
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@51489: imports Finite_Set Option Metis
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@51489: locale comm_monoid_set = comm_monoid
haftmann@51489: begin
haftmann@35816: 
haftmann@51738: interpretation comp_fun_commute f
haftmann@51738:   by default (simp add: fun_eq_iff left_commute)
haftmann@51738: 
haftmann@51738: interpretation comp_fun_commute "f \<circ> g"
haftmann@51738:   by (rule comp_comp_fun_commute)
haftmann@51738: 
haftmann@51489: definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
haftmann@35816: 
haftmann@35816: lemma infinite [simp]:
haftmann@35816:   "\<not> finite A \<Longrightarrow> F g A = 1"
haftmann@51489:   by (simp add: eq_fold)
haftmann@51489: 
haftmann@51489: lemma empty [simp]:
haftmann@51489:   "F g {} = 1"
haftmann@51489:   by (simp add: eq_fold)
haftmann@51489: 
haftmann@51489: lemma insert [simp]:
haftmann@51489:   assumes "finite A" and "x \<notin> A"
haftmann@51489:   shows "F g (insert x A) = g x * F g A"
haftmann@51738:   using assms by (simp add: eq_fold)
haftmann@51489: 
haftmann@51489: lemma remove:
haftmann@51489:   assumes "finite A" and "x \<in> A"
haftmann@51489:   shows "F g A = g x * F g (A - {x})"
haftmann@51489: proof -
haftmann@51489:   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
haftmann@51489:     by (auto dest: mk_disjoint_insert)
wenzelm@53374:   moreover from `finite A` A have "finite B" by simp
haftmann@51489:   ultimately show ?thesis by simp
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma insert_remove:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "F g (insert x A) = g x * F g (A - {x})"
haftmann@51489:   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
haftmann@51489: 
haftmann@51489: lemma neutral:
haftmann@51489:   assumes "\<forall>x\<in>A. g x = 1"
haftmann@51489:   shows "F g A = 1"
haftmann@51738:   using assms by (induct A rule: infinite_finite_induct) simp_all
haftmann@35816: 
haftmann@51489: lemma neutral_const [simp]:
haftmann@51489:   "F (\<lambda>_. 1) A = 1"
haftmann@51489:   by (simp add: neutral)
haftmann@51489: 
haftmann@51489: lemma union_inter:
haftmann@51489:   assumes "finite A" and "finite B"
haftmann@51489:   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
haftmann@51489:   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
haftmann@51489: using assms proof (induct A)
haftmann@51489:   case empty then show ?case by simp
hoelzl@42986: next
haftmann@51489:   case (insert x A) then show ?case
haftmann@51489:     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: corollary union_inter_neutral:
haftmann@51489:   assumes "finite A" and "finite B"
haftmann@51489:   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
haftmann@51489:   shows "F g (A \<union> B) = F g A * F g B"
haftmann@51489:   using assms by (simp add: union_inter [symmetric] neutral)
haftmann@51489: 
haftmann@51489: corollary union_disjoint:
haftmann@51489:   assumes "finite A" and "finite B"
haftmann@51489:   assumes "A \<inter> B = {}"
haftmann@51489:   shows "F g (A \<union> B) = F g A * F g B"
haftmann@51489:   using assms by (simp add: union_inter_neutral)
haftmann@51489: 
haftmann@51489: lemma subset_diff:
haftmann@51489:   "B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> F g A = F g (A - B) * F g B"
haftmann@51489:   by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
haftmann@51489: 
haftmann@51489: lemma reindex:
haftmann@51489:   assumes "inj_on h A"
haftmann@51489:   shows "F g (h ` A) = F (g \<circ> h) A"
haftmann@51489: proof (cases "finite A")
haftmann@51489:   case True
haftmann@51738:   with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
haftmann@51489: next
haftmann@51489:   case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
haftmann@51489:   with False show ?thesis by simp
hoelzl@42986: qed
hoelzl@42986: 
haftmann@51489: lemma cong:
haftmann@51489:   assumes "A = B"
haftmann@51489:   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
haftmann@51489:   shows "F g A = F h B"
haftmann@51489: proof (cases "finite A")
haftmann@51489:   case True
haftmann@51489:   then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
haftmann@51489:   proof induct
haftmann@51489:     case empty then show ?case by simp
haftmann@51489:   next
haftmann@51489:     case (insert x F) then show ?case apply -
haftmann@51489:     apply (simp add: subset_insert_iff, clarify)
haftmann@51489:     apply (subgoal_tac "finite C")
haftmann@51489:       prefer 2 apply (blast dest: finite_subset [rotated])
haftmann@51489:     apply (subgoal_tac "C = insert x (C - {x})")
haftmann@51489:       prefer 2 apply blast
haftmann@51489:     apply (erule ssubst)
haftmann@51489:     apply (simp add: Ball_def del: insert_Diff_single)
haftmann@51489:     done
haftmann@51489:   qed
haftmann@51489:   with `A = B` g_h show ?thesis by simp
haftmann@51489: next
haftmann@51489:   case False
haftmann@51489:   with `A = B` show ?thesis by simp
haftmann@51489: qed
nipkow@48849: 
haftmann@51489: lemma strong_cong [cong]:
haftmann@51489:   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
haftmann@51489:   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
haftmann@51489:   by (rule cong) (insert assms, simp_all add: simp_implies_def)
haftmann@51489: 
haftmann@51489: lemma UNION_disjoint:
haftmann@51489:   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
haftmann@51489:   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
haftmann@51489:   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
haftmann@51489: apply (insert assms)
haftmann@51489: apply (induct rule: finite_induct)
haftmann@51489: apply simp
haftmann@51489: apply atomize
haftmann@51489: apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
haftmann@51489:  prefer 2 apply blast
haftmann@51489: apply (subgoal_tac "A x Int UNION Fa A = {}")
haftmann@51489:  prefer 2 apply blast
haftmann@51489: apply (simp add: union_disjoint)
haftmann@51489: done
haftmann@51489: 
haftmann@51489: lemma Union_disjoint:
haftmann@51489:   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
haftmann@51489:   shows "F g (Union C) = F (F g) C"
haftmann@51489: proof cases
haftmann@51489:   assume "finite C"
haftmann@51489:   from UNION_disjoint [OF this assms]
haftmann@51489:   show ?thesis
haftmann@51489:     by (simp add: SUP_def)
haftmann@51489: qed (auto dest: finite_UnionD intro: infinite)
nipkow@48821: 
haftmann@51489: lemma distrib:
haftmann@51489:   "F (\<lambda>x. g x * h x) A = F g A * F h A"
haftmann@51738:   using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
haftmann@51489: 
haftmann@51489: lemma Sigma:
haftmann@51489:   "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (split g) (SIGMA x:A. B x)"
haftmann@51489: apply (subst Sigma_def)
haftmann@51489: apply (subst UNION_disjoint, assumption, simp)
haftmann@51489:  apply blast
haftmann@51489: apply (rule cong)
haftmann@51489: apply rule
haftmann@51489: apply (simp add: fun_eq_iff)
haftmann@51489: apply (subst UNION_disjoint, simp, simp)
haftmann@51489:  apply blast
haftmann@51489: apply (simp add: comp_def)
haftmann@51489: done
haftmann@51489: 
haftmann@51489: lemma related: 
haftmann@51489:   assumes Re: "R 1 1" 
haftmann@51489:   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
haftmann@51489:   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
haftmann@51489:   shows "R (F h S) (F g S)"
haftmann@51489:   using fS by (rule finite_subset_induct) (insert assms, auto)
nipkow@48849: 
haftmann@51489: lemma eq_general:
haftmann@51489:   assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
haftmann@51489:   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
haftmann@51489:   shows "F f1 S = F f2 S'"
haftmann@51489: proof-
haftmann@51489:   from h f12 have hS: "h ` S = S'" by blast
haftmann@51489:   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
haftmann@51489:     from f12 h H  have "x = y" by auto }
haftmann@51489:   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
haftmann@51489:   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
haftmann@51489:   from hS have "F f2 S' = F f2 (h ` S)" by simp
haftmann@51489:   also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
haftmann@51489:   also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
haftmann@51489:     by blast
haftmann@51489:   finally show ?thesis ..
