haftmann@35719: (*  Title:      HOL/Big_Operators.thy
wenzelm@12396:     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
avigad@16775:                 with contributions by Jeremy Avigad
wenzelm@12396: *)
wenzelm@12396: 
haftmann@35719: header {* Big operators and finite (non-empty) sets *}
haftmann@26041: 
haftmann@35719: theory Big_Operators
haftmann@35722: imports Plain
haftmann@26041: begin
haftmann@26041: 
haftmann@35816: subsection {* Generic monoid operation over a set *}
haftmann@35816: 
haftmann@35816: no_notation times (infixl "*" 70)
haftmann@35816: no_notation Groups.one ("1")
haftmann@35816: 
haftmann@35816: locale comm_monoid_big = comm_monoid +
haftmann@35816:   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@35816:   assumes F_eq: "F g A = (if finite A then fold_image (op *) g 1 A else 1)"
haftmann@35816: 
haftmann@35816: sublocale comm_monoid_big < folding_image proof
haftmann@35816: qed (simp add: F_eq)
haftmann@35816: 
haftmann@35816: context comm_monoid_big
haftmann@35816: begin
haftmann@35816: 
haftmann@35816: lemma infinite [simp]:
haftmann@35816:   "\<not> finite A \<Longrightarrow> F g A = 1"
haftmann@35816:   by (simp add: F_eq)
haftmann@35816: 
hoelzl@42986: lemma F_cong:
hoelzl@42986:   assumes "A = B" "\<And>x. x \<in> B \<Longrightarrow> h x = g x"
hoelzl@42986:   shows "F h A = F g B"
hoelzl@42986: proof cases
hoelzl@42986:   assume "finite A"
hoelzl@42986:   with assms show ?thesis unfolding `A = B` by (simp cong: cong)
hoelzl@42986: next
hoelzl@42986:   assume "\<not> finite A"
hoelzl@42986:   then show ?thesis unfolding `A = B` by simp
hoelzl@42986: qed
hoelzl@42986: 
nipkow@48849: lemma strong_F_cong [cong]:
nipkow@48849:   "\<lbrakk> A = B; !!x. x:B =simp=> g x = h x \<rbrakk>
nipkow@48849:    \<Longrightarrow> F (%x. g x) A = F (%x. h x) B"
nipkow@48849: by (rule F_cong) (simp_all add: simp_implies_def)
nipkow@48849: 
nipkow@48821: lemma F_neutral[simp]: "F (%i. 1) A = 1"
nipkow@48821: by (cases "finite A") (simp_all add: neutral)
nipkow@48821: 
nipkow@48821: lemma F_neutral': "ALL a:A. g a = 1 \<Longrightarrow> F g A = 1"
nipkow@48849: by simp
nipkow@48849: 
nipkow@48849: lemma F_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow> F g A = F g (A - B) * F g B"
nipkow@48849: by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
nipkow@48849: 
nipkow@48849: lemma F_mono_neutral_cong_left:
nipkow@48849:   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
nipkow@48849:   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
nipkow@48849: proof-
nipkow@48849:   have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
nipkow@48849:   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
nipkow@48849:   from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
nipkow@48849:     by (auto intro: finite_subset)
nipkow@48849:   show ?thesis using assms(4)
nipkow@48849:     by (simp add: union_disjoint[OF f d, unfolded eq[symmetric]] F_neutral'[OF assms(3)])
nipkow@48849: qed
nipkow@48849: 
nipkow@48850: lemma F_mono_neutral_cong_right:
nipkow@48850:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
nipkow@48850:    \<Longrightarrow> F g T = F h S"
nipkow@48850: by(auto intro!: F_mono_neutral_cong_left[symmetric])
nipkow@48849: 
nipkow@48849: lemma F_mono_neutral_left:
nipkow@48849:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
nipkow@48849: by(blast intro: F_mono_neutral_cong_left)
nipkow@48849: 
nipkow@48850: lemma F_mono_neutral_right:
nipkow@48850:   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
nipkow@48850: by(blast intro!: F_mono_neutral_left[symmetric])
nipkow@48849: 
nipkow@48849: lemma F_delta: 
nipkow@48849:   assumes fS: "finite S"
nipkow@48849:   shows "F (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
nipkow@48849: proof-
nipkow@48849:   let ?f = "(\<lambda>k. if k=a then b k else 1)"
nipkow@48849:   { assume a: "a \<notin> S"
nipkow@48849:     hence "\<forall>k\<in>S. ?f k = 1" by simp
nipkow@48849:     hence ?thesis  using a by simp }
nipkow@48849:   moreover
nipkow@48849:   { assume a: "a \<in> S"
nipkow@48849:     let ?A = "S - {a}"
nipkow@48849:     let ?B = "{a}"
nipkow@48849:     have eq: "S = ?A \<union> ?B" using a by blast 
nipkow@48849:     have dj: "?A \<inter> ?B = {}" by simp
nipkow@48849:     from fS have fAB: "finite ?A" "finite ?B" by auto  
nipkow@48849:     have "F ?f S = F ?f ?A * F ?f ?B"
nipkow@48849:       using union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
nipkow@48849:       by simp
nipkow@48849:     then have ?thesis  using a by simp }
nipkow@48849:   ultimately show ?thesis by blast
nipkow@48849: qed
nipkow@48849: 
nipkow@48849: lemma F_delta': 
nipkow@48849:   assumes fS: "finite S" shows 
nipkow@48849:   "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
nipkow@48849: using F_delta[OF fS, of a b, symmetric] by (auto intro: F_cong)
nipkow@48821: 
nipkow@48861: lemma F_fun_f: "F (%x. g x * h x) A = (F g A * F h A)"
nipkow@48861: by (cases "finite A") (simp_all add: distrib)
nipkow@48861: 
nipkow@48893: 
nipkow@48893: text {* for ad-hoc proofs for @{const fold_image} *}
nipkow@48893: lemma comm_monoid_mult:  "class.comm_monoid_mult (op *) 1"
nipkow@48893: proof qed (auto intro: assoc commute)
nipkow@48893: 
nipkow@48893: lemma F_Un_neutral:
nipkow@48893:   assumes fS: "finite S" and fT: "finite T"
nipkow@48893:   and I1: "\<forall>x \<in> S\<inter>T. g x = 1"
nipkow@48893:   shows "F g (S \<union> T) = F g S  * F g T"
nipkow@48893: proof -
nipkow@48893:   interpret comm_monoid_mult "op *" 1 by (fact comm_monoid_mult)
nipkow@48893:   show ?thesis
nipkow@48893:   using fS fT
nipkow@48893:   apply (simp add: F_eq)
nipkow@48893:   apply (rule fold_image_Un_one)
nipkow@48893:   using I1 by auto
nipkow@48893: qed
nipkow@48893: 
hoelzl@42986: lemma If_cases:
hoelzl@42986:   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
hoelzl@42986:   assumes fA: "finite A"
hoelzl@42986:   shows "F (\<lambda>x. if P x then h x else g x) A =
hoelzl@42986:          F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
hoelzl@42986: proof-
hoelzl@42986:   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 
hoelzl@42986:           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 
hoelzl@42986:     by blast+
hoelzl@42986:   from fA 
hoelzl@42986:   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
hoelzl@42986:   let ?g = "\<lambda>x. if P x then h x else g x"
hoelzl@42986:   from union_disjoint[OF f a(2), of ?g] a(1)
hoelzl@42986:   show ?thesis
hoelzl@42986:     by (subst (1 2) F_cong) simp_all
hoelzl@42986: qed
hoelzl@42986: 
haftmann@35816: end
haftmann@35816: 
haftmann@35816: text {* for ad-hoc proofs for @{const fold_image} *}
haftmann@35816: 
haftmann@35816: lemma (in comm_monoid_add) comm_monoid_mult:
haftmann@36635:   "class.comm_monoid_mult (op +) 0"
haftmann@35816: proof qed (auto intro: add_assoc add_commute)
haftmann@35816: 
haftmann@35816: notation times (infixl "*" 70)
haftmann@35816: notation Groups.one ("1")
haftmann@35816: 
haftmann@35816: 
nipkow@15402: subsection {* Generalized summation over a set *}
nipkow@15402: 
haftmann@35816: definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
haftmann@35816:   "setsum f A = (if finite A then fold_image (op +) f 0 A else 0)"
haftmann@26041: 
haftmann@35816: sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof
haftmann@35816: qed (fact setsum_def)
nipkow@15402: 
wenzelm@19535: abbreviation
wenzelm@21404:   Setsum  ("\<Sum>_" [1000] 999) where
wenzelm@19535:   "\<Sum>A == setsum (%x. x) A"
wenzelm@19535: 
nipkow@15402: text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402: written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402: 
nipkow@15402: syntax
paulson@17189:   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (xsymbols)
