more algebraic terminology for theories about big operators

--- a/src/Doc/Main/Main_Doc.thy	Sat Dec 14 20:46:36 2013 +0100
+++ b/src/Doc/Main/Main_Doc.thy	Sun Dec 15 15:10:14 2013 +0100
@@ -407,8 +407,8 @@
 @{const Finite_Set.finite} & @{term_type_only Finite_Set.finite "'a set\<Rightarrow>bool"}\\
 @{const Finite_Set.card} & @{term_type_only Finite_Set.card "'a set => nat"}\\
 @{const Finite_Set.fold} & @{term_type_only Finite_Set.fold "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\
-@{const Big_Operators.setsum} & @{term_type_only Big_Operators.setsum "('a => 'b) => 'a set => 'b::comm_monoid_add"}\\
-@{const Big_Operators.setprod} & @{term_type_only Big_Operators.setprod "('a => 'b) => 'a set => 'b::comm_monoid_mult"}\\
+@{const Groups_Big.setsum} & @{term_type_only Groups_Big.setsum "('a => 'b) => 'a set => 'b::comm_monoid_add"}\\
+@{const Groups_Big.setprod} & @{term_type_only Groups_Big.setprod "('a => 'b) => 'a set => 'b::comm_monoid_mult"}\\
 \end{supertabular}
 
 

--- a/src/HOL/Big_Operators.thy	Sat Dec 14 20:46:36 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2193 +0,0 @@
-(*  Title:      HOL/Big_Operators.thy
-    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
-                with contributions by Jeremy Avigad
-*)
-
-header {* Big operators and finite (non-empty) sets *}
-
-theory Big_Operators
-imports Finite_Set Metis
-begin
-
-subsection {* Generic monoid operation over a set *}
-
-no_notation times (infixl "*" 70)
-no_notation Groups.one ("1")
-
-locale comm_monoid_set = comm_monoid
-begin
-
-interpretation comp_fun_commute f
-  by default (simp add: fun_eq_iff left_commute)
-
-interpretation comp_fun_commute "f \<circ> g"
-  by (rule comp_comp_fun_commute)
-
-definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
-where
-  eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
-
-lemma infinite [simp]:
-  "\<not> finite A \<Longrightarrow> F g A = 1"
-  by (simp add: eq_fold)
-
-lemma empty [simp]:
-  "F g {} = 1"
-  by (simp add: eq_fold)
-
-lemma insert [simp]:
-  assumes "finite A" and "x \<notin> A"
-  shows "F g (insert x A) = g x * F g A"
-  using assms by (simp add: eq_fold)
-
-lemma remove:
-  assumes "finite A" and "x \<in> A"
-  shows "F g A = g x * F g (A - {x})"
-proof -
-  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
-    by (auto dest: mk_disjoint_insert)
-  moreover from `finite A` A have "finite B" by simp
-  ultimately show ?thesis by simp
-qed
-
-lemma insert_remove:
-  assumes "finite A"
-  shows "F g (insert x A) = g x * F g (A - {x})"
-  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
-
-lemma neutral:
-  assumes "\<forall>x\<in>A. g x = 1"
-  shows "F g A = 1"
-  using assms by (induct A rule: infinite_finite_induct) simp_all
-
-lemma neutral_const [simp]:
-  "F (\<lambda>_. 1) A = 1"
-  by (simp add: neutral)
-
-lemma union_inter:
-  assumes "finite A" and "finite B"
-  shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
-  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
-using assms proof (induct A)
-  case empty then show ?case by simp
-next
-  case (insert x A) then show ?case
-    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
-qed
-
-corollary union_inter_neutral:
-  assumes "finite A" and "finite B"
-  and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
-  shows "F g (A \<union> B) = F g A * F g B"
-  using assms by (simp add: union_inter [symmetric] neutral)
-
-corollary union_disjoint:
-  assumes "finite A" and "finite B"
-  assumes "A \<inter> B = {}"
-  shows "F g (A \<union> B) = F g A * F g B"
-  using assms by (simp add: union_inter_neutral)
-
-lemma subset_diff:
-  "B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> F g A = F g (A - B) * F g B"
-  by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
-
-lemma reindex:
-  assumes "inj_on h A"
-  shows "F g (h ` A) = F (g \<circ> h) A"
-proof (cases "finite A")
-  case True
-  with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
-next
-  case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
-  with False show ?thesis by simp
-qed
-
-lemma cong:
-  assumes "A = B"
-  assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
-  shows "F g A = F h B"
-proof (cases "finite A")
-  case True
-  then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
-  proof induct
-    case empty then show ?case by simp
-  next
-    case (insert x F) then show ?case apply -
-    apply (simp add: subset_insert_iff, clarify)
-    apply (subgoal_tac "finite C")
-      prefer 2 apply (blast dest: finite_subset [rotated])
-    apply (subgoal_tac "C = insert x (C - {x})")
-      prefer 2 apply blast
-    apply (erule ssubst)
-    apply (simp add: Ball_def del: insert_Diff_single)
-    done
-  qed
-  with `A = B` g_h show ?thesis by simp
-next
-  case False
-  with `A = B` show ?thesis by simp
-qed
-
-lemma strong_cong [cong]:
-  assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
-  shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
-  by (rule cong) (insert assms, simp_all add: simp_implies_def)
-
-lemma UNION_disjoint:
-  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
-  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
-  shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
-apply (insert assms)
-apply (induct rule: finite_induct)
-apply simp
-apply atomize
-apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
- prefer 2 apply blast
-apply (subgoal_tac "A x Int UNION Fa A = {}")
- prefer 2 apply blast
-apply (simp add: union_disjoint)
-done
-
-lemma Union_disjoint:
-  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
-  shows "F g (Union C) = F (F g) C"
-proof cases
-  assume "finite C"
-  from UNION_disjoint [OF this assms]
-  show ?thesis
-    by (simp add: SUP_def)
-qed (auto dest: finite_UnionD intro: infinite)
-
-lemma distrib:
-  "F (\<lambda>x. g x * h x) A = F g A * F h A"
-  using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
-
-lemma Sigma:
-  "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)"
-apply (subst Sigma_def)
-apply (subst UNION_disjoint, assumption, simp)
- apply blast
-apply (rule cong)
-apply rule
-apply (simp add: fun_eq_iff)
-apply (subst UNION_disjoint, simp, simp)
- apply blast
-apply (simp add: comp_def)
-done
-
-lemma related: 
-  assumes Re: "R 1 1" 
-  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
-  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
-  shows "R (F h S) (F g S)"
-  using fS by (rule finite_subset_induct) (insert assms, auto)
-
-lemma eq_general:
-  assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
-  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
-  shows "F f1 S = F f2 S'"
-proof-
-  from h f12 have hS: "h ` S = S'" by blast
-  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
-    from f12 h H  have "x = y" by auto }
-  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
-  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
-  from hS have "F f2 S' = F f2 (h ` S)" by simp
-  also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
-  also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
-    by blast
-  finally show ?thesis ..
-qed
-
-lemma eq_general_reverses:
-  assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
-  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
-  shows "F j S = F g T"
-  (* metis solves it, but not yet available here *)
-  apply (rule eq_general [of T S h g j])
-  apply (rule ballI)
-  apply (frule kh)
-  apply (rule ex1I[])
-  apply blast
-  apply clarsimp
-  apply (drule hk) apply simp
-  apply (rule sym)
-  apply (erule conjunct1[OF conjunct2[OF hk]])
-  apply (rule ballI)
-  apply (drule hk)
-  apply blast
-  done
-
-lemma mono_neutral_cong_left:
-  assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
-  and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
-proof-
-  have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
-  have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
-  from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
-    by (auto intro: finite_subset)
-  show ?thesis using assms(4)
-    by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
-qed
-
-lemma mono_neutral_cong_right:
-  "\<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>
-   \<Longrightarrow> F g T = F h S"
-  by (auto intro!: mono_neutral_cong_left [symmetric])
-
-lemma mono_neutral_left:
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
-  by (blast intro: mono_neutral_cong_left)
-
-lemma mono_neutral_right:
-  "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
-  by (blast intro!: mono_neutral_left [symmetric])
-
-lemma delta: 
-  assumes fS: "finite S"
-  shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else 1)"
-  { assume a: "a \<notin> S"
-    hence "\<forall>k\<in>S. ?f k = 1" by simp
-    hence ?thesis  using a by simp }
-  moreover
-  { assume a: "a \<in> S"
-    let ?A = "S - {a}"
-    let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast 
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto  
-    have "F ?f S = F ?f ?A * F ?f ?B"
-      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
-      by simp
-    then have ?thesis using a by simp }
-  ultimately show ?thesis by blast
-qed
-
-lemma delta': 
-  assumes fS: "finite S"
-  shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
-  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
-
-lemma If_cases:
-  fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
-  assumes fA: "finite A"
-  shows "F (\<lambda>x. if P x then h x else g x) A =
-    F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
-proof -
-  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 
-          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 
-    by blast+
-  from fA 
-  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
-  let ?g = "\<lambda>x. if P x then h x else g x"
-  from union_disjoint [OF f a(2), of ?g] a(1)
-  show ?thesis
-    by (subst (1 2) cong) simp_all
-qed
-
-lemma cartesian_product:
-   "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
-apply (rule sym)
-apply (cases "finite A") 
- apply (cases "finite B") 
-  apply (simp add: Sigma)
- apply (cases "A={}", simp)
- apply simp
-apply (auto intro: infinite dest: finite_cartesian_productD2)
-apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
-done
-
-end
-
-notation times (infixl "*" 70)
-notation Groups.one ("1")
-
-
-subsection {* Generalized summation over a set *}
-
-context comm_monoid_add
-begin
-
-definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
-where
-  "setsum = comm_monoid_set.F plus 0"
-
-sublocale setsum!: comm_monoid_set plus 0
-where
-  "comm_monoid_set.F plus 0 = setsum"
-proof -
-  show "comm_monoid_set plus 0" ..
-  then interpret setsum!: comm_monoid_set plus 0 .