haftmann@51489: qed
nipkow@48849: 
haftmann@51489: lemma eq_general_reverses:
haftmann@51489:   assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@51489:   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
haftmann@51489:   shows "F j S = F g T"
haftmann@51489:   (* metis solves it, but not yet available here *)
haftmann@51489:   apply (rule eq_general [of T S h g j])
haftmann@51489:   apply (rule ballI)
haftmann@51489:   apply (frule kh)
haftmann@51489:   apply (rule ex1I[])
haftmann@51489:   apply blast
haftmann@51489:   apply clarsimp
haftmann@51489:   apply (drule hk) apply simp
haftmann@51489:   apply (rule sym)
haftmann@51489:   apply (erule conjunct1[OF conjunct2[OF hk]])
haftmann@51489:   apply (rule ballI)
haftmann@51489:   apply (drule hk)
haftmann@51489:   apply blast
haftmann@51489:   done
haftmann@51489: 
haftmann@51489: lemma 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)
haftmann@51489:     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
nipkow@48849: qed
nipkow@48849: 
haftmann@51489: lemma 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"
haftmann@51489:   by (auto intro!: mono_neutral_cong_left [symmetric])
nipkow@48849: 
haftmann@51489: lemma 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"
haftmann@51489:   by (blast intro: mono_neutral_cong_left)
nipkow@48849: 
haftmann@51489: lemma 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"
haftmann@51489:   by (blast intro!: mono_neutral_left [symmetric])
nipkow@48849: 
haftmann@51489: lemma delta: 
nipkow@48849:   assumes fS: "finite S"
haftmann@51489:   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"
haftmann@51489:       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
nipkow@48849:       by simp
haftmann@51489:     then have ?thesis using a by simp }
nipkow@48849:   ultimately show ?thesis by blast
nipkow@48849: qed
nipkow@48849: 
haftmann@51489: lemma delta': 
haftmann@51489:   assumes fS: "finite S"
haftmann@51489:   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
haftmann@51489:   using delta [OF fS, of a b, symmetric] by (auto intro: cong)
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 =
haftmann@51489:     F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
haftmann@51489: 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"
haftmann@51489:   from union_disjoint [OF f a(2), of ?g] a(1)
hoelzl@42986:   show ?thesis
haftmann@51489:     by (subst (1 2) cong) simp_all
hoelzl@42986: qed
hoelzl@42986: 
haftmann@51489: lemma cartesian_product:
haftmann@51489:    "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
haftmann@51489: apply (rule sym)
haftmann@51489: apply (cases "finite A") 
haftmann@51489:  apply (cases "finite B") 
haftmann@51489:   apply (simp add: Sigma)
haftmann@51489:  apply (cases "A={}", simp)
haftmann@51489:  apply simp
haftmann@51489: apply (auto intro: infinite dest: finite_cartesian_productD2)
haftmann@51489: apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
haftmann@51489: done
haftmann@51489: 
haftmann@35816: end
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@51738: context comm_monoid_add
haftmann@51738: begin
haftmann@51738: 
haftmann@51738: definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   "setsum = comm_monoid_set.F plus 0"
haftmann@26041: 
haftmann@51738: sublocale setsum!: comm_monoid_set plus 0
haftmann@51489: where
haftmann@51546:   "comm_monoid_set.F plus 0 = setsum"
haftmann@51489: proof -
haftmann@51489:   show "comm_monoid_set plus 0" ..
haftmann@51489:   then interpret setsum!: comm_monoid_set plus 0 .
haftmann@51546:   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
haftmann@51489: qed
nipkow@15402: 
wenzelm@19535: abbreviation
haftmann@51489:   Setsum ("\<Sum>_" [1000] 999) where
haftmann@51489:   "\<Sum>A \<equiv> setsum (%x. x) A"
wenzelm@19535: 
haftmann@51738: end
haftmann@51738: 
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@49660:             val x' = Syntax_Trans.mark_bound_body (x, Tx);
wenzelm@35115:             val t' = subst_bound (x', t);
wenzelm@35115:             val P' = subst_bound (x', P);
wenzelm@49660:           in
wenzelm@49660:             Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
wenzelm@49660:           end
wenzelm@35115:     | setsum_tr' _ = raise Match;
wenzelm@52143: in [(@{const_syntax setsum}, K setsum_tr')] end
nipkow@15402: *}
nipkow@15402: 
haftmann@51489: text {* TODO These are candidates for generalization *}
nipkow@15402: 
haftmann@51489: context comm_monoid_add
haftmann@51489: begin
nipkow@15402: 
haftmann@51489: lemma setsum_reindex_id: 
haftmann@35816:   "inj_on f B ==> setsum f B = setsum id (f ` B)"
haftmann@51489:   by (simp add: setsum.reindex)
nipkow@15402: 
haftmann@51489: lemma setsum_reindex_nonzero:
chaieb@29674:   assumes fS: "finite S"
haftmann@51489:   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"
haftmann@51489:   shows "setsum h (f ` S) = setsum (h \<circ> f) S"
haftmann@51489: using nz 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
haftmann@51489:     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
haftmann@51489:       apply (simp cong del: setsum.strong_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`]
haftmann@51489:       apply (simp cong del: setsum.strong_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: 
haftmann@51489: lemma setsum_cong2:
haftmann@51489:   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
haftmann@51489:   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"
haftmann@51489:   by (simp add: setsum.reindex)
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"
haftmann@51489:   from setsum.mono_neutral_left [OF fA aba, of ?g]
chaieb@30260:   show ?thesis by simp
chaieb@30260: qed
chaieb@30260: 
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"
haftmann@51489:   using assms by (fact setsum.Union_disjoint)
nipkow@15402: 
haftmann@51489: lemma setsum_cartesian_product:
haftmann@51489:   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
haftmann@51489:   by (fact setsum.cartesian_product)
nipkow@15402: 
haftmann@51489: 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 
haftmann@51489:     by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: text {* Commuting outer and inner summation *}
haftmann@51489: 
haftmann@51489: lemma setsum_commute:
haftmann@51489:   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
haftmann@51489: proof (simp add: setsum_cartesian_product)
haftmann@51489:   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
haftmann@51489:     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
haftmann@51489:     (is "?s = _")
haftmann@51489:     apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
haftmann@51489:     apply (simp add: split_def)
haftmann@51489:     done
haftmann@51489:   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
haftmann@51489:     (is "_ = ?t")
haftmann@51489:     apply (simp add: swap_product)
haftmann@51489:     done
haftmann@51489:   finally show "?s = ?t" .