paulson@17189:   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (HTML output)
paulson@17189:   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: 
nipkow@15402: translations -- {* Beware of argument permutation! *}
nipkow@28853:   "SUM i:A. b" == "CONST setsum (%i. b) A"
nipkow@28853:   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
nipkow@15402: 
nipkow@15402: text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402:  @{text"\<Sum>x|P. e"}. *}
nipkow@15402: 
nipkow@15402: syntax
paulson@17189:   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402: syntax (xsymbols)
paulson@17189:   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402: syntax (HTML output)
paulson@17189:   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402: 
nipkow@15402: translations
nipkow@28853:   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
nipkow@28853:   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
nipkow@15402: 
nipkow@15402: print_translation {*
nipkow@15402: let
wenzelm@35115:   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
wenzelm@35115:         if x <> y then raise Match
wenzelm@35115:         else
wenzelm@35115:           let
wenzelm@42284:             val x' = Syntax_Trans.mark_bound x;
wenzelm@35115:             val t' = subst_bound (x', t);
wenzelm@35115:             val P' = subst_bound (x', P);
wenzelm@42284:           in Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound x $ P' $ t' end
wenzelm@35115:     | setsum_tr' _ = raise Match;
wenzelm@35115: in [(@{const_syntax setsum}, setsum_tr')] end
nipkow@15402: *}
nipkow@15402: 
haftmann@35816: lemma setsum_empty:
haftmann@35816:   "setsum f {} = 0"
haftmann@35816:   by (fact setsum.empty)
nipkow@15402: 
haftmann@35816: lemma setsum_insert:
nipkow@28853:   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
haftmann@35816:   by (fact setsum.insert)
haftmann@35816: 
haftmann@35816: lemma setsum_infinite:
haftmann@35816:   "~ finite A ==> setsum f A = 0"
haftmann@35816:   by (fact setsum.infinite)
nipkow@15402: 
haftmann@35816: lemma (in comm_monoid_add) setsum_reindex:
haftmann@35816:   assumes "inj_on f B" shows "setsum h (f ` B) = setsum (h \<circ> f) B"
haftmann@35816: proof -
haftmann@35816:   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
nipkow@48849:   from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex o_def dest!:finite_imageD)
haftmann@35816: qed
paulson@15409: 
nipkow@48849: lemma setsum_reindex_id:
haftmann@35816:   "inj_on f B ==> setsum f B = setsum id (f ` B)"
nipkow@48849: by (simp add: setsum_reindex)
nipkow@15402: 
nipkow@48849: lemma setsum_reindex_nonzero: 
chaieb@29674:   assumes fS: "finite S"
chaieb@29674:   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"
chaieb@29674:   shows "setsum h (f ` S) = setsum (h o f) S"
chaieb@29674: using nz
chaieb@29674: proof(induct rule: finite_induct[OF fS])
chaieb@29674:   case 1 thus ?case by simp
chaieb@29674: next
chaieb@29674:   case (2 x F) 
nipkow@48849:   { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
chaieb@29674:     then obtain y where y: "y \<in> F" "f x = f y" by auto 
chaieb@29674:     from "2.hyps" y have xy: "x \<noteq> y" by auto
chaieb@29674:     
chaieb@29674:     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
chaieb@29674:     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
chaieb@29674:     also have "\<dots> = setsum (h o f) (insert x F)" 
haftmann@35816:       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
haftmann@35816:       using h0
nipkow@48849:       apply (simp cong del:setsum.strong_F_cong)
chaieb@29674:       apply (rule "2.hyps"(3))
chaieb@29674:       apply (rule_tac y="y" in  "2.prems")
chaieb@29674:       apply simp_all
chaieb@29674:       done
nipkow@48849:     finally have ?case . }
chaieb@29674:   moreover
nipkow@48849:   { assume fxF: "f x \<notin> f ` F"
chaieb@29674:     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
chaieb@29674:       using fxF "2.hyps" by simp 
chaieb@29674:     also have "\<dots> = setsum (h o f) (insert x F)"
haftmann@35816:       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
nipkow@48849:       apply (simp cong del:setsum.strong_F_cong)
haftmann@35816:       apply (rule cong [OF refl [of "op + (h (f x))"]])
chaieb@29674:       apply (rule "2.hyps"(3))
chaieb@29674:       apply (rule_tac y="y" in  "2.prems")
chaieb@29674:       apply simp_all
chaieb@29674:       done
nipkow@48849:     finally have ?case . }
chaieb@29674:   ultimately show ?case by blast
chaieb@29674: qed
chaieb@29674: 
nipkow@48849: lemma setsum_cong:
nipkow@15402:   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
nipkow@48849: by (fact setsum.F_cong)
nipkow@15402: 
nipkow@48849: lemma strong_setsum_cong:
nipkow@16733:   "A = B ==> (!!x. x:B =simp=> f x = g x)
nipkow@16733:    ==> setsum (%x. f x) A = setsum (%x. g x) B"
nipkow@48849: by (fact setsum.strong_F_cong)
berghofe@16632: 
nipkow@48849: lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"
nipkow@48849: by (auto intro: setsum_cong)
nipkow@15554: 
nipkow@48849: lemma setsum_reindex_cong:
nipkow@28853:    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
nipkow@28853:     ==> setsum h B = setsum g A"
nipkow@48849: by (simp add: setsum_reindex)
nipkow@15402: 
nipkow@48821: lemmas setsum_0 = setsum.F_neutral
nipkow@48821: lemmas setsum_0' = setsum.F_neutral'
nipkow@15402: 
nipkow@48849: lemma setsum_Un_Int: "finite A ==> finite B ==>
nipkow@15402:   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
nipkow@15402:   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
nipkow@48849: by (fact setsum.union_inter)
nipkow@15402: 
nipkow@48849: lemma setsum_Un_disjoint: "finite A ==> finite B
nipkow@15402:   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
nipkow@48849: by (fact setsum.union_disjoint)
nipkow@48849: 
nipkow@48849: lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
nipkow@48849:     setsum f A = setsum f (A - B) + setsum f B"
nipkow@48849: by(fact setsum.F_subset_diff)
nipkow@15402: 
chaieb@29674: lemma setsum_mono_zero_left: 
nipkow@48849:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
nipkow@48849: by(fact setsum.F_mono_neutral_left)
chaieb@29674: 
nipkow@48849: lemmas setsum_mono_zero_right = setsum.F_mono_neutral_right
chaieb@29674: 
chaieb@29674: lemma setsum_mono_zero_cong_left: 
nipkow@48849:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
nipkow@48849:   \<Longrightarrow> setsum f S = setsum g T"
nipkow@48849: by(fact setsum.F_mono_neutral_cong_left)
chaieb@29674: 
nipkow@48849: lemmas setsum_mono_zero_cong_right = setsum.F_mono_neutral_cong_right
chaieb@29674: 
nipkow@48849: lemma setsum_delta: "finite S \<Longrightarrow>
nipkow@48849:   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
nipkow@48849: by(fact setsum.F_delta)
nipkow@48849: 
nipkow@48849: lemma setsum_delta': "finite S \<Longrightarrow>
nipkow@48849:   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
nipkow@48849: by(fact setsum.F_delta')
chaieb@29674: 
chaieb@30260: lemma setsum_restrict_set:
chaieb@30260:   assumes fA: "finite A"
chaieb@30260:   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
chaieb@30260: proof-
chaieb@30260:   from fA have fab: "finite (A \<inter> B)" by auto
chaieb@30260:   have aba: "A \<inter> B \<subseteq> A" by blast
chaieb@30260:   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
chaieb@30260:   from setsum_mono_zero_left[OF fA aba, of ?g]
chaieb@30260:   show ?thesis by simp
chaieb@30260: qed
chaieb@30260: 
chaieb@30260: lemma setsum_cases:
chaieb@30260:   assumes fA: "finite A"
hoelzl@35577:   shows "setsum (\<lambda>x. if P x then f x else g x) A =
hoelzl@35577:          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
hoelzl@42986:   using setsum.If_cases[OF fA] .