-  from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
-qed
-
-abbreviation
-  Setsum ("\<Sum>_" [1000] 999) where
-  "\<Sum>A \<equiv> setsum (%x. x) A"
-
-end
-
-text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
-written @{text"\<Sum>x\<in>A. e"}. *}
-
-syntax
-  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
-syntax (xsymbols)
-  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
-syntax (HTML output)
-  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
-
-translations -- {* Beware of argument permutation! *}
-  "SUM i:A. b" == "CONST setsum (%i. b) A"
-  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
-
-text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
- @{text"\<Sum>x|P. e"}. *}
-
-syntax
-  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
-syntax (xsymbols)
-  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
-syntax (HTML output)
-  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
-
-translations
-  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
-  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
-
-print_translation {*
-let
-  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
-        if x <> y then raise Match
-        else
-          let
-            val x' = Syntax_Trans.mark_bound_body (x, Tx);
-            val t' = subst_bound (x', t);
-            val P' = subst_bound (x', P);
-          in
-            Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
-          end
-    | setsum_tr' _ = raise Match;
-in [(@{const_syntax setsum}, K setsum_tr')] end
-*}
-
-text {* TODO These are candidates for generalization *}
-
-context comm_monoid_add
-begin
-
-lemma setsum_reindex_id: 
-  "inj_on f B ==> setsum f B = setsum id (f ` B)"
-  by (simp add: setsum.reindex)
-
-lemma setsum_reindex_nonzero:
-  assumes fS: "finite S"
-  and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
-  shows "setsum h (f ` S) = setsum (h \<circ> f) S"
-using nz proof (induct rule: finite_induct [OF fS])
-  case 1 thus ?case by simp
-next
-  case (2 x F) 
-  { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
-    then obtain y where y: "y \<in> F" "f x = f y" by auto 
-    from "2.hyps" y have xy: "x \<noteq> y" by auto
-    from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
-    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
-    also have "\<dots> = setsum (h o f) (insert x F)" 
-      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
-      using h0
-      apply (simp cong del: setsum.strong_cong)
-      apply (rule "2.hyps"(3))
-      apply (rule_tac y="y" in  "2.prems")
-      apply simp_all
-      done
-    finally have ?case . }
-  moreover
-  { assume fxF: "f x \<notin> f ` F"
-    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
-      using fxF "2.hyps" by simp 
-    also have "\<dots> = setsum (h o f) (insert x F)"
-      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
-      apply (simp cong del: setsum.strong_cong)
-      apply (rule cong [OF refl [of "op + (h (f x))"]])
-      apply (rule "2.hyps"(3))
-      apply (rule_tac y="y" in  "2.prems")
-      apply simp_all
-      done
-    finally have ?case . }
-  ultimately show ?case by blast
-qed
-
-lemma setsum_cong2:
-  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
-  by (auto intro: setsum.cong)
-
-lemma setsum_reindex_cong:
-   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
-    ==> setsum h B = setsum g A"
-  by (simp add: setsum.reindex)
-
-lemma setsum_restrict_set:
-  assumes fA: "finite A"
-  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
-proof-
-  from fA have fab: "finite (A \<inter> B)" by auto
-  have aba: "A \<inter> B \<subseteq> A" by blast
-  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
-  from setsum.mono_neutral_left [OF fA aba, of ?g]
-  show ?thesis by simp
-qed
-
-lemma setsum_Union_disjoint:
-  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
-  shows "setsum f (Union C) = setsum (setsum f) C"
-  using assms by (fact setsum.Union_disjoint)
-
-lemma setsum_cartesian_product:
-  "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
-  by (fact setsum.cartesian_product)
-
-lemma setsum_UNION_zero:
-  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
-  and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
-  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
-  using fSS f0
-proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case by simp
-next
-  case (2 T F)
-  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
-    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
-  from fTF have fUF: "finite (\<Union>F)" by auto
-  from "2.prems" TF fTF
-  show ?case 
-    by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
-qed
-
-text {* Commuting outer and inner summation *}
-
-lemma setsum_commute:
-  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
-proof (simp add: setsum_cartesian_product)
-  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
-    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
-    (is "?s = _")
-    apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
-    apply (simp add: split_def)
-    done
-  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
-    (is "_ = ?t")
-    apply (simp add: swap_product)
-    done
-  finally show "?s = ?t" .
-qed
-
-lemma setsum_Plus:
-  fixes A :: "'a set" and B :: "'b set"
-  assumes fin: "finite A" "finite B"
-  shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
-proof -
-  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
-  moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
-    by auto
-  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
-  moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
-  ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
-qed
-
-end
-
-text {* TODO These are legacy *}
-
-lemma setsum_empty:
-  "setsum f {} = 0"
-  by (fact setsum.empty)
-
-lemma setsum_insert:
-  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
-  by (fact setsum.insert)
-
-lemma setsum_infinite:
-  "~ finite A ==> setsum f A = 0"
-  by (fact setsum.infinite)
-
-lemma setsum_reindex:
-  "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
-  by (fact setsum.reindex)
-
-lemma setsum_cong:
-  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
-  by (fact setsum.cong)
-
-lemma strong_setsum_cong:
-  "A = B ==> (!!x. x:B =simp=> f x = g x)
-   ==> setsum (%x. f x) A = setsum (%x. g x) B"
-  by (fact setsum.strong_cong)
-
-lemmas setsum_0 = setsum.neutral_const
-lemmas setsum_0' = setsum.neutral
-
-lemma setsum_Un_Int: "finite A ==> finite B ==>
-  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
-  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
-  by (fact setsum.union_inter)
-
-lemma setsum_Un_disjoint: "finite A ==> finite B
-  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
-  by (fact setsum.union_disjoint)
-
-lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
-    setsum f A = setsum f (A - B) + setsum f B"
-  by (fact setsum.subset_diff)
-
-lemma setsum_mono_zero_left: 
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
-  by (fact setsum.mono_neutral_left)
-
-lemmas setsum_mono_zero_right = setsum.mono_neutral_right
-
-lemma setsum_mono_zero_cong_left: 
-  "\<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>
-  \<Longrightarrow> setsum f S = setsum g T"
-  by (fact setsum.mono_neutral_cong_left)
-
-lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
-
-lemma setsum_delta: "finite S \<Longrightarrow>
-  setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
-  by (fact setsum.delta)
-
-lemma setsum_delta': "finite S \<Longrightarrow>
-  setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
-  by (fact setsum.delta')
-
-lemma setsum_cases:
-  assumes "finite A"
-  shows "setsum (\<lambda>x. if P x then f x else g x) A =
-         setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
-  using assms by (fact setsum.If_cases)
-
-(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
-  the lhs need not be, since UNION I A could still be finite.*)
-lemma setsum_UN_disjoint:
-  assumes "finite I" and "ALL i:I. finite (A i)"
-    and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
-  shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
-  using assms by (fact setsum.UNION_disjoint)
-
-(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
-  the rhs need not be, since SIGMA A B could still be finite.*)
-lemma setsum_Sigma:
-  assumes "finite A" and  "ALL x:A. finite (B x)"
-  shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
-  using assms by (fact setsum.Sigma)
-
-lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
-  by (fact setsum.distrib)
-
-lemma setsum_Un_zero:  
-  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
-  setsum f (S \<union> T) = setsum f S + setsum f T"
-  by (fact setsum.union_inter_neutral)
-
-lemma setsum_eq_general_reverses:
-  assumes fS: "finite S" and fT: "finite T"
-  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
-  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
-  shows "setsum f S = setsum g T"
-  using kh hk by (fact setsum.eq_general_reverses)
-
-
-subsubsection {* Properties in more restricted classes of structures *}
-
-lemma setsum_Un: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: 'a :: ab_group_add) =
-   setsum f A + setsum f B - setsum f (A Int B)"
-by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
-
-lemma setsum_Un2:
-  assumes "finite (A \<union> B)"
-  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
-proof -
-  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
-    by auto
-  with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
-qed
-
-lemma setsum_diff1: "finite A \<Longrightarrow>
-  (setsum f (A - {a}) :: ('a::ab_group_add)) =
-  (if a:A then setsum f A - f a else setsum f A)"
-by (erule finite_induct) (auto simp add: insert_Diff_if)
-
-lemma setsum_diff:
-  assumes le: "finite A" "B \<subseteq> A"
-  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
-proof -
-  from le have finiteB: "finite B" using finite_subset by auto
-  show ?thesis using finiteB le
-  proof induct
-    case empty
-    thus ?case by auto
-  next
-    case (insert x F)
-    thus ?case using le finiteB 
-      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
-  qed
-qed
-
-lemma setsum_mono:
-  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
-  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
-proof (cases "finite K")
-  case True
-  thus ?thesis using le
-  proof induct
-    case empty
-    thus ?case by simp
-  next
-    case insert
-    thus ?case using add_mono by fastforce
-  qed
-next
-  case False then show ?thesis by simp
-qed
-
-lemma setsum_strict_mono:
-  fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
-  assumes "finite A"  "A \<noteq> {}"
-    and "!!x. x:A \<Longrightarrow> f x < g x"
-  shows "setsum f A < setsum g A"
-  using assms
-proof (induct rule: finite_ne_induct)
-  case singleton thus ?case by simp
-next
-  case insert thus ?case by (auto simp: add_strict_mono)
-qed
-
-lemma setsum_strict_mono_ex1:
-fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
-assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
-shows "setsum f A < setsum g A"
-proof-
-  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
-  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
-    by(simp add:insert_absorb[OF `a:A`])
-  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
-    using `finite A` by(subst setsum_Un_disjoint) auto
-  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
-    by(rule setsum_mono)(simp add: assms(2))
-  also have "setsum f {a} < setsum g {a}" using a by simp
-  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
-    using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
-  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
-  finally show ?thesis by (metis add_right_mono add_strict_left_mono)
-qed
-
-lemma setsum_negf:
-  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
-proof (cases "finite A")
-  case True thus ?thesis by (induct set: finite) auto
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_subtractf:
-  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
-    setsum f A - setsum g A"
-  using setsum_addf [of f "- g" A] by (simp add: setsum_negf)
-
-lemma setsum_nonneg:
-  assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
-  shows "0 \<le> setsum f A"
-proof (cases "finite A")
-  case True thus ?thesis using nn
-  proof induct
-    case empty then show ?case by simp
-  next
-    case (insert x F)
-    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
-    with insert show ?case by simp
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_nonpos:
-  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
-  shows "setsum f A \<le> 0"
-proof (cases "finite A")
-  case True thus ?thesis using np
-  proof induct
-    case empty then show ?case by simp
-  next
-    case (insert x F)
-    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
-    with insert show ?case by simp
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_nonneg_leq_bound:
-  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
-  assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
-  shows "f i \<le> B"
-proof -
-  have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
-    using assms by (auto intro!: setsum_nonneg)
-  moreover
-  have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
-    using assms by (simp add: setsum_diff1)
-  ultimately show ?thesis by auto
-qed
-
-lemma setsum_nonneg_0:
-  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
-  assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
-  and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
-  shows "f i = 0"
-  using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
-
-lemma setsum_mono2:
-fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
-assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
-shows "setsum f A \<le> setsum f B"
-proof -
-  have "setsum f A \<le> setsum f A + setsum f (B-A)"
-    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
-  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
-    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
-  also have "A \<union> (B-A) = B" using sub by blast
-  finally show ?thesis .