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma setsum_Plus:
haftmann@51489:   fixes A :: "'a set" and B :: "'b set"
haftmann@51489:   assumes fin: "finite A" "finite B"
haftmann@51489:   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
haftmann@51489: proof -
haftmann@51489:   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
haftmann@51489:   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
haftmann@51489:     by auto
haftmann@51489:   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
haftmann@51489:   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
haftmann@51489:   ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
nipkow@48893: qed
nipkow@48893: 
haftmann@51489: end
haftmann@51489: 
haftmann@51489: text {* TODO These are legacy *}
haftmann@51489: 
haftmann@51489: lemma setsum_empty:
haftmann@51489:   "setsum f {} = 0"
haftmann@51489:   by (fact setsum.empty)
haftmann@51489: 
haftmann@51489: lemma setsum_insert:
haftmann@51489:   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
haftmann@51489:   by (fact setsum.insert)
haftmann@51489: 
haftmann@51489: lemma setsum_infinite:
haftmann@51489:   "~ finite A ==> setsum f A = 0"
haftmann@51489:   by (fact setsum.infinite)
haftmann@51489: 
haftmann@51489: lemma setsum_reindex:
haftmann@51489:   "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
haftmann@51489:   by (fact setsum.reindex)
haftmann@51489: 
haftmann@51489: lemma setsum_cong:
haftmann@51489:   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
haftmann@51489:   by (fact setsum.cong)
haftmann@51489: 
haftmann@51489: lemma strong_setsum_cong:
haftmann@51489:   "A = B ==> (!!x. x:B =simp=> f x = g x)
haftmann@51489:    ==> setsum (%x. f x) A = setsum (%x. g x) B"
haftmann@51489:   by (fact setsum.strong_cong)
haftmann@51489: 
haftmann@51489: lemmas setsum_0 = setsum.neutral_const
haftmann@51489: lemmas setsum_0' = setsum.neutral
haftmann@51489: 
haftmann@51489: lemma setsum_Un_Int: "finite A ==> finite B ==>
haftmann@51489:   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
haftmann@51489:   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
haftmann@51489:   by (fact setsum.union_inter)
haftmann@51489: 
haftmann@51489: lemma setsum_Un_disjoint: "finite A ==> finite B
haftmann@51489:   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
haftmann@51489:   by (fact setsum.union_disjoint)
haftmann@51489: 
haftmann@51489: lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
haftmann@51489:     setsum f A = setsum f (A - B) + setsum f B"
haftmann@51489:   by (fact setsum.subset_diff)
haftmann@51489: 
haftmann@51489: lemma setsum_mono_zero_left: 
haftmann@51489:   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
haftmann@51489:   by (fact setsum.mono_neutral_left)
haftmann@51489: 
haftmann@51489: lemmas setsum_mono_zero_right = setsum.mono_neutral_right
haftmann@51489: 
haftmann@51489: lemma setsum_mono_zero_cong_left: 
haftmann@51489:   "\<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>
haftmann@51489:   \<Longrightarrow> setsum f S = setsum g T"
haftmann@51489:   by (fact setsum.mono_neutral_cong_left)
haftmann@51489: 
haftmann@51489: lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
haftmann@51489: 
haftmann@51489: lemma setsum_delta: "finite S \<Longrightarrow>
haftmann@51489:   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
haftmann@51489:   by (fact setsum.delta)
haftmann@51489: 
haftmann@51489: lemma setsum_delta': "finite S \<Longrightarrow>
haftmann@51489:   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
haftmann@51489:   by (fact setsum.delta')
haftmann@51489: 
haftmann@51489: lemma setsum_cases:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "setsum (\<lambda>x. if P x then f x else g x) A =
haftmann@51489:          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
haftmann@51489:   using assms by (fact setsum.If_cases)
haftmann@51489: 
haftmann@51489: (*But we can't get rid of finite I. If infinite, although the rhs is 0, 
haftmann@51489:   the lhs need not be, since UNION I A could still be finite.*)
haftmann@51489: lemma setsum_UN_disjoint:
haftmann@51489:   assumes "finite I" and "ALL i:I. finite (A i)"
haftmann@51489:     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
haftmann@51489:   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
haftmann@51489:   using assms by (fact setsum.UNION_disjoint)
haftmann@51489: 
haftmann@51489: (*But we can't get rid of finite A. If infinite, although the lhs is 0, 
haftmann@51489:   the rhs need not be, since SIGMA A B could still be finite.*)
haftmann@51489: lemma setsum_Sigma:
haftmann@51489:   assumes "finite A" and  "ALL x:A. finite (B x)"
haftmann@51489:   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@51489:   using assms by (fact setsum.Sigma)
haftmann@51489: 
haftmann@51489: lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
haftmann@51489:   by (fact setsum.distrib)
haftmann@51489: 
haftmann@51489: lemma setsum_Un_zero:  
haftmann@51489:   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
haftmann@51489:   setsum f (S \<union> T) = setsum f S + setsum f T"
haftmann@51489:   by (fact setsum.union_inter_neutral)
haftmann@51489: 
haftmann@51489: lemma setsum_eq_general_reverses:
haftmann@51489:   assumes fS: "finite S" and fT: "finite T"
haftmann@51489:   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
haftmann@51489:   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
haftmann@51489:   shows "setsum f S = setsum g T"
haftmann@51489:   using kh hk by (fact setsum.eq_general_reverses)
haftmann@51489: 
nipkow@15402: 
nipkow@15402: subsubsection {* Properties in more restricted classes of structures *}
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@49715: lemma setsum_Un2:
haftmann@49715:   assumes "finite (A \<union> B)"
haftmann@49715:   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
haftmann@49715: proof -
haftmann@49715:   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@49715:     by auto
haftmann@49715:   with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
haftmann@49715: qed
haftmann@49715: 
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@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
haftmann@51489:   case False then show ?thesis by simp
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
haftmann@51489:   case False thus ?thesis by simp
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"
haftmann@54230:   using setsum_addf [of f "- g" A] by (simp add: setsum_negf)
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
haftmann@51489:   case False thus ?thesis by simp
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
haftmann@51489:   case False thus ?thesis by simp
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
webertj@49962:     case (insert x A) thus ?case by (simp add: distrib_left)
nipkow@15402:   qed
nipkow@15402: next
haftmann@51489:   case False thus ?thesis by simp
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
webertj@49962:     case (insert x A) thus ?case by (simp add: distrib_right)
ballarin@17149:   qed
ballarin@17149: next
haftmann@51489:   case False thus ?thesis by simp
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
haftmann@51489:   case False thus ?thesis by simp
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
haftmann@51489:   case False thus ?thesis by simp
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
haftmann@51489:   case False thus ?thesis by simp
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
haftmann@51489:   case False thus ?thesis by simp
nipkow@31080: qed
nipkow@31080: 
haftmann@51489: lemma setsum_diff1'[rule_format]:
haftmann@51489:   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
haftmann@51489: 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))"])
haftmann@51489: apply (auto simp add: insert_Diff_if add_ac)
haftmann@51489: done
ballarin@17149: 
haftmann@51489: lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
haftmann@51489:   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
haftmann@51489: unfolding setsum_diff1'[OF assms] by auto
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@51489: lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
haftmann@51489: apply (case_tac "finite A")
haftmann@51489:  prefer 2 apply simp
haftmann@51489: apply (erule rev_mp)
haftmann@51489: apply (erule finite_induct, auto)
haftmann@51489: done
haftmann@51489: 
haftmann@51489: lemma setsum_eq_0_iff [simp]:
haftmann@51489:   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
haftmann@51489:   by (induct set: finite) auto
haftmann@51489: 
haftmann@51489: lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
