpret ab_semigroup_idem_mult sup

     by (rule ab_semigroup_idem_mult_sup)

   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto

   moreover with `finite A` have "finite B" by simp

   ultimately show ?thesis  

   by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])

 qed

 end

 subsection {* Versions of @{const min} and @{const max} on non-empty sets *}

 definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where

   "Min = fold1 min"

 definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where

   "Max = fold1 max"

 sublocale linorder < Min!: semilattice_big min Min proof

 qed (simp add: Min_def)

 sublocale linorder < Max!: semilattice_big max Max proof

 qed (simp add: Max_def)

 context linorder

 begin

 lemmas Min_singleton = Min.singleton

 lemmas Max_singleton = Max.singleton

 lemma Min_insert:

   assumes "finite A" and "A \<noteq> {}"

   shows "Min (insert x A) = min x (Min A)"

   using assms by simp

 lemma Max_insert:

   assumes "finite A" and "A \<noteq> {}"

   shows "Max (insert x A) = max x (Max A)"

   using assms by simp

 lemma Min_Un:

   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"

   shows "Min (A \<union> B) = min (Min A) (Min B)"

   using assms by (rule Min.union_idem)

 lemma Max_Un:

   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"

   shows "Max (A \<union> B) = max (Max A) (Max B)"

   using assms by (rule Max.union_idem)

 lemma hom_Min_commute:

   assumes "\<And>x y. h (min x y) = min (h x) (h y)"

     and "finite N" and "N \<noteq> {}"

   shows "h (Min N) = Min (h ` N)"

   using assms by (rule Min.hom_commute)

 lemma hom_Max_commute:

   assumes "\<And>x y. h (max x y) = max (h x) (h y)"

     and "finite N" and "N \<noteq> {}"

   shows "h (Max N) = Max (h ` N)"

   using assms by (rule Max.hom_commute)

 lemma ab_semigroup_idem_mult_min:

   "class.ab_semigroup_idem_mult min"

   proof qed (auto simp add: min_def)

 lemma ab_semigroup_idem_mult_max:

   "class.ab_semigroup_idem_mult max"

   proof qed (auto simp add: max_def)

 lemma max_lattice:

   "class.semilattice_inf max (op \<ge>) (op >)"

   by (fact min_max.dual_semilattice)

 lemma dual_max:

   "ord.max (op \<ge>) = min"

   by (auto simp add: ord.max_def_raw min_def fun_eq_iff)

 lemma dual_min:

   "ord.min (op \<ge>) = max"

   by (auto simp add: ord.min_def_raw max_def fun_eq_iff)

 lemma strict_below_fold1_iff:

   assumes "finite A" and "A \<noteq> {}"

   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"

 proof -

   interpret ab_semigroup_idem_mult min

     by (rule ab_semigroup_idem_mult_min)

   from assms show ?thesis

   by (induct rule: finite_ne_induct)

     (simp_all add: fold1_insert)

 qed

 lemma fold1_below_iff:

   assumes "finite A" and "A \<noteq> {}"

   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"

 proof -

   interpret ab_semigroup_idem_mult min

     by (rule ab_semigroup_idem_mult_min)

   from assms show ?thesis

   by (induct rule: finite_ne_induct)

     (simp_all add: fold1_insert min_le_iff_disj)

 qed

 lemma fold1_strict_below_iff:

   assumes "finite A" and "A \<noteq> {}"

   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"

 proof -

   interpret ab_semigroup_idem_mult min

     by (rule ab_semigroup_idem_mult_min)

   from assms show ?thesis

   by (induct rule: finite_ne_induct)

     (simp_all add: fold1_insert min_less_iff_disj)

 qed

 lemma fold1_antimono:

   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"

   shows "fold1 min B \<le> fold1 min A"

 proof cases

   assume "A = B" thus ?thesis by simp

 next

   interpret ab_semigroup_idem_mult min

     by (rule ab_semigroup_idem_mult_min)

   assume neq: "A \<noteq> B"

   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast

   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)

   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"

   proof -

     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])

     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])

     moreover have "(B-A) \<noteq> {}" using assms neq by blast

     moreover have "A Int (B-A) = {}" using assms by blast

     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)

   qed

   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)

   finally show ?thesis .