-qed
-
-lemma setsum_mono3: "finite B ==> A <= B ==> 
-    ALL x: B - A. 
-      0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
-        setsum f A <= setsum f B"
-  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
-  apply (erule ssubst)
-  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
-  apply simp
-  apply (rule add_left_mono)
-  apply (erule setsum_nonneg)
-  apply (subst setsum_Un_disjoint [THEN sym])
-  apply (erule finite_subset, assumption)
-  apply (rule finite_subset)
-  prefer 2
-  apply assumption
-  apply (auto simp add: sup_absorb2)
-done
-
-lemma setsum_right_distrib: 
-  fixes f :: "'a => ('b::semiring_0)"
-  shows "r * setsum f A = setsum (%n. r * f n) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: distrib_left)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_left_distrib:
-  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
-proof (cases "finite A")
-  case True
-  then show ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: distrib_right)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_divide_distrib:
-  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
-proof (cases "finite A")
-  case True
-  then show ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: add_divide_distrib)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_abs[iff]: 
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A)
-    thus ?case by (auto intro: abs_triangle_ineq order_trans)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_abs_ge_zero[iff]: 
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "0 \<le> setsum (%i. abs(f i)) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by auto
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma abs_setsum_abs[simp]: 
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert a A)
-    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
-    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
-    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
-      by (simp del: abs_of_nonneg)
-    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
-    finally show ?case .
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_diff1'[rule_format]:
-  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
-apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
-apply (auto simp add: insert_Diff_if add_ac)
-done
-
-lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
-  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
-unfolding setsum_diff1'[OF assms] by auto
-
-lemma setsum_product:
-  fixes f :: "'a => ('b::semiring_0)"
-  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
-  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
-
-lemma setsum_mult_setsum_if_inj:
-fixes f :: "'a => ('b::semiring_0)"
-shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
-  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
-by(auto simp: setsum_product setsum_cartesian_product
-        intro!:  setsum_reindex_cong[symmetric])
-
-lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
-apply (case_tac "finite A")
- prefer 2 apply simp
-apply (erule rev_mp)
-apply (erule finite_induct, auto)
-done
-
-lemma setsum_eq_0_iff [simp]:
-  "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
-  by (induct set: finite) auto
-
-lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
-  setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
-apply(erule finite_induct)
-apply (auto simp add:add_is_1)
-done
-
-lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
-
-lemma setsum_Un_nat: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
-  -- {* For the natural numbers, we have subtraction. *}
-by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
-
-lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
-  (if a:A then setsum f A - f a else setsum f A)"
-apply (case_tac "finite A")
- prefer 2 apply simp
-apply (erule finite_induct)
- apply (auto simp add: insert_Diff_if)
-apply (drule_tac a = a in mk_disjoint_insert, auto)
-done
-
-lemma setsum_diff_nat: 
-assumes "finite B" and "B \<subseteq> A"
-shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
-using assms
-proof induct
-  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
-next
-  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
-    and xFinA: "insert x F \<subseteq> A"
-    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
-  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
-  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
-    by (simp add: setsum_diff1_nat)
-  from xFinA have "F \<subseteq> A" by simp
-  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
-  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
-    by simp
-  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
-  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
-    by simp
-  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
-  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
-    by simp
-  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
-qed
-
-lemma setsum_comp_morphism:
-  assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
-  shows "setsum (h \<circ> g) A = h (setsum g A)"
-proof (cases "finite A")
-  case False then show ?thesis by (simp add: assms)
-next
-  case True then show ?thesis by (induct A) (simp_all add: assms)
-qed
-
-
-subsubsection {* Cardinality as special case of @{const setsum} *}
-
-lemma card_eq_setsum:
-  "card A = setsum (\<lambda>x. 1) A"
-proof -
-  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
-    by (simp add: fun_eq_iff)
-  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
-    by (rule arg_cong)
-  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
-    by (blast intro: fun_cong)
-  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
-qed
-
-lemma setsum_constant [simp]:
-  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
-apply (cases "finite A")
-apply (erule finite_induct)
-apply (auto simp add: algebra_simps)
-done
-
-lemma setsum_bounded:
-  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
-  shows "setsum f A \<le> of_nat (card A) * K"
-proof (cases "finite A")
-  case True
-  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
-next
-  case False thus ?thesis by simp
-qed
-
-lemma card_UN_disjoint:
-  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
-    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
-  shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
-proof -
-  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
-  with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
-qed
-
-lemma card_Union_disjoint:
-  "finite C ==> (ALL A:C. finite A) ==>
-   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
-   ==> card (Union C) = setsum card C"
-apply (frule card_UN_disjoint [of C id])
-apply (simp_all add: SUP_def id_def)
-done
-
-
-subsubsection {* Cardinality of products *}
-
-lemma card_SigmaI [simp]:
-  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
-  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
-by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
-
-(*
-lemma SigmaI_insert: "y \<notin> A ==>
-  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
-  by auto
-*)
-
-lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
-  by (cases "finite A \<and> finite B")
-    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
-
-lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
-by (simp add: card_cartesian_product)
-
-
-subsection {* Generalized product over a set *}
-
-context comm_monoid_mult
-begin
-
-definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
-where
-  "setprod = comm_monoid_set.F times 1"
-
-sublocale setprod!: comm_monoid_set times 1
-where
-  "comm_monoid_set.F times 1 = setprod"
-proof -
-  show "comm_monoid_set times 1" ..
-  then interpret setprod!: comm_monoid_set times 1 .
-  from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
-qed
-
-abbreviation
-  Setprod ("\<Prod>_" [1000] 999) where
-  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
-
-end
-
-syntax
-  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
-syntax (xsymbols)
-  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
-syntax (HTML output)
-  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
-
-translations -- {* Beware of argument permutation! *}
-  "PROD i:A. b" == "CONST setprod (%i. b) A" 
-  "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 
-
-text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
- @{text"\<Prod>x|P. e"}. *}
-
-syntax
-  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
-syntax (xsymbols)
-  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
-syntax (HTML output)
-  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
-
-translations
-  "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
-  "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
-
-text {* TODO These are candidates for generalization *}
-
-context comm_monoid_mult
-begin
-
-lemma setprod_reindex_id:
-  "inj_on f B ==> setprod f B = setprod id (f ` B)"
-  by (auto simp add: setprod.reindex)
-
-lemma setprod_reindex_cong:
-  "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
-  by (frule setprod.reindex, simp)
-
-lemma strong_setprod_reindex_cong:
-  assumes i: "inj_on f A"
-  and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
-  shows "setprod h B = setprod g A"
-proof-
-  have "setprod h B = setprod (h o f) A"
-    by (simp add: B setprod.reindex [OF i, of h])
-  then show ?thesis apply simp
-    apply (rule setprod.cong)
-    apply simp
-    by (simp add: eq)
-qed
-
-lemma setprod_Union_disjoint:
-  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 
-  shows "setprod f (Union C) = setprod (setprod f) C"
-  using assms by (fact setprod.Union_disjoint)
-
-text{*Here we can eliminate the finiteness assumptions, by cases.*}
-lemma setprod_cartesian_product:
-  "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
-  by (fact setprod.cartesian_product)
-
-lemma setprod_Un2:
-  assumes "finite (A \<union> B)"
-  shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
-proof -
-  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
-    by auto
-  with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
-qed
-
-end
-
-text {* TODO These are legacy *}
-
-lemma setprod_empty: "setprod f {} = 1"
-  by (fact setprod.empty)
-
-lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
-    setprod f (insert a A) = f a * setprod f A"
-  by (fact setprod.insert)
-
-lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
-  by (fact setprod.infinite)
-
-lemma setprod_reindex:
-  "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
-  by (fact setprod.reindex)
-
-lemma setprod_cong:
-  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
-  by (fact setprod.cong)
-
-lemma strong_setprod_cong:
-  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
-  by (fact setprod.strong_cong)
-
-lemma setprod_Un_one:
-  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
-  \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
-  by (fact setprod.union_inter_neutral)
-
-lemmas setprod_1 = setprod.neutral_const
-lemmas setprod_1' = setprod.neutral
-
-lemma setprod_Un_Int: "finite A ==> finite B
-    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
-  by (fact setprod.union_inter)
-
-lemma setprod_Un_disjoint: "finite A ==> finite B
-  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
-  by (fact setprod.union_disjoint)
-
-lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
-    setprod f A = setprod f (A - B) * setprod f B"
-  by (fact setprod.subset_diff)
-
-lemma setprod_mono_one_left:
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
-  by (fact setprod.mono_neutral_left)
-
-lemmas setprod_mono_one_right = setprod.mono_neutral_right
-
-lemma setprod_mono_one_cong_left: 
-  "\<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>
-  \<Longrightarrow> setprod f S = setprod g T"
-  by (fact setprod.mono_neutral_cong_left)
-
-lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
-
-lemma setprod_delta: "finite S \<Longrightarrow>
-  setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
-  by (fact setprod.delta)
-
-lemma setprod_delta': "finite S \<Longrightarrow>
-  setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
-  by (fact setprod.delta')
-
-lemma setprod_UN_disjoint:
-    "finite I ==> (ALL i:I. finite (A i)) ==>
-        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
-      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
-  by (fact setprod.UNION_disjoint)
-
-lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
-    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
-    (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
-  by (fact setprod.Sigma)
-
-lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
-  by (fact setprod.distrib)
-
-
-subsubsection {* Properties in more restricted classes of structures *}
-
-lemma setprod_zero:
-     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
-apply (induct set: finite, force, clarsimp)
-apply (erule disjE, auto)
-done
-
-lemma setprod_zero_iff[simp]: "finite A ==> 
-  (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