chaieb@29674: 
paulson@15409: (*But we can't get rid of finite I. If infinite, although the rhs is 0, 
paulson@15409:   the lhs need not be, since UNION I A could still be finite.*)
haftmann@35816: lemma (in comm_monoid_add) setsum_UN_disjoint:
haftmann@35816:   assumes "finite I" and "ALL i:I. finite (A i)"
haftmann@35816:     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
haftmann@35816:   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
haftmann@35816: proof -
haftmann@35816:   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
wenzelm@41550:   from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint)
haftmann@35816: qed
nipkow@15402: 
paulson@15409: text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
paulson@15409: directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
nipkow@15402: lemma setsum_Union_disjoint:
hoelzl@44937:   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
hoelzl@44937:   shows "setsum f (Union C) = setsum (setsum f) C"
hoelzl@44937: proof cases
hoelzl@44937:   assume "finite C"
hoelzl@44937:   from setsum_UN_disjoint[OF this assms]
hoelzl@44937:   show ?thesis
hoelzl@44937:     by (simp add: SUP_def)
hoelzl@44937: qed (force dest: finite_UnionD simp add: setsum_def)
nipkow@15402: 
paulson@15409: (*But we can't get rid of finite A. If infinite, although the lhs is 0, 
paulson@15409:   the rhs need not be, since SIGMA A B could still be finite.*)
haftmann@35816: lemma (in comm_monoid_add) setsum_Sigma:
haftmann@35816:   assumes "finite A" and  "ALL x:A. finite (B x)"
haftmann@35816:   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)"
haftmann@35816: proof -
haftmann@35816:   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
wenzelm@41550:   from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def)
haftmann@35816: qed
nipkow@15402: 
paulson@15409: text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409: lemma setsum_cartesian_product: 
paulson@17189:    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
paulson@15409: apply (cases "finite A") 
paulson@15409:  apply (cases "finite B") 
paulson@15409:   apply (simp add: setsum_Sigma)
paulson@15409:  apply (cases "A={}", simp)
nipkow@15543:  apply (simp) 
paulson@15409: apply (auto simp add: setsum_def
paulson@15409:             dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409: done
nipkow@15402: 
nipkow@48861: lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
nipkow@48861: by (fact setsum.F_fun_f)
nipkow@15402: 
nipkow@48893: lemma setsum_Un_zero:  
nipkow@48893:   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
nipkow@48893:   setsum f (S \<union> T) = setsum f S + setsum f T"
nipkow@48893: by(fact setsum.F_Un_neutral)
nipkow@48893: 
nipkow@48893: lemma setsum_UNION_zero: 
nipkow@48893:   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
nipkow@48893:   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"
nipkow@48893:   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
nipkow@48893:   using fSS f0
nipkow@48893: proof(induct rule: finite_induct[OF fS])
nipkow@48893:   case 1 thus ?case by simp
nipkow@48893: next
nipkow@48893:   case (2 T F)
nipkow@48893:   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
nipkow@48893:     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
nipkow@48893:   from fTF have fUF: "finite (\<Union>F)" by auto
nipkow@48893:   from "2.prems" TF fTF
nipkow@48893:   show ?case 
nipkow@48893:     by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
nipkow@48893: qed
nipkow@48893: 
nipkow@15402: 
nipkow@15402: subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402: 
nipkow@15402: lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
nipkow@28853: apply (case_tac "finite A")
nipkow@28853:  prefer 2 apply (simp add: setsum_def)
nipkow@28853: apply (erule rev_mp)
nipkow@28853: apply (erule finite_induct, auto)
nipkow@28853: done
nipkow@15402: 
nipkow@15402: lemma setsum_eq_0_iff [simp]:
nipkow@15402:     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
nipkow@28853: by (induct set: finite) auto
nipkow@15402: 
nipkow@30859: lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
nipkow@30859:   (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
nipkow@30859: apply(erule finite_induct)
nipkow@30859: apply (auto simp add:add_is_1)
nipkow@30859: done
nipkow@30859: 
nipkow@30859: lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
nipkow@30859: 
nipkow@15402: lemma setsum_Un_nat: "finite A ==> finite B ==>
nipkow@28853:   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402:   -- {* For the natural numbers, we have subtraction. *}
nipkow@29667: by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
nipkow@15402: 
nipkow@15402: lemma setsum_Un: "finite A ==> finite B ==>
nipkow@28853:   (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@28853:    setsum f A + setsum f B - setsum f (A Int B)"
nipkow@29667: by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
nipkow@15402: 
haftmann@35816: lemma (in comm_monoid_add) setsum_eq_general_reverses:
chaieb@30260:   assumes fS: "finite S" and fT: "finite T"
chaieb@30260:   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
chaieb@30260:   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
chaieb@30260:   shows "setsum f S = setsum g T"
haftmann@35816: proof -
haftmann@35816:   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
haftmann@35816:   show ?thesis
chaieb@30260:   apply (simp add: setsum_def fS fT)
haftmann@35816:   apply (rule fold_image_eq_general_inverses)
haftmann@35816:   apply (rule fS)
chaieb@30260:   apply (erule kh)
chaieb@30260:   apply (erule hk)
chaieb@30260:   done
haftmann@35816: qed
chaieb@30260: 
nipkow@15402: lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
nipkow@28853:   (if a:A then setsum f A - f a else setsum f A)"
nipkow@28853: apply (case_tac "finite A")
nipkow@28853:  prefer 2 apply (simp add: setsum_def)
nipkow@28853: apply (erule finite_induct)
nipkow@28853:  apply (auto simp add: insert_Diff_if)
nipkow@28853: apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@28853: done
nipkow@15402: 
nipkow@15402: lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402:   (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402:   (if a:A then setsum f A - f a else setsum f A)"
nipkow@28853: by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@28853: 
nipkow@28853: lemma setsum_diff1'[rule_format]:
nipkow@28853:   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
nipkow@28853: 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))"])
nipkow@28853: apply (auto simp add: insert_Diff_if add_ac)
nipkow@28853: done
obua@15552: 
nipkow@31438: lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
nipkow@31438:   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
nipkow@31438: unfolding setsum_diff1'[OF assms] by auto
nipkow@31438: 
nipkow@15402: (* By Jeremy Siek: *)
nipkow@15402: 
nipkow@15402: lemma setsum_diff_nat: 
nipkow@28853: assumes "finite B" and "B \<subseteq> A"
nipkow@28853: shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
nipkow@28853: using assms
wenzelm@19535: proof induct
nipkow@15402:   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402: next
nipkow@15402:   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402:     and xFinA: "insert x F \<subseteq> A"
nipkow@15402:     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402:   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402:   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402:     by (simp add: setsum_diff1_nat)
nipkow@15402:   from xFinA have "F \<subseteq> A" by simp
nipkow@15402:   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402:   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402:     by simp
nipkow@15402:   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402:   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402:     by simp
nipkow@15402:   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402:   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402:     by simp
nipkow@15402:   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_diff:
nipkow@15402:   assumes le: "finite A" "B \<subseteq> A"
nipkow@15402:   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402: proof -
nipkow@15402:   from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402:   show ?thesis using finiteB le
wenzelm@21575:   proof induct
wenzelm@19535:     case empty
wenzelm@19535:     thus ?case by auto
wenzelm@19535:   next
wenzelm@19535:     case (insert x F)
wenzelm@19535:     thus ?case using le finiteB 
wenzelm@19535:       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402:   qed
wenzelm@19535: qed
nipkow@15402: 
nipkow@15402: lemma setsum_mono:
haftmann@35028:   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
nipkow@15402:   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402: proof (cases "finite K")
nipkow@15402:   case True
nipkow@15402:   thus ?thesis using le
wenzelm@19535:   proof induct
nipkow@15402:     case empty
nipkow@15402:     thus ?case by simp
nipkow@15402:   next
nipkow@15402:     case insert
nipkow@44890:     thus ?case using add_mono by fastforce
nipkow@15402:   qed
nipkow@15402: next
nipkow@15402:   case False
nipkow@15402:   thus ?thesis
nipkow@15402:     by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15554: lemma setsum_strict_mono:
haftmann@35028:   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
wenzelm@19535:   assumes "finite A"  "A \<noteq> {}"
wenzelm@19535:     and "!!x. x:A \<Longrightarrow> f x < g x"
wenzelm@19535:   shows "setsum f A < setsum g A"
wenzelm@41550:   using assms
nipkow@15554: proof (induct rule: finite_ne_induct)
nipkow@15554:   case singleton thus ?case by simp
nipkow@15554: next
nipkow@15554:   case insert thus ?case by (auto simp: add_strict_mono)
nipkow@15554: qed
nipkow@15554: 
nipkow@46699: lemma setsum_strict_mono_ex1:
nipkow@46699: fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
nipkow@46699: assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
nipkow@46699: shows "setsum f A < setsum g A"
nipkow@46699: proof-
nipkow@46699:   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
nipkow@46699:   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
nipkow@46699:     by(simp add:insert_absorb[OF `a:A`])
nipkow@46699:   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
nipkow@46699:     using `finite A` by(subst setsum_Un_disjoint) auto
nipkow@46699:   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
nipkow@46699:     by(rule setsum_mono)(simp add: assms(2))
nipkow@46699:   also have "setsum f {a} < setsum g {a}" using a by simp
nipkow@46699:   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
nipkow@46699:     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
nipkow@46699:   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
nipkow@46699:   finally show ?thesis by (metis add_right_mono add_strict_left_mono)
nipkow@46699: qed
nipkow@46699: 
nipkow@15535: lemma setsum_negf:
wenzelm@19535:   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow@15535: proof (cases "finite A")
berghofe@22262:   case True thus ?thesis by (induct set: finite) auto
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15535: lemma setsum_subtractf:
wenzelm@19535:   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
wenzelm@19535:     setsum f A - setsum g A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15535: lemma setsum_nonneg:
haftmann@35028:   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
wenzelm@19535:   shows "0 \<le> setsum f A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True thus ?thesis using nn
wenzelm@21575:   proof induct
wenzelm@19535:     case empty then show ?case by simp
wenzelm@19535:   next
wenzelm@19535:     case (insert x F)
wenzelm@19535:     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
wenzelm@19535:     with insert show ?case by simp
wenzelm@19535:   qed
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15535: lemma setsum_nonpos:
haftmann@35028:   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
wenzelm@19535:   shows "setsum f A \<le> 0"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True thus ?thesis using np
wenzelm@21575:   proof induct
wenzelm@19535:     case empty then show ?case by simp
wenzelm@19535:   next
wenzelm@19535:     case (insert x F)
wenzelm@19535:     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
wenzelm@19535:     with insert show ?case by simp
wenzelm@19535:   qed
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
hoelzl@36622: lemma setsum_nonneg_leq_bound:
hoelzl@36622:   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
hoelzl@36622:   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
hoelzl@36622:   shows "f i \<le> B"
hoelzl@36622: proof -
hoelzl@36622:   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
hoelzl@36622:     using assms by (auto intro!: setsum_nonneg)
hoelzl@36622:   moreover
hoelzl@36622:   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
hoelzl@36622:     using assms by (simp add: setsum_diff1)
hoelzl@36622:   ultimately show ?thesis by auto
hoelzl@36622: qed
hoelzl@36622: 
hoelzl@36622: lemma setsum_nonneg_0:
hoelzl@36622:   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
hoelzl@36622:   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
hoelzl@36622:   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
hoelzl@36622:   shows "f i = 0"
hoelzl@36622:   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
hoelzl@36622: 
nipkow@15539: lemma setsum_mono2:
haftmann@36303: fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
nipkow@15539: assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@15539: shows "setsum f A \<le> setsum f B"
nipkow@15539: proof -
nipkow@15539:   have "setsum f A \<le> setsum f A + setsum f (B-A)"
nipkow@15539:     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
nipkow@15539:   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15539:     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15539:   also have "A \<union> (B-A) = B" using sub by blast
nipkow@15539:   finally show ?thesis .