 qed

 lemma Min_in [simp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "Min A \<in> A"

 proof -

   interpret ab_semigroup_idem_mult min

     by (rule ab_semigroup_idem_mult_min)

   from assms fold1_in show ?thesis by (fastforce simp: Min_def min_def)

 qed

 lemma Max_in [simp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "Max A \<in> A"

 proof -

   interpret ab_semigroup_idem_mult max

     by (rule ab_semigroup_idem_mult_max)

   from assms fold1_in [of A] show ?thesis by (fastforce simp: Max_def max_def)

 qed

 lemma Min_le [simp]:

   assumes "finite A" and "x \<in> A"

   shows "Min A \<le> x"

   using assms by (simp add: Min_def min_max.fold1_belowI)

 lemma Max_ge [simp]:

   assumes "finite A" and "x \<in> A"

   shows "x \<le> Max A"

   by (simp add: Max_def semilattice_inf.fold1_belowI [OF max_lattice] assms)

 lemma Min_ge_iff [simp, no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"

   using assms by (simp add: Min_def min_max.below_fold1_iff)

 lemma Max_le_iff [simp, no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"

   by (simp add: Max_def semilattice_inf.below_fold1_iff [OF max_lattice] assms)

 lemma Min_gr_iff [simp, no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"

   using assms by (simp add: Min_def strict_below_fold1_iff)

 lemma Max_less_iff [simp, no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"

   by (simp add: Max_def linorder.dual_max [OF dual_linorder]

     linorder.strict_below_fold1_iff [OF dual_linorder] assms)

 lemma Min_le_iff [no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"

   using assms by (simp add: Min_def fold1_below_iff)

 lemma Max_ge_iff [no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"

   by (simp add: Max_def linorder.dual_max [OF dual_linorder]

     linorder.fold1_below_iff [OF dual_linorder] assms)

 lemma Min_less_iff [no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"

   using assms by (simp add: Min_def fold1_strict_below_iff)

 lemma Max_gr_iff [no_atp]:

   assumes "finite A" and "A \<noteq> {}"

   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"

   by (simp add: Max_def linorder.dual_max [OF dual_linorder]

     linorder.fold1_strict_below_iff [OF dual_linorder] assms)

 lemma Min_eqI:

   assumes "finite A"

   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"

     and "x \<in> A"

   shows "Min A = x"

 proof (rule antisym)

   from `x \<in> A` have "A \<noteq> {}" by auto

   with assms show "Min A \<ge> x" by simp

 next

   from assms show "x \<ge> Min A" by simp

 qed

 lemma Max_eqI:

   assumes "finite A"

   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"

     and "x \<in> A"

   shows "Max A = x"

 proof (rule antisym)

   from `x \<in> A` have "A \<noteq> {}" by auto

   with assms show "Max A \<le> x" by simp

 next

   from assms show "x \<le> Max A" by simp

 qed

 lemma Min_antimono:

   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"

   shows "Min N \<le> Min M"

   using assms by (simp add: Min_def fold1_antimono)

 lemma Max_mono:

   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"

   shows "Max M \<le> Max N"

   by (simp add: Max_def linorder.dual_max [OF dual_linorder]

     linorder.fold1_antimono [OF dual_linorder] assms)

 lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:

  assumes fin: "finite A"

  and   empty: "P {}" 

  and  insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"

  shows "P A"

 using fin empty insert

 proof (induct rule: finite_psubset_induct)

   case (psubset A)

   have IH: "\<And>B. \<lbrakk>B < A; P {}; (\<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. a<b; P A\<rbrakk> \<Longrightarrow> P (insert b A))\<rbrakk> \<Longrightarrow> P B" by fact 

   have fin: "finite A" by fact 

   have empty: "P {}" by fact

   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact

   show "P A"

   proof (cases "A = {}")

     assume "A = {}" 

     then show "P A" using `P {}` by simp

   next

     let ?B = "A - {Max A}" 

     let ?A = "insert (Max A) ?B"

     have "finite ?B" using `finite A` by simp

     assume "A \<noteq> {}"

     with `finite A` have "Max A : A" by auto

     then have A: "?A = A" using insert_Diff_single insert_absorb by auto

     then have "P ?B" using `P {}` step IH[of ?B] by blast

     moreover 

     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce

     ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastforce

   qed

 qed

 lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:

  "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"

 by(rule linorder.finite_linorder_max_induct[OF dual_linorder])

 end

 context linordered_ab_semigroup_add

 begin

 lemma add_Min_commute:

   fixes k

   assumes "finite N" and "N \<noteq> {}"

   shows "k + Min N = Min {k + m | m. m \<in> N}"

 proof -

   have "\<And>x y. k + min x y = min (k + x) (k + y)"

     by (simp add: min_def not_le)

       (blast intro: antisym less_imp_le add_left_mono)

   with assms show ?thesis

     using hom_Min_commute [of "plus k" N]

     by simp (blast intro: arg_cong [where f = Min])

 qed

 lemma add_Max_commute:

   fixes k

   assumes "finite N" and "N \<noteq> {}"

   shows "k + Max N = Max {k + m | m. m \<in> N}"

 proof -

   have "\<And>x y. k + max x y = max (k + x) (k + y)"

     by (simp add: max_def not_le)

       (blast intro: antisym less_imp_le add_left_mono)

   with assms show ?thesis

     using hom_Max_commute [of "plus k" N]

     by simp (blast intro: arg_cong [where f = Max])

 qed

 end

 context linordered_ab_group_add

 begin

 lemma minus_Max_eq_Min [simp]:

   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"

   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)

 lemma minus_Min_eq_Max [simp]:

   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"

   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)

 end

 end