-  (EX x: A. f x = 0)"
-by (erule finite_induct, auto simp:no_zero_divisors)
-
-lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
-  (setprod f (A Un B) :: 'a ::{field})
-   = setprod f A * setprod f B / setprod f (A Int B)"
-by (subst setprod_Un_Int [symmetric], auto)
-
-lemma setprod_nonneg [rule_format]:
-   "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
-by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
-
-lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
-  --> 0 < setprod f A"
-by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
-
-lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
-  (setprod f (A - {a}) :: 'a :: {field}) =
-  (if a:A then setprod f A / f a else setprod f A)"
-  by (erule finite_induct) (auto simp add: insert_Diff_if)
-
-lemma setprod_inversef: 
-  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
-  shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
-by (erule finite_induct) auto
-
-lemma setprod_dividef:
-  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
-  shows "finite A
-    ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
-apply (subgoal_tac
-         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
-apply (erule ssubst)
-apply (subst divide_inverse)
-apply (subst setprod_timesf)
-apply (subst setprod_inversef, assumption+, rule refl)
-apply (rule setprod_cong, rule refl)
-apply (subst divide_inverse, auto)
-done
-
-lemma setprod_dvd_setprod [rule_format]: 
-    "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
-  apply (cases "finite A")
-  apply (induct set: finite)
-  apply (auto simp add: dvd_def)
-  apply (rule_tac x = "k * ka" in exI)
-  apply (simp add: algebra_simps)
-done
-
-lemma setprod_dvd_setprod_subset:
-  "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
-  apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
-  apply (unfold dvd_def, blast)
-  apply (subst setprod_Un_disjoint [symmetric])
-  apply (auto elim: finite_subset intro: setprod_cong)
-done
-
-lemma setprod_dvd_setprod_subset2:
-  "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 
-      setprod f A dvd setprod g B"
-  apply (rule dvd_trans)
-  apply (rule setprod_dvd_setprod, erule (1) bspec)
-  apply (erule (1) setprod_dvd_setprod_subset)
-done
-
-lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 
-    (f i ::'a::comm_semiring_1) dvd setprod f A"
-by (induct set: finite) (auto intro: dvd_mult)
-
-lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 
-    (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
-  apply (cases "finite A")
-  apply (induct set: finite)
-  apply auto
-done
-
-lemma setprod_mono:
-  fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
-  assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
-  shows "setprod f A \<le> setprod g A"
-proof (cases "finite A")
-  case True
-  hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
-  proof (induct A rule: finite_subset_induct)
-    case (insert a F)
-    thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
-      unfolding setprod_insert[OF insert(1,3)]
-      using assms[rule_format,OF insert(2)] insert
-      by (auto intro: mult_mono mult_nonneg_nonneg)
-  qed auto
-  thus ?thesis by simp
-qed auto
-
-lemma abs_setprod:
-  fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
-  shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
-proof (cases "finite A")
-  case True thus ?thesis
-    by induct (auto simp add: field_simps abs_mult)
-qed auto
-
-lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
-apply (erule finite_induct)
-apply auto
-done
-
-lemma setprod_gen_delta:
-  assumes fS: "finite S"
-  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else c)"
-  {assume a: "a \<notin> S"
-    hence "\<forall> k\<in> S. ?f k = c" by simp
-    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
-  moreover 
-  {assume a: "a \<in> S"
-    let ?A = "S - {a}"
-    let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast 
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto  
-    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
-      apply (rule setprod_cong) by auto
-    have cA: "card ?A = card S - 1" using fS a by auto
-    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
-    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
-      using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
-      by simp
-    then have ?thesis using a cA
-      by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
-  ultimately show ?thesis by blast
-qed
-
-lemma setprod_eq_1_iff [simp]:
-  "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
-  by (induct set: finite) auto
-
-lemma setprod_pos_nat:
-  "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
-using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
-
-lemma setprod_pos_nat_iff[simp]:
-  "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
-using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
-
-
-subsection {* Generic lattice operations over a set *}
-
-no_notation times (infixl "*" 70)
-no_notation Groups.one ("1")
-
-
-subsubsection {* Without neutral element *}
-
-locale semilattice_set = semilattice
-begin
-
-interpretation comp_fun_idem f
-  by default (simp_all add: fun_eq_iff left_commute)
-
-definition F :: "'a set \<Rightarrow> 'a"
-where
-  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)"
-
-lemma eq_fold:
-  assumes "finite A"
-  shows "F (insert x A) = Finite_Set.fold f x A"
-proof (rule sym)
-  let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
-  interpret comp_fun_idem "?f"
-    by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
-  from assms show "Finite_Set.fold f x A = F (insert x A)"
-  proof induct
-    case empty then show ?case by (simp add: eq_fold')
-  next
-    case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
-  qed
-qed
-
-lemma singleton [simp]:
-  "F {x} = x"
-  by (simp add: eq_fold)
-
-lemma insert_not_elem:
-  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
-  shows "F (insert x A) = x * F A"
-proof -
-  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
-  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
-  with `finite A` and `x \<notin> A`
-    have "finite (insert x B)" and "b \<notin> insert x B" by auto
-  then have "F (insert b (insert x B)) = x * F (insert b B)"
-    by (simp add: eq_fold)
-  then show ?thesis by (simp add: * insert_commute)
-qed
-
-lemma in_idem:
-  assumes "finite A" and "x \<in> A"
-  shows "x * F A = F A"
-proof -
-  from assms have "A \<noteq> {}" by auto
-  with `finite A` show ?thesis using `x \<in> A`
-    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
-qed
-
-lemma insert [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "F (insert x A) = x * F A"
-  using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
-
-lemma union:
-  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
-  shows "F (A \<union> B) = F A * F B"
-  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
-
-lemma remove:
-  assumes "finite A" and "x \<in> A"
-  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
-proof -
-  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
-  with assms show ?thesis by simp
-qed
-
-lemma insert_remove:
-  assumes "finite A"
-  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
-  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
-
-lemma subset:
-  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
-  shows "F B * F A = F A"
-proof -
-  from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
-  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
-qed
-
-lemma closed:
-  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
-  shows "F A \<in> A"
-using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
-  case singleton then show ?case by simp
-next
-  case insert with elem show ?case by force
-qed
-
-lemma hom_commute:
-  assumes hom: "\<And>x y. h (x * y) = h x * h y"
-  and N: "finite N" "N \<noteq> {}"
-  shows "h (F N) = F (h ` N)"
-using N proof (induct rule: finite_ne_induct)
-  case singleton thus ?case by simp
-next
-  case (insert n N)
-  then have "h (F (insert n N)) = h (n * F N)" by simp
-  also have "\<dots> = h n * h (F N)" by (rule hom)
-  also have "h (F N) = F (h ` N)" by (rule insert)
-  also have "h n * \<dots> = F (insert (h n) (h ` N))"
-    using insert by simp
-  also have "insert (h n) (h ` N) = h ` insert n N" by simp
-  finally show ?case .
-qed
-
-end
-
-locale semilattice_order_set = semilattice_order + semilattice_set
-begin
-
-lemma bounded_iff:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
-  using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
-
-lemma boundedI:
-  assumes "finite A"
-  assumes "A \<noteq> {}"
-  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
-  shows "x \<preceq> F A"
-  using assms by (simp add: bounded_iff)
-
-lemma boundedE:
-  assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
-  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
-  using assms by (simp add: bounded_iff)
-
-lemma coboundedI:
-  assumes "finite A"
-    and "a \<in> A"
-  shows "F A \<preceq> a"
-proof -
-  from assms have "A \<noteq> {}" by auto
-  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
-  proof (induct rule: finite_ne_induct)
-    case singleton thus ?case by (simp add: refl)
-  next
-    case (insert x B)
-    from insert have "a = x \<or> a \<in> B" by simp
-    then show ?case using insert by (auto intro: coboundedI2)
-  qed
-qed
-
-lemma antimono:
-  assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
-  shows "F B \<preceq> F A"
-proof (cases "A = B")
-  case True then show ?thesis by (simp add: refl)
-next
-  case False
-  have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
-  then have "F B = F (A \<union> (B - A))" by simp
-  also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
-  also have "\<dots> \<preceq> F A" by simp
-  finally show ?thesis .
-qed
-
-end
-
-
-subsubsection {* With neutral element *}
-
-locale semilattice_neutr_set = semilattice_neutr
-begin
-
-interpretation comp_fun_idem f
-  by default (simp_all add: fun_eq_iff left_commute)
-
-definition F :: "'a set \<Rightarrow> 'a"
-where
-  eq_fold: "F A = Finite_Set.fold f 1 A"
-
-lemma infinite [simp]:
-  "\<not> finite A \<Longrightarrow> F A = 1"
-  by (simp add: eq_fold)
-
-lemma empty [simp]:
-  "F {} = 1"
-  by (simp add: eq_fold)
-
-lemma insert [simp]:
-  assumes "finite A"
-  shows "F (insert x A) = x * F A"
-  using assms by (simp add: eq_fold)
-
-lemma in_idem:
-  assumes "finite A" and "x \<in> A"
-  shows "x * F A = F A"
-proof -
-  from assms have "A \<noteq> {}" by auto
-  with `finite A` show ?thesis using `x \<in> A`
-    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
-qed
-
-lemma union:
-  assumes "finite A" and "finite B"
-  shows "F (A \<union> B) = F A * F B"
-  using assms by (induct A) (simp_all add: ac_simps)
-
-lemma remove:
-  assumes "finite A" and "x \<in> A"
-  shows "F A = x * F (A - {x})"
-proof -
-  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
-  with assms show ?thesis by simp
-qed
-
-lemma insert_remove:
-  assumes "finite A"
-  shows "F (insert x A) = x * F (A - {x})"
-  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
-
-lemma subset:
-  assumes "finite A" and "B \<subseteq> A"
-  shows "F B * F A = F A"
-proof -
-  from assms have "finite B" by (auto dest: finite_subset)
-  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
-qed
-
-lemma closed:
-  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
-  shows "F A \<in> A"
-using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
-  case singleton then show ?case by simp
-next
-  case insert with elem show ?case by force
-qed
-
-end
-
-locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
-begin
-
-lemma bounded_iff:
-  assumes "finite A"
-  shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
-  using assms by (induct A) (simp_all add: bounded_iff)
-
-lemma boundedI:
-  assumes "finite A"
-  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
-  shows "x \<preceq> F A"
-  using assms by (simp add: bounded_iff)
-
-lemma boundedE:
-  assumes "finite A" and "x \<preceq> F A"
-  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
-  using assms by (simp add: bounded_iff)
-
-lemma coboundedI:
-  assumes "finite A"
-    and "a \<in> A"
-  shows "F A \<preceq> a"
-proof -
-  from assms have "A \<noteq> {}" by auto
-  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
-  proof (induct rule: finite_ne_induct)
-    case singleton thus ?case by (simp add: refl)
-  next
-    case (insert x B)
-    from insert have "a = x \<or> a \<in> B" by simp
-    then show ?case using insert by (auto intro: coboundedI2)
-  qed
-qed
-
-lemma antimono:
-  assumes "A \<subseteq> B" and "finite B"
-  shows "F B \<preceq> F A"
-proof (cases "A = B")
-  case True then show ?thesis by (simp add: refl)
-next
-  case False
-  have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
-  then have "F B = F (A \<union> (B - A))" by simp
-  also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
-  also have "\<dots> \<preceq> F A" by simp
-  finally show ?thesis .