nipkow@15539: qed
nipkow@15542: 
avigad@16775: lemma setsum_mono3: "finite B ==> A <= B ==> 
avigad@16775:     ALL x: B - A. 
haftmann@35028:       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
avigad@16775:         setsum f A <= setsum f B"
avigad@16775:   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
avigad@16775:   apply (erule ssubst)
avigad@16775:   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
avigad@16775:   apply simp
avigad@16775:   apply (rule add_left_mono)
avigad@16775:   apply (erule setsum_nonneg)
avigad@16775:   apply (subst setsum_Un_disjoint [THEN sym])
avigad@16775:   apply (erule finite_subset, assumption)
avigad@16775:   apply (rule finite_subset)
avigad@16775:   prefer 2
avigad@16775:   apply assumption
haftmann@32698:   apply (auto simp add: sup_absorb2)
avigad@16775: done
avigad@16775: 
ballarin@19279: lemma setsum_right_distrib: 
huffman@22934:   fixes f :: "'a => ('b::semiring_0)"
nipkow@15402:   shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402: proof (cases "finite A")
nipkow@15402:   case True
nipkow@15402:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15402:     case empty thus ?case by simp
nipkow@15402:   next
nipkow@15402:     case (insert x A) thus ?case by (simp add: right_distrib)
nipkow@15402:   qed
nipkow@15402: next
nipkow@15402:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
ballarin@17149: lemma setsum_left_distrib:
huffman@22934:   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
ballarin@17149: proof (cases "finite A")
ballarin@17149:   case True
ballarin@17149:   then show ?thesis
ballarin@17149:   proof induct
ballarin@17149:     case empty thus ?case by simp
ballarin@17149:   next
ballarin@17149:     case (insert x A) thus ?case by (simp add: left_distrib)
ballarin@17149:   qed
ballarin@17149: next
ballarin@17149:   case False thus ?thesis by (simp add: setsum_def)
ballarin@17149: qed
ballarin@17149: 
ballarin@17149: lemma setsum_divide_distrib:
ballarin@17149:   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
ballarin@17149: proof (cases "finite A")
ballarin@17149:   case True
ballarin@17149:   then show ?thesis
ballarin@17149:   proof induct
ballarin@17149:     case empty thus ?case by simp
ballarin@17149:   next
ballarin@17149:     case (insert x A) thus ?case by (simp add: add_divide_distrib)
ballarin@17149:   qed
ballarin@17149: next
ballarin@17149:   case False thus ?thesis by (simp add: setsum_def)
ballarin@17149: qed
ballarin@17149: 
nipkow@15535: lemma setsum_abs[iff]: 
haftmann@35028:   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
nipkow@15402:   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True
nipkow@15535:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15535:     case empty thus ?case by simp
nipkow@15535:   next
nipkow@15535:     case (insert x A)
nipkow@15535:     thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15535:   qed
nipkow@15402: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15535: lemma setsum_abs_ge_zero[iff]: 
haftmann@35028:   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
nipkow@15402:   shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True
nipkow@15535:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15535:     case empty thus ?case by simp
nipkow@15535:   next
huffman@36977:     case (insert x A) thus ?case by auto
nipkow@15535:   qed
nipkow@15402: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15539: lemma abs_setsum_abs[simp]: 
haftmann@35028:   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
nipkow@15539:   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
nipkow@15539: proof (cases "finite A")
nipkow@15539:   case True
nipkow@15539:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15539:     case empty thus ?case by simp
nipkow@15539:   next
nipkow@15539:     case (insert a A)
nipkow@15539:     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
nipkow@15539:     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
avigad@16775:     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
avigad@16775:       by (simp del: abs_of_nonneg)
nipkow@15539:     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
nipkow@15539:     finally show ?case .
nipkow@15539:   qed
nipkow@15539: next
nipkow@15539:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15539: qed
nipkow@15539: 
nipkow@31080: lemma setsum_Plus:
nipkow@31080:   fixes A :: "'a set" and B :: "'b set"
nipkow@31080:   assumes fin: "finite A" "finite B"
nipkow@31080:   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
nipkow@31080: proof -
nipkow@31080:   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
nipkow@31080:   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
nipkow@40786:     by auto
nipkow@31080:   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
nipkow@31080:   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
nipkow@31080:   ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
nipkow@31080: qed
nipkow@31080: 
nipkow@31080: 
ballarin@17149: text {* Commuting outer and inner summation *}
ballarin@17149: 
ballarin@17149: lemma setsum_commute:
ballarin@17149:   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
ballarin@17149: proof (simp add: setsum_cartesian_product)
paulson@17189:   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
paulson@17189:     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
ballarin@17149:     (is "?s = _")
ballarin@17149:     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
ballarin@17149:     apply (simp add: split_def)
ballarin@17149:     done
paulson@17189:   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
ballarin@17149:     (is "_ = ?t")
ballarin@17149:     apply (simp add: swap_product)
ballarin@17149:     done
ballarin@17149:   finally show "?s = ?t" .
ballarin@17149: qed
ballarin@17149: 
ballarin@19279: lemma setsum_product:
huffman@22934:   fixes f :: "'a => ('b::semiring_0)"
ballarin@19279:   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
ballarin@19279:   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
ballarin@19279: 
nipkow@34223: lemma setsum_mult_setsum_if_inj:
nipkow@34223: fixes f :: "'a => ('b::semiring_0)"
nipkow@34223: shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
nipkow@34223:   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
nipkow@34223: by(auto simp: setsum_product setsum_cartesian_product
nipkow@34223:         intro!:  setsum_reindex_cong[symmetric])
nipkow@34223: 
haftmann@35722: lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
haftmann@35722: apply (cases "finite A")
haftmann@35722: apply (erule finite_induct)
haftmann@35722: apply (auto simp add: algebra_simps)
haftmann@35722: done
haftmann@35722: 
haftmann@35722: lemma setsum_bounded:
haftmann@35722:   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
haftmann@35722:   shows "setsum f A \<le> of_nat(card A) * K"
haftmann@35722: proof (cases "finite A")
haftmann@35722:   case True
haftmann@35722:   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
haftmann@35722: next
haftmann@35722:   case False thus ?thesis by (simp add: setsum_def)
haftmann@35722: qed
haftmann@35722: 
haftmann@35722: 
haftmann@35722: subsubsection {* Cardinality as special case of @{const setsum} *}
haftmann@35722: 
haftmann@35722: lemma card_eq_setsum:
haftmann@35722:   "card A = setsum (\<lambda>x. 1) A"
haftmann@35722:   by (simp only: card_def setsum_def)
haftmann@35722: 
haftmann@35722: lemma card_UN_disjoint:
haftmann@46629:   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@46629:     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@46629:   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
haftmann@46629: proof -
haftmann@46629:   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
haftmann@46629:   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
haftmann@46629: qed
haftmann@35722: 
haftmann@35722: lemma card_Union_disjoint:
haftmann@35722:   "finite C ==> (ALL A:C. finite A) ==>
haftmann@35722:    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
haftmann@35722:    ==> card (Union C) = setsum card C"
haftmann@35722: apply (frule card_UN_disjoint [of C id])
hoelzl@44937: apply (simp_all add: SUP_def id_def)
haftmann@35722: done
haftmann@35722: 
haftmann@35722: text{*The image of a finite set can be expressed using @{term fold_image}.*}
haftmann@35722: lemma image_eq_fold_image:
haftmann@35722:   "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
haftmann@35722: proof (induct rule: finite_induct)
haftmann@35722:   case empty then show ?case by simp
haftmann@35722: next
haftmann@35722:   interpret ab_semigroup_mult "op Un"
haftmann@35722:     proof qed auto
haftmann@35722:   case insert 
haftmann@35722:   then show ?case by simp
haftmann@35722: qed
haftmann@35722: 
haftmann@35722: subsubsection {* Cardinality of products *}
haftmann@35722: 
haftmann@35722: lemma card_SigmaI [simp]:
haftmann@35722:   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
haftmann@35722:   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
haftmann@35722: by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
haftmann@35722: 
haftmann@35722: (*
haftmann@35722: lemma SigmaI_insert: "y \<notin> A ==>
haftmann@35722:   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
haftmann@35722:   by auto
haftmann@35722: *)
haftmann@35722: 
haftmann@35722: lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
haftmann@35722:   by (cases "finite A \<and> finite B")
haftmann@35722:     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
haftmann@35722: 
haftmann@35722: lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
haftmann@35722: by (simp add: card_cartesian_product)
haftmann@35722: 
ballarin@17149: 
nipkow@15402: subsection {* Generalized product over a set *}
nipkow@15402: 
haftmann@35816: definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
haftmann@35816:   "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"
haftmann@35816: 
huffman@35938: sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof
haftmann@35816: qed (fact setprod_def)
nipkow@15402: 
wenzelm@19535: abbreviation
wenzelm@21404:   Setprod  ("\<Prod>_" [1000] 999) where
wenzelm@19535:   "\<Prod>A == setprod (%x. x) A"
wenzelm@19535: 
nipkow@15402: syntax
paulson@17189:   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (xsymbols)
paulson@17189:   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (HTML output)
paulson@17189:   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@16550: 
nipkow@16550: translations -- {* Beware of argument permutation! *}
nipkow@28853:   "PROD i:A. b" == "CONST setprod (%i. b) A" 
nipkow@28853:   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 
nipkow@16550: 
nipkow@16550: text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
nipkow@16550:  @{text"\<Prod>x|P. e"}. *}
nipkow@16550: 
nipkow@16550: syntax
paulson@17189:   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