haftmann@51489:   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
haftmann@51489: apply(erule finite_induct)
haftmann@51489: apply (auto simp add:add_is_1)
haftmann@51489: done
haftmann@51489: 
haftmann@51489: lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
haftmann@51489: 
haftmann@51489: lemma setsum_Un_nat: "finite A ==> finite B ==>
haftmann@51489:   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
haftmann@51489:   -- {* For the natural numbers, we have subtraction. *}
haftmann@51489: by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
haftmann@51489: 
haftmann@51489: lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
haftmann@51489:   (if a:A then setsum f A - f a else setsum f A)"
haftmann@51489: apply (case_tac "finite A")
haftmann@51489:  prefer 2 apply simp
haftmann@51489: apply (erule finite_induct)
haftmann@51489:  apply (auto simp add: insert_Diff_if)
haftmann@51489: apply (drule_tac a = a in mk_disjoint_insert, auto)
haftmann@51489: done
haftmann@51489: 
haftmann@51489: lemma setsum_diff_nat: 
haftmann@51489: assumes "finite B" and "B \<subseteq> A"
haftmann@51489: shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
haftmann@51489: using assms
haftmann@51489: proof induct
haftmann@51489:   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
haftmann@51489: next
haftmann@51489:   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
haftmann@51489:     and xFinA: "insert x F \<subseteq> A"
haftmann@51489:     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
haftmann@51489:   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
haftmann@51489:   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
haftmann@51489:     by (simp add: setsum_diff1_nat)
haftmann@51489:   from xFinA have "F \<subseteq> A" by simp
haftmann@51489:   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
haftmann@51489:   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
haftmann@51489:     by simp
haftmann@51489:   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
haftmann@51489:   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
haftmann@51489:     by simp
haftmann@51489:   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
haftmann@51489:   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
haftmann@51489:     by simp
haftmann@51489:   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
haftmann@51489: qed
haftmann@51489: 
haftmann@51600: lemma setsum_comp_morphism:
haftmann@51600:   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
haftmann@51600:   shows "setsum (h \<circ> g) A = h (setsum g A)"
haftmann@51600: proof (cases "finite A")
haftmann@51600:   case False then show ?thesis by (simp add: assms)
haftmann@51600: next
haftmann@51600:   case True then show ?thesis by (induct A) (simp_all add: assms)
haftmann@51600: qed
haftmann@51600: 
haftmann@51489: 
haftmann@51489: subsubsection {* Cardinality as special case of @{const setsum} *}
haftmann@51489: 
haftmann@51489: lemma card_eq_setsum:
haftmann@51489:   "card A = setsum (\<lambda>x. 1) A"
haftmann@51489: proof -
haftmann@51489:   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
haftmann@51489:     by (simp add: fun_eq_iff)
haftmann@51489:   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
haftmann@51489:     by (rule arg_cong)
haftmann@51489:   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
haftmann@51489:     by (blast intro: fun_cong)
haftmann@51489:   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma setsum_constant [simp]:
haftmann@51489:   "(\<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@51489:   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@51489:   case False thus ?thesis by simp
haftmann@35722: qed
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: 
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@51738: context comm_monoid_mult
haftmann@51738: begin
haftmann@51738: 
haftmann@51738: definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   "setprod = comm_monoid_set.F times 1"
haftmann@35816: 
haftmann@51738: sublocale setprod!: comm_monoid_set times 1
haftmann@51489: where
haftmann@51546:   "comm_monoid_set.F times 1 = setprod"
haftmann@51489: proof -
haftmann@51489:   show "comm_monoid_set times 1" ..
haftmann@51489:   then interpret setprod!: comm_monoid_set times 1 .
haftmann@51546:   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
haftmann@51489: qed
nipkow@15402: 
wenzelm@19535: abbreviation
haftmann@51489:   Setprod ("\<Prod>_" [1000] 999) where
haftmann@51489:   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
wenzelm@19535: 
haftmann@51738: end
haftmann@51738: 
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@51489: text {* TODO These are candidates for generalization *}
haftmann@51489: 
haftmann@51489: context comm_monoid_mult
haftmann@51489: begin
haftmann@51489: 
haftmann@51489: lemma setprod_reindex_id:
haftmann@51489:   "inj_on f B ==> setprod f B = setprod id (f ` B)"
haftmann@51489:   by (auto simp add: setprod.reindex)
haftmann@51489: 
haftmann@51489: lemma setprod_reindex_cong:
haftmann@51489:   "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
haftmann@51489:   by (frule setprod.reindex, simp)
haftmann@51489: 
haftmann@51489: lemma strong_setprod_reindex_cong:
haftmann@51489:   assumes i: "inj_on f A"
haftmann@51489:   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
haftmann@51489:   shows "setprod h B = setprod g A"
haftmann@51489: proof-
haftmann@51489:   have "setprod h B = setprod (h o f) A"
haftmann@51489:     by (simp add: B setprod.reindex [OF i, of h])
haftmann@51489:   then show ?thesis apply simp
haftmann@51489:     apply (rule setprod.cong)
haftmann@51489:     apply simp
haftmann@51489:     by (simp add: eq)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma setprod_Union_disjoint:
haftmann@51489:   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 
haftmann@51489:   shows "setprod f (Union C) = setprod (setprod f) C"
haftmann@51489:   using assms by (fact setprod.Union_disjoint)
haftmann@51489: 
haftmann@51489: text{*Here we can eliminate the finiteness assumptions, by cases.*}
haftmann@51489: lemma setprod_cartesian_product:
haftmann@51489:   "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
haftmann@51489:   by (fact setprod.cartesian_product)
haftmann@51489: 
haftmann@51489: lemma setprod_Un2:
haftmann@51489:   assumes "finite (A \<union> B)"
haftmann@51489:   shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
haftmann@51489: proof -
haftmann@51489:   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
haftmann@51489:     by auto
haftmann@51489:   with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: end
haftmann@51489: 
haftmann@51489: text {* TODO These are legacy *}
haftmann@51489: 
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:
haftmann@51489:   "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
haftmann@51489:   by (fact 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"
haftmann@51489:   by (fact setprod.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"
haftmann@51489:   by (fact setprod.strong_cong)
nipkow@15402: 
haftmann@51489: lemma setprod_Un_one:
haftmann@51489:   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
haftmann@51489:   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
haftmann@51489:   by (fact setprod.union_inter_neutral)
chaieb@29674: 
haftmann@51489: lemmas setprod_1 = setprod.neutral_const
haftmann@51489: lemmas setprod_1' = setprod.neutral
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"
haftmann@51489:   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"
haftmann@51489:   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"
haftmann@51489:   by (fact setprod.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"
haftmann@51489:   by (fact setprod.mono_neutral_left)
nipkow@30837: 
haftmann@51489: lemmas setprod_mono_one_right = setprod.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"
haftmann@51489:   by (fact setprod.mono_neutral_cong_left)
nipkow@48849: 
haftmann@51489: lemmas setprod_mono_one_cong_right = setprod.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)"
haftmann@51489:   by (fact setprod.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)"
haftmann@51489:   by (fact setprod.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"
haftmann@51489:   by (fact setprod.UNION_disjoint)
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)"
haftmann@51489:   by (fact setprod.Sigma)
nipkow@15402: 
haftmann@51489: lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
haftmann@51489:   by (fact setprod.distrib)
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsubsection {* Properties in more restricted classes of structures *}