-qed
-
-end
-
-notation times (infixl "*" 70)
-notation Groups.one ("1")
-
-
-subsection {* Lattice operations on finite sets *}
-
-text {*
-  For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
-  to @{class linorder}.  This is badly designed: both should depend on a common abstract
-  distributive lattice rather than having this non-subclass dependecy between two
-  classes.  But for the moment we have to live with it.  This forces us to setup
-  this sublocale dependency simultaneously with the lattice operations on finite
-  sets, to avoid garbage.
-*}
-
-definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
-where
-  "Inf_fin = semilattice_set.F inf"
-
-definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
-where
-  "Sup_fin = semilattice_set.F sup"
-
-context linorder
-begin
-
-definition Min :: "'a set \<Rightarrow> 'a"
-where
-  "Min = semilattice_set.F min"
-
-definition Max :: "'a set \<Rightarrow> 'a"
-where
-  "Max = semilattice_set.F max"
-
-sublocale Min!: semilattice_order_set min less_eq less
-  + Max!: semilattice_order_set max greater_eq greater
-where
-  "semilattice_set.F min = Min"
-  and "semilattice_set.F max = Max"
-proof -
-  show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
-  then interpret Min!: semilattice_order_set min less_eq less .
-  show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
-  then interpret Max!: semilattice_order_set max greater_eq greater .
-  from Min_def show "semilattice_set.F min = Min" by rule
-  from Max_def show "semilattice_set.F max = Max" by rule
-qed
-
-
-text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
-
-sublocale min_max!: distrib_lattice min less_eq less max
-where
-  "semilattice_inf.Inf_fin min = Min"
-  and "semilattice_sup.Sup_fin max = Max"
-proof -
-  show "class.distrib_lattice min less_eq less max"
-  proof
-    fix x y z
-    show "max x (min y z) = min (max x y) (max x z)"
-      by (auto simp add: min_def max_def)
-  qed (auto simp add: min_def max_def not_le less_imp_le)
-  then interpret min_max!: distrib_lattice min less_eq less max .
-  show "semilattice_inf.Inf_fin min = Min"
-    by (simp only: min_max.Inf_fin_def Min_def)
-  show "semilattice_sup.Sup_fin max = Max"
-    by (simp only: min_max.Sup_fin_def Max_def)
-qed
-
-lemmas le_maxI1 = min_max.sup_ge1
-lemmas le_maxI2 = min_max.sup_ge2
- 
-lemmas min_ac = min_max.inf_assoc min_max.inf_commute
-  min.left_commute
-
-lemmas max_ac = min_max.sup_assoc min_max.sup_commute
-  max.left_commute
-
-end
-
-
-text {* Lattice operations proper *}
-
-sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
-where
-  "semilattice_set.F inf = Inf_fin"
-proof -
-  show "semilattice_order_set inf less_eq less" ..
-  then interpret Inf_fin!: semilattice_order_set inf less_eq less .
-  from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
-qed
-
-sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
-where
-  "semilattice_set.F sup = Sup_fin"
-proof -
-  show "semilattice_order_set sup greater_eq greater" ..
-  then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
-  from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
-qed
-
-
-text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
-
-lemma Inf_fin_Min:
-  "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
-  by (simp add: Inf_fin_def Min_def inf_min)
-
-lemma Sup_fin_Max:
-  "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
-  by (simp add: Sup_fin_def Max_def sup_max)
-
-
-
-subsection {* Infimum and Supremum over non-empty sets *}
-
-text {*
-  After this non-regular bootstrap, things continue canonically.
-*}
-
-context lattice
-begin
-
-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"
-apply(subgoal_tac "EX a. a:A")
-prefer 2 apply blast
-apply(erule exE)
-apply(rule order_trans)
-apply(erule (1) Inf_fin.coboundedI)
-apply(erule (1) Sup_fin.coboundedI)
-done
-
-lemma sup_Inf_absorb [simp]:
-  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = a"
-apply(subst sup_commute)
-apply(simp add: sup_absorb2 Inf_fin.coboundedI)
-done
-
-lemma inf_Sup_absorb [simp]:
-  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = a"
-by (simp add: inf_absorb1 Sup_fin.coboundedI)
-
-end
-
-context distrib_lattice
-begin
-
-lemma sup_Inf1_distrib:
-  assumes "finite A"
-    and "A \<noteq> {}"
-  shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
-using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
-  (rule arg_cong [where f="Inf_fin"], blast)
-
-lemma sup_Inf2_distrib:
-  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
-  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}"
-using A proof (induct rule: finite_ne_induct)
-  case singleton then show ?case
-    by (simp add: sup_Inf1_distrib [OF B])
-next
-  case (insert x A)
-  have finB: "finite {sup x b |b. b \<in> B}"
-    by (rule finite_surj [where f = "sup x", OF B(1)], auto)
-  have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
-  proof -
-    have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
-      by blast
-    thus ?thesis by(simp add: insert(1) B(1))
-  qed
-  have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
-  have "sup (\<Sqinter>\<^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)"
-    using insert by simp
-  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)
-  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})"
-    using insert by(simp add:sup_Inf1_distrib[OF B])
-  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})"
-    (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
-    using B insert
-    by (simp add: Inf_fin.union [OF finB _ finAB ne])
-  also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
-    by blast
-  finally show ?case .
-qed
-
-lemma inf_Sup1_distrib:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
-using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
-  (rule arg_cong [where f="Sup_fin"], blast)
-
-lemma inf_Sup2_distrib:
-  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
-  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}"
-using A proof (induct rule: finite_ne_induct)
-  case singleton thus ?case
-    by(simp add: inf_Sup1_distrib [OF B])
-next
-  case (insert x A)
-  have finB: "finite {inf x b |b. b \<in> B}"
-    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
-  have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
-  proof -
-    have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
-      by blast
-    thus ?thesis by(simp add: insert(1) B(1))
-  qed
-  have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
-  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)"
-    using insert by simp
-  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)
-  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})"
-    using insert by(simp add:inf_Sup1_distrib[OF B])
-  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})"
-    (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
-    using B insert
-    by (simp add: Sup_fin.union [OF finB _ finAB ne])
-  also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
-    by blast
-  finally show ?case .
-qed
-
-end
-
-context complete_lattice
-begin
-
-lemma Inf_fin_Inf:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = Inf A"
-proof -
-  from assms obtain b B where "A = insert b B" and "finite B" by auto
-  then show ?thesis
-    by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
-qed
-
-lemma Sup_fin_Sup:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = Sup A"
-proof -
-  from assms obtain b B where "A = insert b B" and "finite B" by auto
-  then show ?thesis
-    by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
-qed
-
-end
-
-
-subsection {* Minimum and Maximum over non-empty sets *}
-
-context linorder
-begin
-
-lemma dual_min:
-  "ord.min greater_eq = max"
-  by (auto simp add: ord.min_def max_def fun_eq_iff)
-
-lemma dual_max:
-  "ord.max greater_eq = min"
-  by (auto simp add: ord.max_def min_def fun_eq_iff)
-
-lemma dual_Min:
-  "linorder.Min greater_eq = Max"
-proof -
-  interpret dual!: linorder greater_eq greater by (fact dual_linorder)
-  show ?thesis by (simp add: dual.Min_def dual_min Max_def)
-qed
-
-lemma dual_Max:
-  "linorder.Max greater_eq = Min"
-proof -
-  interpret dual!: linorder greater_eq greater by (fact dual_linorder)
-  show ?thesis by (simp add: dual.Max_def dual_max Min_def)
-qed
-
-lemmas Min_singleton = Min.singleton
-lemmas Max_singleton = Max.singleton
-lemmas Min_insert = Min.insert
-lemmas Max_insert = Max.insert
-lemmas Min_Un = Min.union
-lemmas Max_Un = Max.union
-lemmas hom_Min_commute = Min.hom_commute
-lemmas hom_Max_commute = Max.hom_commute
-
-lemma Min_in [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Min A \<in> A"
-  using assms by (auto simp add: min_def Min.closed)
-
-lemma Max_in [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Max A \<in> A"
-  using assms by (auto simp add: max_def Max.closed)
-
-lemma Min_le [simp]:
-  assumes "finite A" and "x \<in> A"
-  shows "Min A \<le> x"
-  using assms by (fact Min.coboundedI)
-
-lemma Max_ge [simp]:
-  assumes "finite A" and "x \<in> A"
-  shows "x \<le> Max A"
-  using assms by (fact Max.coboundedI)
-
-lemma Min_eqI:
-  assumes "finite A"
-  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
-    and "x \<in> A"
-  shows "Min A = x"
-proof (rule antisym)
-  from `x \<in> A` have "A \<noteq> {}" by auto
-  with assms show "Min A \<ge> x" by simp
-next
-  from assms show "x \<ge> Min A" by simp
-qed
-
-lemma Max_eqI:
-  assumes "finite A"
-  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
-    and "x \<in> A"
-  shows "Max A = x"
-proof (rule antisym)
-  from `x \<in> A` have "A \<noteq> {}" by auto
-  with assms show "Max A \<le> x" by simp
-next
-  from assms show "x \<le> Max A" by simp
-qed
-
-context
-  fixes A :: "'a set"
-  assumes fin_nonempty: "finite A" "A \<noteq> {}"
-begin
-
-lemma Min_ge_iff [simp]:
-  "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
-  using fin_nonempty by (fact Min.bounded_iff)
-
-lemma Max_le_iff [simp]:
-  "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
-  using fin_nonempty by (fact Max.bounded_iff)
-
-lemma Min_gr_iff [simp]:
-  "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
-  using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
-
-lemma Max_less_iff [simp]:
-  "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
-  using fin_nonempty by (induct rule: finite_ne_induct) simp_all
-
-lemma Min_le_iff:
-  "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
-  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
-
-lemma Max_ge_iff:
-  "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
-  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
-
-lemma Min_less_iff:
-  "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
-  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
-
-lemma Max_gr_iff:
-  "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
-  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
-
-end
-
-lemma Min_antimono:
-  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
-  shows "Min N \<le> Min M"
-  using assms by (fact Min.antimono)
-
-lemma Max_mono:
-  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
-  shows "Max M \<le> Max N"
-  using assms by (fact Max.antimono)
-
-lemma mono_Min_commute:
-  assumes "mono f"
-  assumes "finite A" and "A \<noteq> {}"
-  shows "f (Min A) = Min (f ` A)"
-proof (rule linorder_class.Min_eqI [symmetric])
-  from `finite A` show "finite (f ` A)" by simp
-  from assms show "f (Min A) \<in> f ` A" by simp
-  fix x
-  assume "x \<in> f ` A"
-  then obtain y where "y \<in> A" and "x = f y" ..