nipkow@16550: syntax (xsymbols)
paulson@17189:   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550: syntax (HTML output)
paulson@17189:   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550: 
nipkow@15402: translations
nipkow@28853:   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
nipkow@28853:   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
nipkow@16550: 
haftmann@35816: lemma setprod_empty: "setprod f {} = 1"
haftmann@35816:   by (fact setprod.empty)
nipkow@15402: 
haftmann@35816: lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
nipkow@15402:     setprod f (insert a A) = f a * setprod f A"
haftmann@35816:   by (fact setprod.insert)
nipkow@15402: 
haftmann@35816: lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
haftmann@35816:   by (fact setprod.infinite)
paulson@15409: 
nipkow@15402: lemma setprod_reindex:
nipkow@28853:    "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
nipkow@48849: by(auto simp: setprod_def fold_image_reindex o_def dest!:finite_imageD)
nipkow@15402: 
nipkow@15402: lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
nipkow@15402: by (auto simp add: setprod_reindex)
nipkow@15402: 
nipkow@15402: lemma setprod_cong:
nipkow@15402:   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
nipkow@48849: by(fact setprod.F_cong)
nipkow@15402: 
nipkow@48849: lemma strong_setprod_cong:
berghofe@16632:   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
nipkow@48849: by(fact setprod.strong_F_cong)
berghofe@16632: 
nipkow@15402: lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402:     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@28853: by (frule setprod_reindex, simp)
nipkow@15402: 
chaieb@29674: lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
chaieb@29674:   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
chaieb@29674:   shows "setprod h B = setprod g A"
chaieb@29674: proof-
chaieb@29674:     have "setprod h B = setprod (h o f) A"
chaieb@29674:       by (simp add: B setprod_reindex[OF i, of h])
chaieb@29674:     then show ?thesis apply simp
chaieb@29674:       apply (rule setprod_cong)
chaieb@29674:       apply simp
nipkow@30837:       by (simp add: eq)
chaieb@29674: qed
chaieb@29674: 
nipkow@48893: lemma setprod_Un_one: "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
nipkow@48893:   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
nipkow@48893: by(fact setprod.F_Un_neutral)
nipkow@15402: 
nipkow@48821: lemmas setprod_1 = setprod.F_neutral
nipkow@48821: lemmas setprod_1' = setprod.F_neutral'
nipkow@15402: 
nipkow@15402: 
nipkow@15402: lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402:     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
nipkow@48849: by (fact setprod.union_inter)
nipkow@15402: 
nipkow@15402: lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402:   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
nipkow@48849: by (fact setprod.union_disjoint)
nipkow@48849: 
nipkow@48849: lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
nipkow@48849:     setprod f A = setprod f (A - B) * setprod f B"
nipkow@48849: by(fact setprod.F_subset_diff)
nipkow@15402: 
nipkow@48849: lemma setprod_mono_one_left:
nipkow@48849:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
nipkow@48849: by(fact setprod.F_mono_neutral_left)
nipkow@30837: 
nipkow@48849: lemmas setprod_mono_one_right = setprod.F_mono_neutral_right
nipkow@30837: 
nipkow@48849: lemma setprod_mono_one_cong_left: 
nipkow@48849:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
nipkow@48849:   \<Longrightarrow> setprod f S = setprod g T"
nipkow@48849: by(fact setprod.F_mono_neutral_cong_left)
nipkow@48849: 
nipkow@48849: lemmas setprod_mono_one_cong_right = setprod.F_mono_neutral_cong_right
chaieb@29674: 
nipkow@48849: lemma setprod_delta: "finite S \<Longrightarrow>
nipkow@48849:   setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
nipkow@48849: by(fact setprod.F_delta)
chaieb@29674: 
nipkow@48849: lemma setprod_delta': "finite S \<Longrightarrow>
nipkow@48849:   setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
nipkow@48849: by(fact setprod.F_delta')
chaieb@29674: 
nipkow@15402: lemma setprod_UN_disjoint:
nipkow@15402:     "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402:         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402:       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
wenzelm@41550:   by (simp add: setprod_def fold_image_UN_disjoint)
nipkow@15402: 
nipkow@15402: lemma setprod_Union_disjoint:
hoelzl@44937:   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 
hoelzl@44937:   shows "setprod f (Union C) = setprod (setprod f) C"
hoelzl@44937: proof cases
hoelzl@44937:   assume "finite C"
hoelzl@44937:   from setprod_UN_disjoint[OF this assms]
hoelzl@44937:   show ?thesis
hoelzl@44937:     by (simp add: SUP_def)
hoelzl@44937: qed (force dest: finite_UnionD simp add: setprod_def)
nipkow@15402: 
nipkow@15402: lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@16550:     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
paulson@17189:     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
wenzelm@41550: by(simp add:setprod_def fold_image_Sigma split_def)
nipkow@15402: 
paulson@15409: text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409: lemma setprod_cartesian_product: 
paulson@17189:      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
paulson@15409: apply (cases "finite A") 
paulson@15409:  apply (cases "finite B") 
paulson@15409:   apply (simp add: setprod_Sigma)
paulson@15409:  apply (cases "A={}", simp)
nipkow@48849:  apply (simp) 
paulson@15409: apply (auto simp add: setprod_def
paulson@15409:             dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409: done
nipkow@15402: 
nipkow@48861: lemma setprod_timesf: "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
nipkow@48861: by (fact setprod.F_fun_f)
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402: 
nipkow@15402: lemma setprod_eq_1_iff [simp]:
nipkow@28853:   "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
nipkow@28853: by (induct set: finite) auto
nipkow@15402: 
nipkow@15402: lemma setprod_zero:
huffman@23277:      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
nipkow@28853: apply (induct set: finite, force, clarsimp)
nipkow@28853: apply (erule disjE, auto)
nipkow@28853: done
nipkow@15402: 
nipkow@15402: lemma setprod_nonneg [rule_format]:
haftmann@35028:    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
huffman@30841: by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
huffman@30841: 
haftmann@35028: lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
nipkow@28853:   --> 0 < setprod f A"
huffman@30841: by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
nipkow@15402: 
nipkow@30843: lemma setprod_zero_iff[simp]: "finite A ==> 
nipkow@30843:   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
nipkow@30843:   (EX x: A. f x = 0)"
nipkow@30843: by (erule finite_induct, auto simp:no_zero_divisors)
nipkow@30843: 
nipkow@30843: lemma setprod_pos_nat:
nipkow@30843:   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
nipkow@30843: using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
nipkow@15402: 
nipkow@30863: lemma setprod_pos_nat_iff[simp]:
nipkow@30863:   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
nipkow@30863: using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
nipkow@30863: 
nipkow@15402: lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
nipkow@28853:   (setprod f (A Un B) :: 'a ::{field})
nipkow@28853:    = setprod f A * setprod f B / setprod f (A Int B)"
nipkow@30843: by (subst setprod_Un_Int [symmetric], auto)
nipkow@15402: 
nipkow@15402: lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@28853:   (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@28853:   (if a:A then setprod f A / f a else setprod f A)"
haftmann@36303:   by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402: 
paulson@31906: lemma setprod_inversef: 
haftmann@36409:   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
paulson@31906:   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@28853: by (erule finite_induct) auto
nipkow@15402: 
nipkow@15402: lemma setprod_dividef:
haftmann@36409:   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
wenzelm@31916:   shows "finite A
nipkow@28853:     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@28853: apply (subgoal_tac
nipkow@15402:          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@28853: apply (erule ssubst)
nipkow@28853: apply (subst divide_inverse)
nipkow@28853: apply (subst setprod_timesf)
nipkow@28853: apply (subst setprod_inversef, assumption+, rule refl)
nipkow@28853: apply (rule setprod_cong, rule refl)
nipkow@28853: apply (subst divide_inverse, auto)
nipkow@28853: done
nipkow@28853: 
nipkow@29925: lemma setprod_dvd_setprod [rule_format]: 
nipkow@29925:     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
nipkow@29925:   apply (cases "finite A")
nipkow@29925:   apply (induct set: finite)
nipkow@29925:   apply (auto simp add: dvd_def)
nipkow@29925:   apply (rule_tac x = "k * ka" in exI)
nipkow@29925:   apply (simp add: algebra_simps)
nipkow@29925: done
nipkow@29925: 
nipkow@29925: lemma setprod_dvd_setprod_subset:
nipkow@29925:   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
nipkow@29925:   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
nipkow@29925:   apply (unfold dvd_def, blast)
nipkow@29925:   apply (subst setprod_Un_disjoint [symmetric])
nipkow@29925:   apply (auto elim: finite_subset intro: setprod_cong)
nipkow@29925: done
nipkow@29925: 
nipkow@29925: lemma setprod_dvd_setprod_subset2:
nipkow@29925:   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 
nipkow@29925:       setprod f A dvd setprod g B"
nipkow@29925:   apply (rule dvd_trans)
nipkow@29925:   apply (rule setprod_dvd_setprod, erule (1) bspec)
nipkow@29925:   apply (erule (1) setprod_dvd_setprod_subset)
nipkow@29925: done
nipkow@29925: 
nipkow@29925: lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 
nipkow@29925:     (f i ::'a::comm_semiring_1) dvd setprod f A"
nipkow@29925: by (induct set: finite) (auto intro: dvd_mult)
nipkow@29925: 
nipkow@29925: lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 
nipkow@29925:     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
nipkow@29925:   apply (cases "finite A")
nipkow@29925:   apply (induct set: finite)
nipkow@29925:   apply auto
nipkow@29925: done
nipkow@29925: 
hoelzl@35171: lemma setprod_mono:
hoelzl@35171:   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
hoelzl@35171:   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
hoelzl@35171:   shows "setprod f A \<le> setprod g A"
hoelzl@35171: proof (cases "finite A")
hoelzl@35171:   case True
hoelzl@35171:   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
hoelzl@35171:   proof (induct A rule: finite_subset_induct)