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: 
haftmann@51489: lemma setprod_zero_iff[simp]: "finite A ==> 
haftmann@51489:   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
haftmann@51489:   (EX x: A. f x = 0)"
haftmann@51489: by (erule finite_induct, auto simp:no_zero_divisors)
haftmann@51489: 
haftmann@51489: lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
haftmann@51489:   (setprod f (A Un B) :: 'a ::{field})
haftmann@51489:    = setprod f A * setprod f B / setprod f (A Int B)"
haftmann@51489: by (subst setprod_Un_Int [symmetric], auto)
haftmann@51489: 
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@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@51489:   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: 
haftmann@51489: lemma setprod_eq_1_iff [simp]:
haftmann@51489:   "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
haftmann@51489:   by (induct set: finite) auto
chaieb@29674: 
haftmann@51489: lemma setprod_pos_nat:
haftmann@51489:   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
haftmann@51489: using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
haftmann@51489: 
haftmann@51489: lemma setprod_pos_nat_iff[simp]:
haftmann@51489:   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
haftmann@51489: using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
haftmann@51489: 
haftmann@51489: 
haftmann@51489: subsection {* Generic lattice operations over a set *}
haftmann@35816: 
haftmann@35816: no_notation times (infixl "*" 70)
haftmann@35816: no_notation Groups.one ("1")
haftmann@35816: 
haftmann@51489: 
haftmann@51489: subsubsection {* Without neutral element *}
haftmann@51489: 
haftmann@51489: locale semilattice_set = semilattice
haftmann@51489: begin
haftmann@51489: 
haftmann@51738: interpretation comp_fun_idem f
haftmann@51738:   by default (simp_all add: fun_eq_iff left_commute)
haftmann@51738: 
haftmann@51489: definition F :: "'a set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   eq_fold': "F A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)) None A)"
haftmann@51489: 
haftmann@51489: lemma eq_fold:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "F (insert x A) = Finite_Set.fold f x A"
haftmann@51489: proof (rule sym)
haftmann@51489:   let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
haftmann@51489:   interpret comp_fun_idem "?f"
haftmann@51489:     by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
haftmann@51489:   from assms show "Finite_Set.fold f x A = F (insert x A)"
haftmann@51489:   proof induct
haftmann@51489:     case empty then show ?case by (simp add: eq_fold')
haftmann@51489:   next
haftmann@51489:     case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
haftmann@51489:   qed
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma singleton [simp]:
haftmann@51489:   "F {x} = x"
haftmann@51489:   by (simp add: eq_fold)
haftmann@51489: 
haftmann@51489: lemma insert_not_elem:
haftmann@51489:   assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
haftmann@51489:   shows "F (insert x A) = x * F A"
haftmann@51489: proof -
haftmann@51489:   from `A \<noteq> {}` obtain b where "b \<in> A" by blast
haftmann@51489:   then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@51489:   with `finite A` and `x \<notin> A`
haftmann@51489:     have "finite (insert x B)" and "b \<notin> insert x B" by auto
haftmann@51489:   then have "F (insert b (insert x B)) = x * F (insert b B)"
haftmann@51489:     by (simp add: eq_fold)
haftmann@51489:   then show ?thesis by (simp add: * insert_commute)
haftmann@51489: qed
haftmann@51489: 
haftmann@51586: lemma in_idem:
haftmann@51489:   assumes "finite A" and "x \<in> A"
haftmann@51489:   shows "x * F A = F A"
haftmann@51489: proof -
haftmann@51489:   from assms have "A \<noteq> {}" by auto
haftmann@51489:   with `finite A` show ?thesis using `x \<in> A`
haftmann@51489:     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma insert [simp]:
haftmann@51489:   assumes "finite A" and "A \<noteq> {}"
haftmann@51489:   shows "F (insert x A) = x * F A"
haftmann@51586:   using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
haftmann@51489: 
haftmann@51489: lemma union:
haftmann@51489:   assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
haftmann@51489:   shows "F (A \<union> B) = F A * F B"
haftmann@51489:   using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
haftmann@51489: 
haftmann@51489: lemma remove:
haftmann@51489:   assumes "finite A" and "x \<in> A"
haftmann@51489:   shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@51489: proof -
haftmann@51489:   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@51489:   with assms show ?thesis by simp
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma insert_remove:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
haftmann@51489:   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@51489: 
haftmann@51489: lemma subset:
haftmann@51489:   assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
haftmann@51489:   shows "F B * F A = F A"
haftmann@51489: proof -
haftmann@51489:   from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
haftmann@51489:   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma closed:
haftmann@51489:   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@51489:   shows "F A \<in> A"
haftmann@51489: using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@51489:   case singleton then show ?case by simp
haftmann@51489: next
haftmann@51489:   case insert with elem show ?case by force
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma hom_commute:
haftmann@51489:   assumes hom: "\<And>x y. h (x * y) = h x * h y"
haftmann@51489:   and N: "finite N" "N \<noteq> {}"
haftmann@51489:   shows "h (F N) = F (h ` N)"
haftmann@51489: using N proof (induct rule: finite_ne_induct)
haftmann@51489:   case singleton thus ?case by simp
haftmann@51489: next
haftmann@51489:   case (insert n N)
haftmann@51489:   then have "h (F (insert n N)) = h (n * F N)" by simp
haftmann@51489:   also have "\<dots> = h n * h (F N)" by (rule hom)
haftmann@51489:   also have "h (F N) = F (h ` N)" by (rule insert)
haftmann@51489:   also have "h n * \<dots> = F (insert (h n) (h ` N))"
haftmann@51489:     using insert by simp
haftmann@51489:   also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@51489:   finally show ?case .
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: end
haftmann@51489: 
haftmann@51489: locale semilattice_order_set = semilattice_order + semilattice_set
haftmann@51489: begin
haftmann@51489: 
haftmann@51489: lemma bounded_iff:
haftmann@51489:   assumes "finite A" and "A \<noteq> {}"
haftmann@51489:   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
haftmann@51489:   using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
haftmann@51489: 
haftmann@51489: lemma boundedI:
haftmann@51489:   assumes "finite A"
haftmann@51489:   assumes "A \<noteq> {}"
haftmann@51489:   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489:   shows "x \<preceq> F A"
haftmann@51489:   using assms by (simp add: bounded_iff)
haftmann@51489: 
haftmann@51489: lemma boundedE:
haftmann@51489:   assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
haftmann@51489:   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489:   using assms by (simp add: bounded_iff)
haftmann@35816: 
haftmann@51489: lemma coboundedI:
haftmann@51489:   assumes "finite A"
haftmann@51489:     and "a \<in> A"
haftmann@51489:   shows "F A \<preceq> a"
haftmann@51489: proof -
haftmann@51489:   from assms have "A \<noteq> {}" by auto
haftmann@51489:   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
haftmann@51489:   proof (induct rule: finite_ne_induct)
haftmann@51489:     case singleton thus ?case by (simp add: refl)
haftmann@51489:   next
haftmann@51489:     case (insert x B)
haftmann@51489:     from insert have "a = x \<or> a \<in> B" by simp
haftmann@51489:     then show ?case using insert by (auto intro: coboundedI2)
haftmann@51489:   qed
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma antimono:
haftmann@51489:   assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
haftmann@51489:   shows "F B \<preceq> F A"
haftmann@51489: proof (cases "A = B")
haftmann@51489:   case True then show ?thesis by (simp add: refl)
haftmann@51489: next
haftmann@51489:   case False
haftmann@51489:   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
haftmann@51489:   then have "F B = F (A \<union> (B - A))" by simp
haftmann@51489:   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
haftmann@51489:   also have "\<dots> \<preceq> F A" by simp
haftmann@51489:   finally show ?thesis .