-  with assms have "Min A \<le> y" by auto
-  with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
-  with `x = f y` show "f (Min A) \<le> x" by simp
-qed
-
-lemma mono_Max_commute:
-  assumes "mono f"
-  assumes "finite A" and "A \<noteq> {}"
-  shows "f (Max A) = Max (f ` A)"
-proof (rule linorder_class.Max_eqI [symmetric])
-  from `finite A` show "finite (f ` A)" by simp
-  from assms show "f (Max A) \<in> f ` A" by simp
-  fix x
-  assume "x \<in> f ` A"
-  then obtain y where "y \<in> A" and "x = f y" ..
-  with assms have "y \<le> Max A" by auto
-  with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
-  with `x = f y` show "x \<le> f (Max A)" by simp
-qed
-
-lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
-  assumes fin: "finite A"
-  and empty: "P {}" 
-  and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
-  shows "P A"
-using fin empty insert
-proof (induct rule: finite_psubset_induct)
-  case (psubset A)
-  have IH: "\<And>B. \<lbrakk>B < A; P {}; (\<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. a<b; P A\<rbrakk> \<Longrightarrow> P (insert b A))\<rbrakk> \<Longrightarrow> P B" by fact 
-  have fin: "finite A" by fact 
-  have empty: "P {}" by fact
-  have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
-  show "P A"
-  proof (cases "A = {}")
-    assume "A = {}" 
-    then show "P A" using `P {}` by simp
-  next
-    let ?B = "A - {Max A}" 
-    let ?A = "insert (Max A) ?B"
-    have "finite ?B" using `finite A` by simp
-    assume "A \<noteq> {}"
-    with `finite A` have "Max A : A" by auto
-    then have A: "?A = A" using insert_Diff_single insert_absorb by auto
-    then have "P ?B" using `P {}` step IH [of ?B] by blast
-    moreover 
-    have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
-    ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
-  qed
-qed
-
-lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
-  "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
-  by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
-
-lemma Least_Min:
-  assumes "finite {a. P a}" and "\<exists>a. P a"
-  shows "(LEAST a. P a) = Min {a. P a}"
-proof -
-  { fix A :: "'a set"
-    assume A: "finite A" "A \<noteq> {}"
-    have "(LEAST a. a \<in> A) = Min A"
-    using A proof (induct A rule: finite_ne_induct)
-      case singleton show ?case by (rule Least_equality) simp_all
-    next
-      case (insert a A)
-      have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
-        by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
-      with insert show ?case by simp
-    qed
-  } from this [of "{a. P a}"] assms show ?thesis by simp
-qed
-
-end
-
-context linordered_ab_semigroup_add
-begin
-
-lemma add_Min_commute:
-  fixes k
-  assumes "finite N" and "N \<noteq> {}"
-  shows "k + Min N = Min {k + m | m. m \<in> N}"
-proof -
-  have "\<And>x y. k + min x y = min (k + x) (k + y)"
-    by (simp add: min_def not_le)
-      (blast intro: antisym less_imp_le add_left_mono)
-  with assms show ?thesis
-    using hom_Min_commute [of "plus k" N]
-    by simp (blast intro: arg_cong [where f = Min])
-qed
-
-lemma add_Max_commute:
-  fixes k
-  assumes "finite N" and "N \<noteq> {}"
-  shows "k + Max N = Max {k + m | m. m \<in> N}"
-proof -
-  have "\<And>x y. k + max x y = max (k + x) (k + y)"
-    by (simp add: max_def not_le)
-      (blast intro: antisym less_imp_le add_left_mono)
-  with assms show ?thesis
-    using hom_Max_commute [of "plus k" N]
-    by simp (blast intro: arg_cong [where f = Max])
-qed
-
-end
-
-context linordered_ab_group_add
-begin
-
-lemma minus_Max_eq_Min [simp]:
-  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
-  by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
-
-lemma minus_Min_eq_Max [simp]:
-  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
-  by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
-
-end
-
-context complete_linorder
-begin
-
-lemma Min_Inf:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Min A = Inf A"
-proof -
-  from assms obtain b B where "A = insert b B" and "finite B" by auto
-  then show ?thesis
-    by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
-qed
-
-lemma Max_Sup:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Max A = Sup A"
-proof -
-  from assms obtain b B where "A = insert b B" and "finite B" by auto
-  then show ?thesis
-    by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
-qed
-
-end
-
-end

--- a/src/HOL/Equiv_Relations.thy	Sat Dec 14 20:46:36 2013 +0100
+++ b/src/HOL/Equiv_Relations.thy	Sun Dec 15 15:10:14 2013 +0100
@@ -5,7 +5,7 @@
 header {* Equivalence Relations in Higher-Order Set Theory *}
 
 theory Equiv_Relations
-imports Big_Operators Relation
+imports Groups_Big Relation
 begin
 
 subsection {* Equivalence relations -- set version *}

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Groups_Big.thy	Sun Dec 15 15:10:14 2013 +0100
@@ -0,0 +1,1379 @@
+(*  Title:      HOL/Groups_Big.thy
+    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
+                with contributions by Jeremy Avigad
+*)
+
+header {* Big sum and product over finite (non-empty) sets *}
+
+theory Groups_Big
+imports Finite_Set
+begin
+
+subsection {* Generic monoid operation over a set *}
+
+no_notation times (infixl "*" 70)
+no_notation Groups.one ("1")
+
+locale comm_monoid_set = comm_monoid
+begin
+
+interpretation comp_fun_commute f
+  by default (simp add: fun_eq_iff left_commute)
+
+interpretation comp_fun_commute "f \<circ> g"
+  by (rule comp_comp_fun_commute)
+
+definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+where
+  eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
+
+lemma infinite [simp]:
+  "\<not> finite A \<Longrightarrow> F g A = 1"
+  by (simp add: eq_fold)
+
+lemma empty [simp]:
+  "F g {} = 1"
+  by (simp add: eq_fold)
+
+lemma insert [simp]:
+  assumes "finite A" and "x \<notin> A"
+  shows "F g (insert x A) = g x * F g A"
+  using assms by (simp add: eq_fold)
+
+lemma remove:
+  assumes "finite A" and "x \<in> A"
+  shows "F g A = g x * F g (A - {x})"
+proof -
+  from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
+    by (auto dest: mk_disjoint_insert)
+  moreover from `finite A` A have "finite B" by simp
+  ultimately show ?thesis by simp
+qed
+
+lemma insert_remove:
+  assumes "finite A"
+  shows "F g (insert x A) = g x * F g (A - {x})"
+  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
+
+lemma neutral:
+  assumes "\<forall>x\<in>A. g x = 1"
+  shows "F g A = 1"
+  using assms by (induct A rule: infinite_finite_induct) simp_all
+
+lemma neutral_const [simp]:
+  "F (\<lambda>_. 1) A = 1"
+  by (simp add: neutral)
+
+lemma union_inter:
+  assumes "finite A" and "finite B"
+  shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
+  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
+using assms proof (induct A)
+  case empty then show ?case by simp
+next
+  case (insert x A) then show ?case
+    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
+qed
+
+corollary union_inter_neutral:
+  assumes "finite A" and "finite B"
+  and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
+  shows "F g (A \<union> B) = F g A * F g B"
+  using assms by (simp add: union_inter [symmetric] neutral)
+
+corollary union_disjoint:
+  assumes "finite A" and "finite B"
+  assumes "A \<inter> B = {}"
+  shows "F g (A \<union> B) = F g A * F g B"
+  using assms by (simp add: union_inter_neutral)
+
+lemma subset_diff:
+  assumes "B \<subseteq> A" and "finite A"
+  shows "F g A = F g (A - B) * F g B"
+proof -
+  from assms have "finite (A - B)" by auto
+  moreover from assms have "finite B" by (rule finite_subset)
+  moreover from assms have "(A - B) \<inter> B = {}" by auto
+  ultimately have "F g (A - B \<union> B) = F g (A - B) * F g B" by (rule union_disjoint)
+  moreover from assms have "A \<union> B = A" by auto
+  ultimately show ?thesis by simp
+qed
+
+lemma reindex:
+  assumes "inj_on h A"
+  shows "F g (h ` A) = F (g \<circ> h) A"
+proof (cases "finite A")
+  case True
+  with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
+next
+  case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
+  with False show ?thesis by simp
+qed
+
+lemma cong:
+  assumes "A = B"
+  assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
+  shows "F g A = F h B"
+proof (cases "finite A")
+  case True
+  then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
+  proof induct
+    case empty then show ?case by simp
+  next
+    case (insert x F) then show ?case apply -
+    apply (simp add: subset_insert_iff, clarify)
+    apply (subgoal_tac "finite C")
+      prefer 2 apply (blast dest: finite_subset [rotated])
+    apply (subgoal_tac "C = insert x (C - {x})")
+      prefer 2 apply blast
+    apply (erule ssubst)
+    apply (simp add: Ball_def del: insert_Diff_single)
+    done
+  qed
+  with `A = B` g_h show ?thesis by simp
+next
+  case False
+  with `A = B` show ?thesis by simp
+qed
+
+lemma strong_cong [cong]:
+  assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
+  shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
+  by (rule cong) (insert assms, simp_all add: simp_implies_def)
+
+lemma UNION_disjoint:
+  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
+  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
+  shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
+apply (insert assms)
+apply (induct rule: finite_induct)
+apply simp
+apply atomize
+apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
+ prefer 2 apply blast
+apply (subgoal_tac "A x Int UNION Fa A = {}")
+ prefer 2 apply blast
+apply (simp add: union_disjoint)
+done
+
+lemma Union_disjoint:
+  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
+  shows "F g (Union C) = F (F g) C"
+proof cases
+  assume "finite C"
+  from UNION_disjoint [OF this assms]
+  show ?thesis
+    by (simp add: SUP_def)
+qed (auto dest: finite_UnionD intro: infinite)
+
+lemma distrib:
+  "F (\<lambda>x. g x * h x) A = F g A * F h A"
+  using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
+
+lemma Sigma:
+  "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)"
+apply (subst Sigma_def)
+apply (subst UNION_disjoint, assumption, simp)
+ apply blast
+apply (rule cong)
+apply rule
+apply (simp add: fun_eq_iff)
+apply (subst UNION_disjoint, simp, simp)
+ apply blast
+apply (simp add: comp_def)
+done
+
+lemma related: 
+  assumes Re: "R 1 1" 
+  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 
+  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
+  shows "R (F h S) (F g S)"
+  using fS by (rule finite_subset_induct) (insert assms, auto)
+
+lemma eq_general:
+  assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 
+  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
+  shows "F f1 S = F f2 S'"
+proof-
+  from h f12 have hS: "h ` S = S'" by blast
+  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
+    from f12 h H  have "x = y" by auto }
+  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
+  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 
+  from hS have "F f2 S' = F f2 (h ` S)" by simp
+  also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
+  also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
+    by blast
+  finally show ?thesis ..