hoelzl@35171:     case (insert a F)
hoelzl@35171:     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
hoelzl@35171:       unfolding setprod_insert[OF insert(1,3)]
hoelzl@35171:       using assms[rule_format,OF insert(2)] insert
hoelzl@35171:       by (auto intro: mult_mono mult_nonneg_nonneg)
hoelzl@35171:   qed auto
hoelzl@35171:   thus ?thesis by simp
hoelzl@35171: qed auto
hoelzl@35171: 
hoelzl@35171: lemma abs_setprod:
hoelzl@35171:   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
hoelzl@35171:   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
hoelzl@35171: proof (cases "finite A")
hoelzl@35171:   case True thus ?thesis
huffman@35216:     by induct (auto simp add: field_simps abs_mult)
hoelzl@35171: qed auto
hoelzl@35171: 
haftmann@31017: lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
nipkow@28853: apply (erule finite_induct)
huffman@35216: apply auto
nipkow@28853: done
nipkow@15402: 
chaieb@29674: lemma setprod_gen_delta:
chaieb@29674:   assumes fS: "finite S"
haftmann@31017:   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)"
chaieb@29674: proof-
chaieb@29674:   let ?f = "(\<lambda>k. if k=a then b k else c)"
chaieb@29674:   {assume a: "a \<notin> S"
chaieb@29674:     hence "\<forall> k\<in> S. ?f k = c" by simp
nipkow@48849:     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
chaieb@29674:   moreover 
chaieb@29674:   {assume a: "a \<in> S"
chaieb@29674:     let ?A = "S - {a}"
chaieb@29674:     let ?B = "{a}"
chaieb@29674:     have eq: "S = ?A \<union> ?B" using a by blast 
chaieb@29674:     have dj: "?A \<inter> ?B = {}" by simp
chaieb@29674:     from fS have fAB: "finite ?A" "finite ?B" by auto  
chaieb@29674:     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
chaieb@29674:       apply (rule setprod_cong) by auto
chaieb@29674:     have cA: "card ?A = card S - 1" using fS a by auto
chaieb@29674:     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
chaieb@29674:     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
chaieb@29674:       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
chaieb@29674:       by simp
chaieb@29674:     then have ?thesis using a cA
haftmann@36349:       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
chaieb@29674:   ultimately show ?thesis by blast
chaieb@29674: qed
chaieb@29674: 
chaieb@29674: 
haftmann@35816: subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}
haftmann@35816: 
haftmann@35816: no_notation times (infixl "*" 70)
haftmann@35816: no_notation Groups.one ("1")
haftmann@35816: 
haftmann@35816: locale semilattice_big = semilattice +
haftmann@35816:   fixes F :: "'a set \<Rightarrow> 'a"
haftmann@35816:   assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"
haftmann@35816: 
haftmann@35816: sublocale semilattice_big < folding_one_idem proof
haftmann@35816: qed (simp_all add: F_eq)
haftmann@35816: 
haftmann@35816: notation times (infixl "*" 70)
haftmann@35816: notation Groups.one ("1")
haftmann@22917: 
haftmann@35816: context lattice
haftmann@35816: begin
haftmann@35816: 
haftmann@35816: definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900) where
haftmann@35816:   "Inf_fin = fold1 inf"
haftmann@35816: 
haftmann@35816: definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900) where
haftmann@35816:   "Sup_fin = fold1 sup"
haftmann@35816: 
haftmann@35816: end
haftmann@35816: 
haftmann@35816: sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof
haftmann@35816: qed (simp add: Inf_fin_def)
haftmann@35816: 
haftmann@35816: sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof
haftmann@35816: qed (simp add: Sup_fin_def)
haftmann@22917: 
haftmann@35028: context semilattice_inf
haftmann@26041: begin
haftmann@26041: 
haftmann@36635: lemma ab_semigroup_idem_mult_inf:
haftmann@36635:   "class.ab_semigroup_idem_mult inf"
haftmann@35816: proof qed (rule inf_assoc inf_commute inf_idem)+
haftmann@35816: 
haftmann@46033: lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold inf b (insert a A) = inf a (Finite_Set.fold inf b A)"
haftmann@42871: by(rule comp_fun_idem.fold_insert_idem[OF ab_semigroup_idem_mult.comp_fun_idem[OF ab_semigroup_idem_mult_inf]])
haftmann@35816: 
haftmann@46033: 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"
haftmann@35816: by (induct pred: finite) (auto intro: le_infI1)
haftmann@35816: 
haftmann@46033: lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> Finite_Set.fold inf b A \<le> inf a b"
haftmann@35816: proof(induct arbitrary: a pred:finite)
haftmann@35816:   case empty thus ?case by simp
haftmann@35816: next
haftmann@35816:   case (insert x A)
haftmann@35816:   show ?case
haftmann@35816:   proof cases
haftmann@35816:     assume "A = {}" thus ?thesis using insert by simp
haftmann@35816:   next
haftmann@35816:     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
haftmann@35816:   qed
haftmann@35816: qed
haftmann@35816: 
haftmann@26041: lemma below_fold1_iff:
haftmann@26041:   assumes "finite A" "A \<noteq> {}"
haftmann@26041:   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult inf
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
haftmann@26041:   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_belowI:
haftmann@26757:   assumes "finite A"
haftmann@26041:     and "a \<in> A"
haftmann@26041:   shows "fold1 inf A \<le> a"
haftmann@26757: proof -
haftmann@26757:   from assms have "A \<noteq> {}" by auto
haftmann@26757:   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
haftmann@26757:   proof (induct rule: finite_ne_induct)
haftmann@26757:     case singleton thus ?case by simp
haftmann@26041:   next
haftmann@29509:     interpret ab_semigroup_idem_mult inf
haftmann@26757:       by (rule ab_semigroup_idem_mult_inf)
haftmann@26757:     case (insert x F)
haftmann@26757:     from insert(5) have "a = x \<or> a \<in> F" by simp
haftmann@26757:     thus ?case
haftmann@26757:     proof
haftmann@26757:       assume "a = x" thus ?thesis using insert
nipkow@29667:         by (simp add: mult_ac)
haftmann@26757:     next
haftmann@26757:       assume "a \<in> F"
haftmann@26757:       hence bel: "fold1 inf F \<le> a" by (rule insert)
haftmann@26757:       have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
nipkow@29667:         using insert by (simp add: mult_ac)
haftmann@26757:       also have "inf (fold1 inf F) a = fold1 inf F"
haftmann@26757:         using bel by (auto intro: antisym)
haftmann@26757:       also have "inf x \<dots> = fold1 inf (insert x F)"
nipkow@29667:         using insert by (simp add: mult_ac)
haftmann@26757:       finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
haftmann@26757:       moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
haftmann@26757:       ultimately show ?thesis by simp
haftmann@26757:     qed
haftmann@26041:   qed
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@35816: context semilattice_sup
haftmann@22917: begin
haftmann@22917: 
haftmann@36635: lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"
haftmann@35816: by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
haftmann@35816: 
haftmann@46033: lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold sup b (insert a A) = sup a (Finite_Set.fold sup b A)"
haftmann@35816: by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
haftmann@22917: 
haftmann@46033: 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"
haftmann@35816: by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
haftmann@35816: 
haftmann@46033: lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> Finite_Set.fold sup b A"
haftmann@35816: by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
haftmann@35816: 
haftmann@35816: end
haftmann@35816: 
haftmann@35816: context lattice
haftmann@35816: begin
haftmann@25062: 
wenzelm@31916: lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
haftmann@24342: apply(unfold Sup_fin_def Inf_fin_def)
nipkow@15500: apply(subgoal_tac "EX a. a:A")
nipkow@15500: prefer 2 apply blast
nipkow@15500: apply(erule exE)
haftmann@22388: apply(rule order_trans)
haftmann@26757: apply(erule (1) fold1_belowI)
haftmann@35028: apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
nipkow@15500: done
nipkow@15500: 
haftmann@24342: lemma sup_Inf_absorb [simp]:
wenzelm@31916:   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
nipkow@15512: apply(subst sup_commute)
haftmann@26041: apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
nipkow@15504: done
nipkow@15504: 
haftmann@24342: lemma inf_Sup_absorb [simp]:
wenzelm@31916:   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
haftmann@26041: by (simp add: Sup_fin_def inf_absorb1
haftmann@35028:   semilattice_inf.fold1_belowI [OF dual_semilattice])
haftmann@24342: 
haftmann@24342: end
haftmann@24342: 
haftmann@24342: context distrib_lattice
haftmann@24342: begin
haftmann@24342: 
haftmann@24342: lemma sup_Inf1_distrib:
haftmann@26041:   assumes "finite A"
haftmann@26041:     and "A \<noteq> {}"
wenzelm@31916:   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult inf
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Inf_fin_def image_def
haftmann@26041:       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
berghofe@26792:         (rule arg_cong [where f="fold1 inf"], blast)
haftmann@26041: qed
nipkow@18423: 
haftmann@24342: lemma sup_Inf2_distrib:
haftmann@24342:   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
wenzelm@31916:   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}"
haftmann@24342: using A proof (induct rule: finite_ne_induct)
nipkow@15500:   case singleton thus ?case
wenzelm@41550:     by (simp add: sup_Inf1_distrib [OF B])
nipkow@15500: next
haftmann@29509:   interpret ab_semigroup_idem_mult inf
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
nipkow@15500:   case (insert x A)
haftmann@25062:   have finB: "finite {sup x b |b. b \<in> B}"
haftmann@25062:     by(rule finite_surj[where f = "sup x", OF B(1)], auto)
haftmann@25062:   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
nipkow@15500:   proof -
haftmann@25062:     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
nipkow@15500:       by blast
berghofe@15517:     thus ?thesis by(simp add: insert(1) B(1))
nipkow@15500:   qed
haftmann@25062:   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
wenzelm@31916:   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)"
wenzelm@41550:     using insert by simp
wenzelm@31916:   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)
wenzelm@31916:   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})"
nipkow@15500:     using insert by(simp add:sup_Inf1_distrib[OF B])
wenzelm@31916:   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})"
wenzelm@31916:     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
nipkow@15500:     using B insert
haftmann@26041:     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
haftmann@25062:   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@15500:     by blast
nipkow@15500:   finally show ?case .