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: end
haftmann@51489: 
haftmann@51489: 
haftmann@51489: subsubsection {* With neutral element *}
haftmann@51489: 
haftmann@51489: locale semilattice_neutr_set = semilattice_neutr
haftmann@51489: begin
haftmann@51489: 
haftmann@51738: interpretation comp_fun_idem f
haftmann@51738:   by default (simp_all add: fun_eq_iff left_commute)
haftmann@51738: 
haftmann@51489: definition F :: "'a set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   eq_fold: "F A = Finite_Set.fold f 1 A"
haftmann@51489: 
haftmann@51489: lemma infinite [simp]:
haftmann@51489:   "\<not> finite A \<Longrightarrow> F A = 1"
haftmann@51489:   by (simp add: eq_fold)
haftmann@51489: 
haftmann@51489: lemma empty [simp]:
haftmann@51489:   "F {} = 1"
haftmann@51489:   by (simp add: eq_fold)
haftmann@51489: 
haftmann@51489: lemma insert [simp]:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "F (insert x A) = x * F A"
haftmann@51738:   using assms by (simp add: eq_fold)
haftmann@51489: 
haftmann@51586: lemma in_idem:
haftmann@51489:   assumes "finite A" and "x \<in> A"
haftmann@51489:   shows "x * F A = F A"
haftmann@51489: proof -
haftmann@51489:   from assms have "A \<noteq> {}" by auto
haftmann@51489:   with `finite A` show ?thesis using `x \<in> A`
haftmann@51489:     by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma union:
haftmann@51489:   assumes "finite A" and "finite B"
haftmann@51489:   shows "F (A \<union> B) = F A * F B"
haftmann@51489:   using assms by (induct A) (simp_all add: ac_simps)
haftmann@51489: 
haftmann@51489: lemma remove:
haftmann@51489:   assumes "finite A" and "x \<in> A"
haftmann@51489:   shows "F A = x * F (A - {x})"
haftmann@51489: proof -
haftmann@51489:   from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
haftmann@51489:   with assms show ?thesis by simp
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma insert_remove:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "F (insert x A) = x * F (A - {x})"
haftmann@51489:   using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
haftmann@51489: 
haftmann@51489: lemma subset:
haftmann@51489:   assumes "finite A" and "B \<subseteq> A"
haftmann@51489:   shows "F B * F A = F A"
haftmann@51489: proof -
haftmann@51489:   from assms have "finite B" by (auto dest: finite_subset)
haftmann@51489:   with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma closed:
haftmann@51489:   assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
haftmann@51489:   shows "F A \<in> A"
haftmann@51489: using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
haftmann@51489:   case singleton then show ?case by simp
haftmann@51489: next
haftmann@51489:   case insert with elem show ?case by force
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: end
haftmann@51489: 
haftmann@51489: locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
haftmann@51489: begin
haftmann@51489: 
haftmann@51489: lemma bounded_iff:
haftmann@51489:   assumes "finite A"
haftmann@51489:   shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
haftmann@51489:   using assms by (induct A) (simp_all add: bounded_iff)
haftmann@51489: 
haftmann@51489: lemma boundedI:
haftmann@51489:   assumes "finite A"
haftmann@51489:   assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489:   shows "x \<preceq> F A"
haftmann@51489:   using assms by (simp add: bounded_iff)
haftmann@51489: 
haftmann@51489: lemma boundedE:
haftmann@51489:   assumes "finite A" and "x \<preceq> F A"
haftmann@51489:   obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
haftmann@51489:   using assms by (simp add: bounded_iff)
haftmann@51489: 
haftmann@51489: lemma coboundedI:
haftmann@51489:   assumes "finite A"
haftmann@51489:     and "a \<in> A"
haftmann@51489:   shows "F A \<preceq> a"
haftmann@51489: proof -
haftmann@51489:   from assms have "A \<noteq> {}" by auto
haftmann@51489:   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
haftmann@51489:   proof (induct rule: finite_ne_induct)
haftmann@51489:     case singleton thus ?case by (simp add: refl)
haftmann@51489:   next
haftmann@51489:     case (insert x B)
haftmann@51489:     from insert have "a = x \<or> a \<in> B" by simp
haftmann@51489:     then show ?case using insert by (auto intro: coboundedI2)
haftmann@51489:   qed
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma antimono:
haftmann@51489:   assumes "A \<subseteq> B" and "finite B"
haftmann@51489:   shows "F B \<preceq> F A"
haftmann@51489: proof (cases "A = B")
haftmann@51489:   case True then show ?thesis by (simp add: refl)
haftmann@51489: next
haftmann@51489:   case False
haftmann@51489:   have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
haftmann@51489:   then have "F B = F (A \<union> (B - A))" by simp
haftmann@51489:   also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
haftmann@51489:   also have "\<dots> \<preceq> F A" by simp
haftmann@51489:   finally show ?thesis .
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: end
haftmann@35816: 
haftmann@35816: notation times (infixl "*" 70)
haftmann@35816: notation Groups.one ("1")
haftmann@22917: 
haftmann@35816: 
haftmann@51489: subsection {* Lattice operations on finite sets *}
haftmann@35816: 
haftmann@51489: text {*
haftmann@51489:   For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
haftmann@51489:   to @{class linorder}.  This is badly designed: both should depend on a common abstract
haftmann@51489:   distributive lattice rather than having this non-subclass dependecy between two
haftmann@51489:   classes.  But for the moment we have to live with it.  This forces us to setup
haftmann@51489:   this sublocale dependency simultaneously with the lattice operations on finite
haftmann@51489:   sets, to avoid garbage.
haftmann@51489: *}
haftmann@22917: 
wenzelm@53174: definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
haftmann@51489: where
haftmann@51489:   "Inf_fin = semilattice_set.F inf"
haftmann@26041: 
wenzelm@53174: definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
haftmann@51489: where
haftmann@51489:   "Sup_fin = semilattice_set.F sup"
haftmann@35816: 
haftmann@51738: context linorder
haftmann@51738: begin
haftmann@51738: 
haftmann@51738: definition Min :: "'a set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   "Min = semilattice_set.F min"
haftmann@35816: 
haftmann@51738: definition Max :: "'a set \<Rightarrow> 'a"
haftmann@51489: where
haftmann@51489:   "Max = semilattice_set.F max"
haftmann@51489: 
haftmann@51738: sublocale Min!: semilattice_order_set min less_eq less
haftmann@51540:   + Max!: semilattice_order_set max greater_eq greater
haftmann@51540: where
haftmann@51540:   "semilattice_set.F min = Min"
haftmann@51540:   and "semilattice_set.F max = Max"
haftmann@51540: proof -
haftmann@51540:   show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
haftmann@52364:   then interpret Min!: semilattice_order_set min less_eq less .
haftmann@51540:   show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
haftmann@51540:   then interpret Max!: semilattice_order_set max greater_eq greater .
haftmann@51540:   from Min_def show "semilattice_set.F min = Min" by rule
haftmann@51540:   from Max_def show "semilattice_set.F max = Max" by rule
haftmann@51540: qed
haftmann@51540: 
haftmann@51540: 
haftmann@51489: text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
haftmann@35816: 
haftmann@51738: sublocale min_max!: distrib_lattice min less_eq less max
haftmann@51489: where
haftmann@51489:   "semilattice_inf.Inf_fin min = Min"
haftmann@51489:   and "semilattice_sup.Sup_fin max = Max"
haftmann@26041: proof -
haftmann@51489:   show "class.distrib_lattice min less_eq less max"
haftmann@51489:   proof
haftmann@51489:     fix x y z
haftmann@51489:     show "max x (min y z) = min (max x y) (max x z)"
haftmann@51489:       by (auto simp add: min_def max_def)
haftmann@51489:   qed (auto simp add: min_def max_def not_le less_imp_le)
haftmann@51489:   then interpret min_max!: distrib_lattice min less_eq less max .