+qed
+
+lemma eq_general_reverses:
+  assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
+  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
+  shows "F j S = F g T"
+  (* metis solves it, but not yet available here *)
+  apply (rule eq_general [of T S h g j])
+  apply (rule ballI)
+  apply (frule kh)
+  apply (rule ex1I[])
+  apply blast
+  apply clarsimp
+  apply (drule hk) apply simp
+  apply (rule sym)
+  apply (erule conjunct1[OF conjunct2[OF hk]])
+  apply (rule ballI)
+  apply (drule hk)
+  apply blast
+  done
+
+lemma mono_neutral_cong_left:
+  assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
+  and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
+proof-
+  have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
+  have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
+  from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
+    by (auto intro: finite_subset)
+  show ?thesis using assms(4)
+    by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
+qed
+
+lemma mono_neutral_cong_right:
+  "\<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>
+   \<Longrightarrow> F g T = F h S"
+  by (auto intro!: mono_neutral_cong_left [symmetric])
+
+lemma mono_neutral_left:
+  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
+  by (blast intro: mono_neutral_cong_left)
+
+lemma mono_neutral_right:
+  "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
+  by (blast intro!: mono_neutral_left [symmetric])
+
+lemma delta: 
+  assumes fS: "finite S"
+  shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
+proof-
+  let ?f = "(\<lambda>k. if k=a then b k else 1)"
+  { assume a: "a \<notin> S"
+    hence "\<forall>k\<in>S. ?f k = 1" by simp
+    hence ?thesis  using a by simp }
+  moreover
+  { assume a: "a \<in> S"
+    let ?A = "S - {a}"
+    let ?B = "{a}"
+    have eq: "S = ?A \<union> ?B" using a by blast 
+    have dj: "?A \<inter> ?B = {}" by simp
+    from fS have fAB: "finite ?A" "finite ?B" by auto  
+    have "F ?f S = F ?f ?A * F ?f ?B"
+      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
+      by simp
+    then have ?thesis using a by simp }
+  ultimately show ?thesis by blast
+qed
+
+lemma delta': 
+  assumes fS: "finite S"
+  shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
+  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
+
+lemma If_cases:
+  fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
+  assumes fA: "finite A"
+  shows "F (\<lambda>x. if P x then h x else g x) A =
+    F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
+proof -
+  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" 
+          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" 
+    by blast+
+  from fA 
+  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
+  let ?g = "\<lambda>x. if P x then h x else g x"
+  from union_disjoint [OF f a(2), of ?g] a(1)
+  show ?thesis
+    by (subst (1 2) cong) simp_all
+qed
+
+lemma cartesian_product:
+   "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
+apply (rule sym)
+apply (cases "finite A") 
+ apply (cases "finite B") 
+  apply (simp add: Sigma)
+ apply (cases "A={}", simp)
+ apply simp
+apply (auto intro: infinite dest: finite_cartesian_productD2)
+apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
+done
+
+end
+
+notation times (infixl "*" 70)
+notation Groups.one ("1")
+
+
+subsection {* Generalized summation over a set *}
+
+context comm_monoid_add
+begin
+
+definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+where
+  "setsum = comm_monoid_set.F plus 0"
+
+sublocale setsum!: comm_monoid_set plus 0
+where
+  "comm_monoid_set.F plus 0 = setsum"
+proof -
+  show "comm_monoid_set plus 0" ..
+  then interpret setsum!: comm_monoid_set plus 0 .
+  from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule
+qed
+
+abbreviation
+  Setsum ("\<Sum>_" [1000] 999) where
+  "\<Sum>A \<equiv> setsum (%x. x) A"
+
+end
+
+text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
+written @{text"\<Sum>x\<in>A. e"}. *}
+
+syntax
+  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
+syntax (xsymbols)
+  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
+syntax (HTML output)
+  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
+
+translations -- {* Beware of argument permutation! *}
+  "SUM i:A. b" == "CONST setsum (%i. b) A"
+  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
+
+text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
+ @{text"\<Sum>x|P. e"}. *}
+
+syntax
+  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
+syntax (xsymbols)
+  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
+syntax (HTML output)
+  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
+
+translations
+  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
+  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
+
+print_translation {*
+let
+  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =
+        if x <> y then raise Match
+        else
+          let
+            val x' = Syntax_Trans.mark_bound_body (x, Tx);
+            val t' = subst_bound (x', t);
+            val P' = subst_bound (x', P);
+          in
+            Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
+          end
+    | setsum_tr' _ = raise Match;
+in [(@{const_syntax setsum}, K setsum_tr')] end
+*}
+
+text {* TODO These are candidates for generalization *}
+
+context comm_monoid_add
+begin
+
+lemma setsum_reindex_id: 
+  "inj_on f B ==> setsum f B = setsum id (f ` B)"
+  by (simp add: setsum.reindex)
+
+lemma setsum_reindex_nonzero:
+  assumes fS: "finite S"
+  and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
+  shows "setsum h (f ` S) = setsum (h \<circ> f) S"
+using nz proof (induct rule: finite_induct [OF fS])
+  case 1 thus ?case by simp
+next
+  case (2 x F) 
+  { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
+    then obtain y where y: "y \<in> F" "f x = f y" by auto 
+    from "2.hyps" y have xy: "x \<noteq> y" by auto
+    from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
+    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
+    also have "\<dots> = setsum (h o f) (insert x F)" 
+      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
+      using h0
+      apply (simp cong del: setsum.strong_cong)
+      apply (rule "2.hyps"(3))
+      apply (rule_tac y="y" in  "2.prems")
+      apply simp_all
+      done
+    finally have ?case . }
+  moreover
+  { assume fxF: "f x \<notin> f ` F"
+    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 
+      using fxF "2.hyps" by simp 
+    also have "\<dots> = setsum (h o f) (insert x F)"
+      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
+      apply (simp cong del: setsum.strong_cong)
+      apply (rule cong [OF refl [of "op + (h (f x))"]])
+      apply (rule "2.hyps"(3))
+      apply (rule_tac y="y" in  "2.prems")
+      apply simp_all
+      done
+    finally have ?case . }
+  ultimately show ?case by blast
+qed
+
+lemma setsum_cong2:
+  "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
+  by (auto intro: setsum.cong)
+
+lemma setsum_reindex_cong:
+   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 
+    ==> setsum h B = setsum g A"
+  by (simp add: setsum.reindex)
+
+lemma setsum_restrict_set:
+  assumes fA: "finite A"
+  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
+proof-
+  from fA have fab: "finite (A \<inter> B)" by auto
+  have aba: "A \<inter> B \<subseteq> A" by blast
+  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
+  from setsum.mono_neutral_left [OF fA aba, of ?g]
+  show ?thesis by simp
+qed
+
+lemma setsum_Union_disjoint:
+  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
+  shows "setsum f (Union C) = setsum (setsum f) C"
+  using assms by (fact setsum.Union_disjoint)
+
+lemma setsum_cartesian_product:
+  "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
+  by (fact setsum.cartesian_product)
+
+lemma setsum_UNION_zero:
+  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
+  and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
+  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
+  using fSS f0
+proof(induct rule: finite_induct[OF fS])
+  case 1 thus ?case by simp
+next
+  case (2 T F)
+  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 
+    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
+  from fTF have fUF: "finite (\<Union>F)" by auto
+  from "2.prems" TF fTF
+  show ?case 
+    by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
+qed
+
+text {* Commuting outer and inner summation *}
+
+lemma setsum_commute:
+  "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
+proof (simp add: setsum_cartesian_product)
+  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
+    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
+    (is "?s = _")
+    apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
+    apply (simp add: split_def)
+    done
+  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
+    (is "_ = ?t")
+    apply (simp add: swap_product)
+    done
+  finally show "?s = ?t" .