nipkow@15500: qed
nipkow@15500: 
haftmann@24342: lemma inf_Sup1_distrib:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
wenzelm@31916:   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult sup
haftmann@26041:     by (rule ab_semigroup_idem_mult_sup)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
berghofe@26792:       (rule arg_cong [where f="fold1 sup"], blast)
haftmann@26041: qed
nipkow@18423: 
haftmann@24342: lemma inf_Sup2_distrib:
haftmann@24342:   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
wenzelm@31916:   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}"
haftmann@24342: using A proof (induct rule: finite_ne_induct)
nipkow@18423:   case singleton thus ?case
huffman@44921:     by(simp add: inf_Sup1_distrib [OF B])
nipkow@18423: next
nipkow@18423:   case (insert x A)
haftmann@25062:   have finB: "finite {inf x b |b. b \<in> B}"
haftmann@25062:     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
haftmann@25062:   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
nipkow@18423:   proof -
haftmann@25062:     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
nipkow@18423:       by blast
nipkow@18423:     thus ?thesis by(simp add: insert(1) B(1))
nipkow@18423:   qed
haftmann@25062:   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@29509:   interpret ab_semigroup_idem_mult sup
haftmann@26041:     by (rule ab_semigroup_idem_mult_sup)
wenzelm@31916:   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)"
wenzelm@41550:     using insert by simp
wenzelm@31916:   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)
wenzelm@31916:   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})"
nipkow@18423:     using insert by(simp add:inf_Sup1_distrib[OF B])
wenzelm@31916:   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})"
wenzelm@31916:     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
nipkow@18423:     using B insert
haftmann@26041:     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
haftmann@25062:   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@18423:     by blast
nipkow@18423:   finally show ?case .
nipkow@18423: qed
nipkow@18423: 
haftmann@24342: end
haftmann@24342: 
haftmann@35719: context complete_lattice
haftmann@35719: begin
haftmann@35719: 
haftmann@35719: lemma Inf_fin_Inf:
haftmann@35719:   assumes "finite A" and "A \<noteq> {}"
haftmann@35719:   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
haftmann@35719: proof -
haftmann@35719:   interpret ab_semigroup_idem_mult inf
haftmann@35719:     by (rule ab_semigroup_idem_mult_inf)
noschinl@44918:   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
haftmann@35719:   moreover with `finite A` have "finite B" by simp
noschinl@44918:   ultimately show ?thesis
noschinl@44918:     by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
haftmann@35719: qed
haftmann@35719: 
haftmann@35719: lemma Sup_fin_Sup:
haftmann@35719:   assumes "finite A" and "A \<noteq> {}"
haftmann@35719:   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
haftmann@35719: proof -
haftmann@35719:   interpret ab_semigroup_idem_mult sup
haftmann@35719:     by (rule ab_semigroup_idem_mult_sup)
noschinl@44918:   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
haftmann@35719:   moreover with `finite A` have "finite B" by simp
haftmann@35719:   ultimately show ?thesis  
haftmann@35719:   by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
haftmann@35719: qed
haftmann@35719: 
haftmann@35719: end
haftmann@35719: 
haftmann@22917: 
haftmann@35816: subsection {* Versions of @{const min} and @{const max} on non-empty sets *}
haftmann@35816: 
haftmann@35816: definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where
haftmann@35816:   "Min = fold1 min"
haftmann@22917: 
haftmann@35816: definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where
haftmann@35816:   "Max = fold1 max"
haftmann@35816: 
haftmann@35816: sublocale linorder < Min!: semilattice_big min Min proof
haftmann@35816: qed (simp add: Min_def)
haftmann@35816: 
haftmann@35816: sublocale linorder < Max!: semilattice_big max Max proof
haftmann@35816: qed (simp add: Max_def)
haftmann@22917: 
haftmann@24342: context linorder
haftmann@22917: begin
haftmann@22917: 
haftmann@35816: lemmas Min_singleton = Min.singleton
haftmann@35816: lemmas Max_singleton = Max.singleton
haftmann@35816: 
haftmann@35816: lemma Min_insert:
haftmann@35816:   assumes "finite A" and "A \<noteq> {}"
haftmann@35816:   shows "Min (insert x A) = min x (Min A)"
haftmann@35816:   using assms by simp
haftmann@35816: 
haftmann@35816: lemma Max_insert:
haftmann@35816:   assumes "finite A" and "A \<noteq> {}"
haftmann@35816:   shows "Max (insert x A) = max x (Max A)"
haftmann@35816:   using assms by simp
haftmann@35816: 
haftmann@35816: lemma Min_Un:
haftmann@35816:   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
haftmann@35816:   shows "Min (A \<union> B) = min (Min A) (Min B)"
haftmann@35816:   using assms by (rule Min.union_idem)
haftmann@35816: 
haftmann@35816: lemma Max_Un:
haftmann@35816:   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
haftmann@35816:   shows "Max (A \<union> B) = max (Max A) (Max B)"
haftmann@35816:   using assms by (rule Max.union_idem)
haftmann@35816: 
haftmann@35816: lemma hom_Min_commute:
haftmann@35816:   assumes "\<And>x y. h (min x y) = min (h x) (h y)"
haftmann@35816:     and "finite N" and "N \<noteq> {}"
haftmann@35816:   shows "h (Min N) = Min (h ` N)"
haftmann@35816:   using assms by (rule Min.hom_commute)
haftmann@35816: 
haftmann@35816: lemma hom_Max_commute:
haftmann@35816:   assumes "\<And>x y. h (max x y) = max (h x) (h y)"
haftmann@35816:     and "finite N" and "N \<noteq> {}"
haftmann@35816:   shows "h (Max N) = Max (h ` N)"
haftmann@35816:   using assms by (rule Max.hom_commute)
haftmann@35816: 
haftmann@26041: lemma ab_semigroup_idem_mult_min:
haftmann@36635:   "class.ab_semigroup_idem_mult min"
haftmann@28823:   proof qed (auto simp add: min_def)
haftmann@26041: 
haftmann@26041: lemma ab_semigroup_idem_mult_max:
haftmann@36635:   "class.ab_semigroup_idem_mult max"
haftmann@28823:   proof qed (auto simp add: max_def)
haftmann@26041: 
haftmann@26041: lemma max_lattice:
krauss@44845:   "class.semilattice_inf max (op \<ge>) (op >)"
haftmann@32203:   by (fact min_max.dual_semilattice)
haftmann@26041: 
haftmann@26041: lemma dual_max:
haftmann@26041:   "ord.max (op \<ge>) = min"
wenzelm@46904:   by (auto simp add: ord.max_def min_def fun_eq_iff)
haftmann@26041: 
haftmann@26041: lemma dual_min:
haftmann@26041:   "ord.min (op \<ge>) = max"
wenzelm@46904:   by (auto simp add: ord.min_def max_def fun_eq_iff)
haftmann@26041: 
haftmann@26041: lemma strict_below_fold1_iff:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult min
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   by (induct rule: finite_ne_induct)
haftmann@26041:     (simp_all add: fold1_insert)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_below_iff:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult min
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   by (induct rule: finite_ne_induct)
haftmann@26041:     (simp_all add: fold1_insert min_le_iff_disj)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_strict_below_iff:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult min
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   by (induct rule: finite_ne_induct)
haftmann@26041:     (simp_all add: fold1_insert min_less_iff_disj)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_antimono:
haftmann@26041:   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
haftmann@26041:   shows "fold1 min B \<le> fold1 min A"
haftmann@26041: proof cases
haftmann@26041:   assume "A = B" thus ?thesis by simp
haftmann@26041: next
haftmann@29509:   interpret ab_semigroup_idem_mult min
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
wenzelm@41550:   assume neq: "A \<noteq> B"
haftmann@26041:   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
haftmann@26041:   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
haftmann@26041:   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
haftmann@26041:   proof -
haftmann@26041:     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
wenzelm@41550:     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])
wenzelm@41550:     moreover have "(B-A) \<noteq> {}" using assms neq by blast
wenzelm@41550:     moreover have "A Int (B-A) = {}" using assms by blast
haftmann@26041:     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
haftmann@26041:   qed
haftmann@26041:   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
haftmann@26041:   finally show ?thesis .