haftmann@51489:   show "semilattice_inf.Inf_fin min = Min"
haftmann@51489:     by (simp only: min_max.Inf_fin_def Min_def)
haftmann@51489:   show "semilattice_sup.Sup_fin max = Max"
haftmann@51489:     by (simp only: min_max.Sup_fin_def Max_def)
haftmann@26041: qed
haftmann@26041: 
haftmann@51489: lemmas le_maxI1 = min_max.sup_ge1
haftmann@51489: lemmas le_maxI2 = min_max.sup_ge2
haftmann@51489:  
haftmann@51489: lemmas min_ac = min_max.inf_assoc min_max.inf_commute
haftmann@51540:   min.left_commute
haftmann@51489: 
haftmann@51489: lemmas max_ac = min_max.sup_assoc min_max.sup_commute
haftmann@51540:   max.left_commute
haftmann@51489: 
haftmann@51738: end
haftmann@51738: 
haftmann@51489: 
haftmann@51489: text {* Lattice operations proper *}
haftmann@51489: 
haftmann@51489: sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
haftmann@51489: where
haftmann@51546:   "semilattice_set.F inf = Inf_fin"
haftmann@26757: proof -
haftmann@51489:   show "semilattice_order_set inf less_eq less" ..
haftmann@52364:   then interpret Inf_fin!: semilattice_order_set inf less_eq less .
haftmann@51546:   from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
haftmann@26041: qed
haftmann@26041: 
haftmann@51489: sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
haftmann@51489: where
haftmann@51546:   "semilattice_set.F sup = Sup_fin"
haftmann@51489: proof -
haftmann@51489:   show "semilattice_order_set sup greater_eq greater" ..
haftmann@51489:   then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
haftmann@51546:   from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
haftmann@51489: qed
haftmann@35816: 
haftmann@51489: 
haftmann@51540: text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
haftmann@51540: 
haftmann@51540: lemma Inf_fin_Min:
haftmann@51540:   "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
haftmann@51540:   by (simp add: Inf_fin_def Min_def inf_min)
haftmann@51540: 
haftmann@51540: lemma Sup_fin_Max:
haftmann@51540:   "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
haftmann@51540:   by (simp add: Sup_fin_def Max_def sup_max)
haftmann@51540: 
haftmann@51540: 
haftmann@51540: 
haftmann@51489: subsection {* Infimum and Supremum over non-empty sets *}
haftmann@22917: 
haftmann@51489: text {*
haftmann@51489:   After this non-regular bootstrap, things continue canonically.
haftmann@51489: *}
haftmann@35816: 
haftmann@35816: context lattice
haftmann@35816: begin
haftmann@25062: 
wenzelm@53174: lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^sub>f\<^sub>i\<^sub>nA \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>nA"
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@51489: apply(erule (1) Inf_fin.coboundedI)
haftmann@51489: apply(erule (1) Sup_fin.coboundedI)
nipkow@15500: done
nipkow@15500: 
haftmann@24342: lemma sup_Inf_absorb [simp]:
wenzelm@53174:   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = a"
nipkow@15512: apply(subst sup_commute)
haftmann@51489: apply(simp add: sup_absorb2 Inf_fin.coboundedI)
nipkow@15504: done
nipkow@15504: 
haftmann@24342: lemma inf_Sup_absorb [simp]:
wenzelm@53174:   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = a"
haftmann@51489: by (simp add: inf_absorb1 Sup_fin.coboundedI)
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@53174:   shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
haftmann@51489: using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
haftmann@51489:   (rule arg_cong [where f="Inf_fin"], blast)
nipkow@18423: 
haftmann@24342: lemma sup_Inf2_distrib:
haftmann@24342:   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
wenzelm@53174:   shows "sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup a b|a b. a \<in> A \<and> b \<in> B}"
haftmann@24342: using A proof (induct rule: finite_ne_induct)
haftmann@51489:   case singleton then show ?case
wenzelm@41550:     by (simp add: sup_Inf1_distrib [OF B])
nipkow@15500: next
nipkow@15500:   case (insert x A)
haftmann@25062:   have finB: "finite {sup x b |b. b \<in> B}"
haftmann@51489:     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@53174:   have "sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n(insert x A)) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB) = sup (inf x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA)) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB)"
wenzelm@41550:     using insert by simp
wenzelm@53174:   also have "\<dots> = inf (sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB)) (sup (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nB))" by(rule sup_inf_distrib2)
wenzelm@53174:   also have "\<dots> = inf (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x b|b. b \<in> B}) (\<Sqinter>\<^sub>f\<^sub>i\<^sub>n{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@53174:   also have "\<dots> = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
wenzelm@53174:     (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
nipkow@15500:     using B insert
haftmann@51489:     by (simp add: Inf_fin.union [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@53174:   shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
haftmann@51489: using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
haftmann@51489:   (rule arg_cong [where f="Sup_fin"], blast)
nipkow@18423: 
haftmann@24342: lemma inf_Sup2_distrib:
haftmann@24342:   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
wenzelm@53174:   shows "inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{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
wenzelm@53174:   have "inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>n(insert x A)) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB) = inf (sup x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA)) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB)"
wenzelm@41550:     using insert by simp
wenzelm@53174:   also have "\<dots> = sup (inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB)) (inf (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) (\<Squnion>\<^sub>f\<^sub>i\<^sub>nB))" by(rule inf_sup_distrib2)
wenzelm@53174:   also have "\<dots> = sup (\<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x b|b. b \<in> B}) (\<Squnion>\<^sub>f\<^sub>i\<^sub>n{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@53174:   also have "\<dots> = \<Squnion>\<^sub>f\<^sub>i\<^sub>n({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
wenzelm@53174:     (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
nipkow@18423:     using B insert
haftmann@51489:     by (simp add: Sup_fin.union [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> {}"
wenzelm@53174:   shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = Inf A"
haftmann@35719: proof -
haftmann@51489:   from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51489:   then show ?thesis
haftmann@51489:     by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
haftmann@35719: qed
haftmann@35719: 
haftmann@35719: lemma Sup_fin_Sup:
haftmann@35719:   assumes "finite A" and "A \<noteq> {}"
wenzelm@53174:   shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = Sup A"
haftmann@35719: proof -
haftmann@51489:   from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51489:   then show ?thesis
haftmann@51489:     by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
haftmann@35719: qed
haftmann@35719: 
haftmann@35719: end
haftmann@35719: 
haftmann@22917: 
haftmann@51489: subsection {* Minimum and Maximum over non-empty sets *}
haftmann@22917: 
haftmann@24342: context linorder
haftmann@22917: begin
haftmann@22917: 
haftmann@26041: lemma dual_min:
haftmann@51489:   "ord.min greater_eq = max"
wenzelm@46904:   by (auto simp add: ord.min_def max_def fun_eq_iff)
haftmann@26041: 
haftmann@51489: lemma dual_max:
haftmann@51489:   "ord.max greater_eq = min"
haftmann@51489:   by (auto simp add: ord.max_def min_def fun_eq_iff)