haftmann@26041: qed
haftmann@26041: 
paulson@24427: lemma Min_in [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A \<in> A"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult min
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
nipkow@44890:   from assms fold1_in show ?thesis by (fastforce simp: Min_def min_def)
haftmann@26041: qed
nipkow@15392: 
paulson@24427: lemma Max_in [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A \<in> A"
haftmann@26041: proof -
haftmann@29509:   interpret ab_semigroup_idem_mult max
haftmann@26041:     by (rule ab_semigroup_idem_mult_max)
nipkow@44890:   from assms fold1_in [of A] show ?thesis by (fastforce simp: Max_def max_def)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Min_le [simp]:
haftmann@26757:   assumes "finite A" and "x \<in> A"
haftmann@26041:   shows "Min A \<le> x"
haftmann@32203:   using assms by (simp add: Min_def min_max.fold1_belowI)
haftmann@26041: 
haftmann@26041: lemma Max_ge [simp]:
haftmann@26757:   assumes "finite A" and "x \<in> A"
haftmann@26041:   shows "x \<le> Max A"
huffman@44921:   by (simp add: Max_def semilattice_inf.fold1_belowI [OF max_lattice] assms)
haftmann@26041: 
blanchet@35828: lemma Min_ge_iff [simp, no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@32203:   using assms by (simp add: Min_def min_max.below_fold1_iff)
haftmann@26041: 
blanchet@35828: lemma Max_le_iff [simp, no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
huffman@44921:   by (simp add: Max_def semilattice_inf.below_fold1_iff [OF max_lattice] assms)
haftmann@26041: 
blanchet@35828: lemma Min_gr_iff [simp, no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@32203:   using assms by (simp add: Min_def strict_below_fold1_iff)
haftmann@26041: 
blanchet@35828: lemma Max_less_iff [simp, no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
huffman@44921:   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
huffman@44921:     linorder.strict_below_fold1_iff [OF dual_linorder] assms)
nipkow@18493: 
blanchet@35828: lemma Min_le_iff [no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@32203:   using assms by (simp add: Min_def fold1_below_iff)
nipkow@15497: 
blanchet@35828: lemma Max_ge_iff [no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
huffman@44921:   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
huffman@44921:     linorder.fold1_below_iff [OF dual_linorder] assms)
haftmann@22917: 
blanchet@35828: lemma Min_less_iff [no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@32203:   using assms by (simp add: Min_def fold1_strict_below_iff)
haftmann@22917: 
blanchet@35828: lemma Max_gr_iff [no_atp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
huffman@44921:   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
huffman@44921:     linorder.fold1_strict_below_iff [OF dual_linorder] assms)
haftmann@26041: 
haftmann@30325: lemma Min_eqI:
haftmann@30325:   assumes "finite A"
haftmann@30325:   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
haftmann@30325:     and "x \<in> A"
haftmann@30325:   shows "Min A = x"
haftmann@30325: proof (rule antisym)
haftmann@30325:   from `x \<in> A` have "A \<noteq> {}" by auto
haftmann@30325:   with assms show "Min A \<ge> x" by simp
haftmann@30325: next
haftmann@30325:   from assms show "x \<ge> Min A" by simp
haftmann@30325: qed
haftmann@30325: 
haftmann@30325: lemma Max_eqI:
haftmann@30325:   assumes "finite A"
haftmann@30325:   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
haftmann@30325:     and "x \<in> A"
haftmann@30325:   shows "Max A = x"
haftmann@30325: proof (rule antisym)
haftmann@30325:   from `x \<in> A` have "A \<noteq> {}" by auto
haftmann@30325:   with assms show "Max A \<le> x" by simp
haftmann@30325: next
haftmann@30325:   from assms show "x \<le> Max A" by simp
haftmann@30325: qed
haftmann@30325: 
haftmann@26041: lemma Min_antimono:
haftmann@26041:   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@26041:   shows "Min N \<le> Min M"
haftmann@32203:   using assms by (simp add: Min_def fold1_antimono)
haftmann@26041: 
haftmann@26041: lemma Max_mono:
haftmann@26041:   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@26041:   shows "Max M \<le> Max N"
huffman@44921:   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
huffman@44921:     linorder.fold1_antimono [OF dual_linorder] assms)
haftmann@22917: 
nipkow@32006: lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
urbanc@36079:  assumes fin: "finite A"
urbanc@36079:  and   empty: "P {}" 
urbanc@36079:  and  insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"
urbanc@36079:  shows "P A"
urbanc@36079: using fin empty insert
nipkow@32006: proof (induct rule: finite_psubset_induct)
urbanc@36079:   case (psubset A)
urbanc@36079:   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 
urbanc@36079:   have fin: "finite A" by fact 
urbanc@36079:   have empty: "P {}" by fact
urbanc@36079:   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
krauss@26748:   show "P A"
haftmann@26757:   proof (cases "A = {}")
urbanc@36079:     assume "A = {}" 
urbanc@36079:     then show "P A" using `P {}` by simp
krauss@26748:   next
urbanc@36079:     let ?B = "A - {Max A}" 
urbanc@36079:     let ?A = "insert (Max A) ?B"
urbanc@36079:     have "finite ?B" using `finite A` by simp
krauss@26748:     assume "A \<noteq> {}"
krauss@26748:     with `finite A` have "Max A : A" by auto
urbanc@36079:     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
urbanc@36079:     then have "P ?B" using `P {}` step IH[of ?B] by blast
urbanc@36079:     moreover 
nipkow@44890:     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
nipkow@44890:     ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastforce
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
nipkow@32006: lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
nipkow@33434:  "\<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"
nipkow@32006: by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
nipkow@32006: 
haftmann@22917: end
haftmann@22917: 
haftmann@35028: context linordered_ab_semigroup_add
haftmann@22917: begin
haftmann@22917: 
haftmann@22917: lemma add_Min_commute:
haftmann@22917:   fixes k
haftmann@25062:   assumes "finite N" and "N \<noteq> {}"
haftmann@25062:   shows "k + Min N = Min {k + m | m. m \<in> N}"
haftmann@25062: proof -
haftmann@25062:   have "\<And>x y. k + min x y = min (k + x) (k + y)"
haftmann@25062:     by (simp add: min_def not_le)
haftmann@25062:       (blast intro: antisym less_imp_le add_left_mono)
haftmann@25062:   with assms show ?thesis
haftmann@25062:     using hom_Min_commute [of "plus k" N]
haftmann@25062:     by simp (blast intro: arg_cong [where f = Min])
haftmann@25062: qed
haftmann@22917: 
haftmann@22917: lemma add_Max_commute:
haftmann@22917:   fixes k
haftmann@25062:   assumes "finite N" and "N \<noteq> {}"
haftmann@25062:   shows "k + Max N = Max {k + m | m. m \<in> N}"
haftmann@25062: proof -
haftmann@25062:   have "\<And>x y. k + max x y = max (k + x) (k + y)"
haftmann@25062:     by (simp add: max_def not_le)
haftmann@25062:       (blast intro: antisym less_imp_le add_left_mono)
haftmann@25062:   with assms show ?thesis
haftmann@25062:     using hom_Max_commute [of "plus k" N]
haftmann@25062:     by simp (blast intro: arg_cong [where f = Max])
haftmann@25062: qed
haftmann@22917: 
haftmann@22917: end
haftmann@22917: 
haftmann@35034: context linordered_ab_group_add
haftmann@35034: begin
haftmann@35034: 
haftmann@35034: lemma minus_Max_eq_Min [simp]:
haftmann@35034:   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
haftmann@35034:   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
haftmann@35034: 
haftmann@35034: lemma minus_Min_eq_Max [simp]:
haftmann@35034:   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
haftmann@35034:   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
haftmann@35034: 
haftmann@35034: end
haftmann@35034: 
haftmann@25571: end