haftmann@51489: 
haftmann@51489: lemma dual_Min:
haftmann@51489:   "linorder.Min greater_eq = Max"
haftmann@26041: proof -
haftmann@51489:   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
haftmann@51489:   show ?thesis by (simp add: dual.Min_def dual_min Max_def)
haftmann@26041: qed
haftmann@26041: 
haftmann@51489: lemma dual_Max:
haftmann@51489:   "linorder.Max greater_eq = Min"
haftmann@26041: proof -
haftmann@51489:   interpret dual!: linorder greater_eq greater by (fact dual_linorder)
haftmann@51489:   show ?thesis by (simp add: dual.Max_def dual_max Min_def)
haftmann@26041: qed
haftmann@26041: 
haftmann@51540: lemmas Min_singleton = Min.singleton
haftmann@51540: lemmas Max_singleton = Max.singleton
haftmann@51540: lemmas Min_insert = Min.insert
haftmann@51540: lemmas Max_insert = Max.insert
haftmann@51540: lemmas Min_Un = Min.union
haftmann@51540: lemmas Max_Un = Max.union
haftmann@51540: lemmas hom_Min_commute = Min.hom_commute
haftmann@51540: lemmas hom_Max_commute = Max.hom_commute
haftmann@26041: 
paulson@24427: lemma Min_in [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A \<in> A"
haftmann@51540:   using assms by (auto simp add: min_def Min.closed)
nipkow@15392: 
paulson@24427: lemma Max_in [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A \<in> A"
haftmann@51540:   using assms by (auto simp add: max_def Max.closed)
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@51540:   using assms by (fact Min.coboundedI)
haftmann@26041: 
haftmann@26041: lemma Max_ge [simp]:
haftmann@26757:   assumes "finite A" and "x \<in> A"
haftmann@26041:   shows "x \<le> Max A"
haftmann@51540:   using assms by (fact Max.coboundedI)
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@51738: context
haftmann@51738:   fixes A :: "'a set"
haftmann@51738:   assumes fin_nonempty: "finite A" "A \<noteq> {}"
haftmann@51738: begin
haftmann@51738: 
blanchet@54147: lemma Min_ge_iff [simp]:
haftmann@51738:   "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@51738:   using fin_nonempty by (fact Min.bounded_iff)
haftmann@51489: 
blanchet@54147: lemma Max_le_iff [simp]:
haftmann@51738:   "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
haftmann@51738:   using fin_nonempty by (fact Max.bounded_iff)
haftmann@51489: 
blanchet@54147: lemma Min_gr_iff [simp]:
haftmann@51738:   "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@51738:   using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
haftmann@51489: 
blanchet@54147: lemma Max_less_iff [simp]:
haftmann@51738:   "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
haftmann@51738:   using fin_nonempty by (induct rule: finite_ne_induct) simp_all
haftmann@51489: 
blanchet@54147: lemma Min_le_iff:
haftmann@51738:   "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@51738:   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
haftmann@51489: 
blanchet@54147: lemma Max_ge_iff:
haftmann@51738:   "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
haftmann@51738:   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
haftmann@51489: 
blanchet@54147: lemma Min_less_iff:
haftmann@51738:   "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@51738:   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
haftmann@51489: 
blanchet@54147: lemma Max_gr_iff:
haftmann@51738:   "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
haftmann@51738:   using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
haftmann@51738: 
haftmann@51738: end
haftmann@51489: 
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@51540:   using assms by (fact Min.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"
haftmann@51540:   using assms by (fact Max.antimono)
haftmann@51489: 
haftmann@51489: lemma mono_Min_commute:
haftmann@51489:   assumes "mono f"
haftmann@51489:   assumes "finite A" and "A \<noteq> {}"
haftmann@51489:   shows "f (Min A) = Min (f ` A)"
haftmann@51489: proof (rule linorder_class.Min_eqI [symmetric])
haftmann@51489:   from `finite A` show "finite (f ` A)" by simp
haftmann@51489:   from assms show "f (Min A) \<in> f ` A" by simp
haftmann@51489:   fix x
haftmann@51489:   assume "x \<in> f ` A"
haftmann@51489:   then obtain y where "y \<in> A" and "x = f y" ..
haftmann@51489:   with assms have "Min A \<le> y" by auto
haftmann@51489:   with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
haftmann@51489:   with `x = f y` show "f (Min A) \<le> x" by simp
haftmann@51489: qed
haftmann@22917: 
haftmann@51489: lemma mono_Max_commute:
haftmann@51489:   assumes "mono f"
haftmann@51489:   assumes "finite A" and "A \<noteq> {}"
haftmann@51489:   shows "f (Max A) = Max (f ` A)"
haftmann@51489: proof (rule linorder_class.Max_eqI [symmetric])
haftmann@51489:   from `finite A` show "finite (f ` A)" by simp
haftmann@51489:   from assms show "f (Max A) \<in> f ` A" by simp
haftmann@51489:   fix x
haftmann@51489:   assume "x \<in> f ` A"
haftmann@51489:   then obtain y where "y \<in> A" and "x = f y" ..
haftmann@51489:   with assms have "y \<le> Max A" by auto
haftmann@51489:   with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
haftmann@51489:   with `x = f y` show "x \<le> f (Max A)" by simp
haftmann@51489: qed
haftmann@51489: 
haftmann@51489: lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
haftmann@51489:   assumes fin: "finite A"
haftmann@51489:   and empty: "P {}" 
haftmann@51489:   and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
haftmann@51489:   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
haftmann@51489:     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
haftmann@51489:     ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
haftmann@51489: lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
haftmann@51489:   "\<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"
haftmann@51489:   by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
nipkow@32006: 
haftmann@52379: lemma Least_Min:
haftmann@52379:   assumes "finite {a. P a}" and "\<exists>a. P a"
haftmann@52379:   shows "(LEAST a. P a) = Min {a. P a}"
haftmann@52379: proof -
haftmann@52379:   { fix A :: "'a set"
haftmann@52379:     assume A: "finite A" "A \<noteq> {}"
haftmann@52379:     have "(LEAST a. a \<in> A) = Min A"
haftmann@52379:     using A proof (induct A rule: finite_ne_induct)
haftmann@52379:       case singleton show ?case by (rule Least_equality) simp_all
haftmann@52379:     next
haftmann@52379:       case (insert a A)
haftmann@52379:       have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
haftmann@52379:         by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
haftmann@52379:       with insert show ?case by simp
haftmann@52379:     qed
haftmann@52379:   } from this [of "{a. P a}"] assms show ?thesis by simp
haftmann@52379: qed
haftmann@52379: 
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@51489:   "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@51489:   "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@51540: context complete_linorder
haftmann@51540: begin
haftmann@51540: 
haftmann@51540: lemma Min_Inf:
haftmann@51540:   assumes "finite A" and "A \<noteq> {}"
haftmann@51540:   shows "Min A = Inf A"
haftmann@51540: proof -
haftmann@51540:   from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51540:   then show ?thesis
haftmann@51540:     by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
haftmann@51540: qed
haftmann@51540: 
haftmann@51540: lemma Max_Sup:
haftmann@51540:   assumes "finite A" and "A \<noteq> {}"
haftmann@51540:   shows "Max A = Sup A"
haftmann@51540: proof -
haftmann@51540:   from assms obtain b B where "A = insert b B" and "finite B" by auto
haftmann@51540:   then show ?thesis
haftmann@51540:     by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
haftmann@51540: qed
haftmann@51540: 
haftmann@25571: end
haftmann@51263: 
haftmann@51540: end