+qed
+
+lemma setsum_Plus:
+  fixes A :: "'a set" and B :: "'b set"
+  assumes fin: "finite A" "finite B"
+  shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
+proof -
+  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
+  moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
+    by auto
+  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
+  moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
+  ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
+qed
+
+end
+
+text {* TODO These are legacy *}
+
+lemma setsum_empty:
+  "setsum f {} = 0"
+  by (fact setsum.empty)
+
+lemma setsum_insert:
+  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
+  by (fact setsum.insert)
+
+lemma setsum_infinite:
+  "~ finite A ==> setsum f A = 0"
+  by (fact setsum.infinite)
+
+lemma setsum_reindex:
+  "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
+  by (fact setsum.reindex)
+
+lemma setsum_cong:
+  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
+  by (fact setsum.cong)
+
+lemma strong_setsum_cong:
+  "A = B ==> (!!x. x:B =simp=> f x = g x)
+   ==> setsum (%x. f x) A = setsum (%x. g x) B"
+  by (fact setsum.strong_cong)
+
+lemmas setsum_0 = setsum.neutral_const
+lemmas setsum_0' = setsum.neutral
+
+lemma setsum_Un_Int: "finite A ==> finite B ==>
+  setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
+  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
+  by (fact setsum.union_inter)
+
+lemma setsum_Un_disjoint: "finite A ==> finite B
+  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
+  by (fact setsum.union_disjoint)
+
+lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
+    setsum f A = setsum f (A - B) + setsum f B"
+  by (fact setsum.subset_diff)
+
+lemma setsum_mono_zero_left: 
+  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
+  by (fact setsum.mono_neutral_left)
+
+lemmas setsum_mono_zero_right = setsum.mono_neutral_right
+
+lemma setsum_mono_zero_cong_left: 
+  "\<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>
+  \<Longrightarrow> setsum f S = setsum g T"
+  by (fact setsum.mono_neutral_cong_left)
+
+lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
+
+lemma setsum_delta: "finite S \<Longrightarrow>
+  setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
+  by (fact setsum.delta)
+
+lemma setsum_delta': "finite S \<Longrightarrow>
+  setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
+  by (fact setsum.delta')
+
+lemma setsum_cases:
+  assumes "finite A"
+  shows "setsum (\<lambda>x. if P x then f x else g x) A =
+         setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
+  using assms by (fact setsum.If_cases)
+
+(*But we can't get rid of finite I. If infinite, although the rhs is 0, 
+  the lhs need not be, since UNION I A could still be finite.*)
+lemma setsum_UN_disjoint:
+  assumes "finite I" and "ALL i:I. finite (A i)"
+    and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
+  shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
+  using assms by (fact setsum.UNION_disjoint)
+
+(*But we can't get rid of finite A. If infinite, although the lhs is 0, 
+  the rhs need not be, since SIGMA A B could still be finite.*)
+lemma setsum_Sigma:
+  assumes "finite A" and  "ALL x:A. finite (B x)"
+  shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
+  using assms by (fact setsum.Sigma)
+
+lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
+  by (fact setsum.distrib)
+
+lemma setsum_Un_zero:  
+  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
+  setsum f (S \<union> T) = setsum f S + setsum f T"
+  by (fact setsum.union_inter_neutral)
+
+lemma setsum_eq_general_reverses:
+  assumes fS: "finite S" and fT: "finite T"
+  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
+  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
+  shows "setsum f S = setsum g T"
+  using kh hk by (fact setsum.eq_general_reverses)
+
+
+subsubsection {* Properties in more restricted classes of structures *}
+
+lemma setsum_Un: "finite A ==> finite B ==>
+  (setsum f (A Un B) :: 'a :: ab_group_add) =
+   setsum f A + setsum f B - setsum f (A Int B)"
+by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
+
+lemma setsum_Un2:
+  assumes "finite (A \<union> B)"
+  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
+proof -
+  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
+    by auto
+  with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
+qed
+
+lemma setsum_diff1: "finite A \<Longrightarrow>
+  (setsum f (A - {a}) :: ('a::ab_group_add)) =
+  (if a:A then setsum f A - f a else setsum f A)"
+by (erule finite_induct) (auto simp add: insert_Diff_if)
+
+lemma setsum_diff:
+  assumes le: "finite A" "B \<subseteq> A"
+  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
+proof -
+  from le have finiteB: "finite B" using finite_subset by auto
+  show ?thesis using finiteB le
+  proof induct
+    case empty
+    thus ?case by auto
+  next
+    case (insert x F)
+    thus ?case using le finiteB 
+      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
+  qed
+qed
+
+lemma setsum_mono:
+  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
+  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
+proof (cases "finite K")
+  case True
+  thus ?thesis using le
+  proof induct
+    case empty
+    thus ?case by simp
+  next
+    case insert
+    thus ?case using add_mono by fastforce
+  qed
+next
+  case False then show ?thesis by simp
+qed
+
+lemma setsum_strict_mono:
+  fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
+  assumes "finite A"  "A \<noteq> {}"
+    and "!!x. x:A \<Longrightarrow> f x < g x"
+  shows "setsum f A < setsum g A"
+  using assms
+proof (induct rule: finite_ne_induct)
+  case singleton thus ?case by simp
+next
+  case insert thus ?case by (auto simp: add_strict_mono)
+qed
+
+lemma setsum_strict_mono_ex1:
+fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
+assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
+shows "setsum f A < setsum g A"
+proof-
+  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
+  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
+    by(simp add:insert_absorb[OF `a:A`])
+  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
+    using `finite A` by(subst setsum_Un_disjoint) auto
+  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
+    by(rule setsum_mono)(simp add: assms(2))
+  also have "setsum f {a} < setsum g {a}" using a by simp
+  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
+    using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
+  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
+  finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
+qed
+
+lemma setsum_negf:
+  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
+proof (cases "finite A")
+  case True thus ?thesis by (induct set: finite) auto
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_subtractf:
+  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
+    setsum f A - setsum g A"
+  using setsum_addf [of f "- g" A] by (simp add: setsum_negf)
+
+lemma setsum_nonneg:
+  assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
+  shows "0 \<le> setsum f A"
+proof (cases "finite A")
+  case True thus ?thesis using nn
+  proof induct
+    case empty then show ?case by simp
+  next
+    case (insert x F)
+    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
+    with insert show ?case by simp
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_nonpos:
+  assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
+  shows "setsum f A \<le> 0"
+proof (cases "finite A")
+  case True thus ?thesis using np
+  proof induct
+    case empty then show ?case by simp
+  next
+    case (insert x F)
+    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
+    with insert show ?case by simp
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_nonneg_leq_bound:
+  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
+  assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
+  shows "f i \<le> B"
+proof -
+  have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
+    using assms by (auto intro!: setsum_nonneg)
+  moreover
+  have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
+    using assms by (simp add: setsum_diff1)
+  ultimately show ?thesis by auto
+qed
+
+lemma setsum_nonneg_0:
+  fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
+  assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
+  and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
+  shows "f i = 0"
+  using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
+
+lemma setsum_mono2:
+fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
+assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
+shows "setsum f A \<le> setsum f B"
+proof -
+  have "setsum f A \<le> setsum f A + setsum f (B-A)"
+    by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
+  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
+    by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
+  also have "A \<union> (B-A) = B" using sub by blast
+  finally show ?thesis .
+qed
+
+lemma setsum_mono3: "finite B ==> A <= B ==> 
+    ALL x: B - A. 
+      0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
+        setsum f A <= setsum f B"
+  apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
+  apply (erule ssubst)
+  apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
+  apply simp
+  apply (rule add_left_mono)
+  apply (erule setsum_nonneg)
+  apply (subst setsum_Un_disjoint [THEN sym])
+  apply (erule finite_subset, assumption)
+  apply (rule finite_subset)
+  prefer 2
+  apply assumption
+  apply (auto simp add: sup_absorb2)
+done
+
+lemma setsum_right_distrib: 
+  fixes f :: "'a => ('b::semiring_0)"
+  shows "r * setsum f A = setsum (%n. r * f n) A"
+proof (cases "finite A")
+  case True
+  thus ?thesis
+  proof induct
+    case empty thus ?case by simp
+  next
+    case (insert x A) thus ?case by (simp add: distrib_left)
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_left_distrib:
+  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+  proof induct
+    case empty thus ?case by simp
+  next
+    case (insert x A) thus ?case by (simp add: distrib_right)
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_divide_distrib:
+  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+  proof induct
+    case empty thus ?case by simp
+  next
+    case (insert x A) thus ?case by (simp add: add_divide_distrib)
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_abs[iff]: 
+  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
+  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
+proof (cases "finite A")
+  case True
+  thus ?thesis
+  proof induct
+    case empty thus ?case by simp
+  next
+    case (insert x A)
+    thus ?case by (auto intro: abs_triangle_ineq order_trans)
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_abs_ge_zero[iff]: 
+  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
+  shows "0 \<le> setsum (%i. abs(f i)) A"
+proof (cases "finite A")
+  case True
+  thus ?thesis
+  proof induct
+    case empty thus ?case by simp
+  next
+    case (insert x A) thus ?case by auto
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma abs_setsum_abs[simp]: 
+  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
+  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
+proof (cases "finite A")
+  case True
+  thus ?thesis
+  proof induct
+    case empty thus ?case by simp
+  next
+    case (insert a A)
+    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
+    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
+    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
+      by (simp del: abs_of_nonneg)
+    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
+    finally show ?case .
+  qed
+next
+  case False thus ?thesis by simp
+qed
+
+lemma setsum_diff1'[rule_format]:
+  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
+apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
+apply (auto simp add: insert_Diff_if add_ac)
+done
+
+lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
+  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
+unfolding setsum_diff1'[OF assms] by auto
+
+lemma setsum_product:
+  fixes f :: "'a => ('b::semiring_0)"
+  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
+  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
+
+lemma setsum_mult_setsum_if_inj:
+fixes f :: "'a => ('b::semiring_0)"
+shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
+  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
+by(auto simp: setsum_product setsum_cartesian_product
+        intro!:  setsum_reindex_cong[symmetric])
+
+lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
+apply (case_tac "finite A")
+ prefer 2 apply simp
+apply (erule rev_mp)
+apply (erule finite_induct, auto)
+done
+
+lemma setsum_eq_0_iff [simp]:
+  "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
+  by (induct set: finite) auto
+
+lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
+  setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
+apply(erule finite_induct)
+apply (auto simp add:add_is_1)
+done
+
+lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
+
+lemma setsum_Un_nat: "finite A ==> finite B ==>
+  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
+  -- {* For the natural numbers, we have subtraction. *}
+by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
+
+lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
+  (if a:A then setsum f A - f a else setsum f A)"
+apply (case_tac "finite A")
+ prefer 2 apply simp
+apply (erule finite_induct)
+ apply (auto simp add: insert_Diff_if)
+apply (drule_tac a = a in mk_disjoint_insert, auto)
+done
+
+lemma setsum_diff_nat: 
+assumes "finite B" and "B \<subseteq> A"
+shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
+using assms
+proof induct
+  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
+next
+  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
+    and xFinA: "insert x F \<subseteq> A"
+    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
+  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
+  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
+    by (simp add: setsum_diff1_nat)
+  from xFinA have "F \<subseteq> A" by simp
+  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
+  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
+    by simp
+  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
+  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
+    by simp
+  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
+  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
+    by simp
+  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
+qed
+
+lemma setsum_comp_morphism:
+  assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
+  shows "setsum (h \<circ> g) A = h (setsum g A)"
+proof (cases "finite A")
+  case False then show ?thesis by (simp add: assms)
+next
+  case True then show ?thesis by (induct A) (simp_all add: assms)
+qed
+
+
+subsubsection {* Cardinality as special case of @{const setsum} *}
+
+lemma card_eq_setsum:
+  "card A = setsum (\<lambda>x. 1) A"
+proof -
+  have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
+    by (simp add: fun_eq_iff)
+  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
+    by (rule arg_cong)
+  then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
+    by (blast intro: fun_cong)
+  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
+qed
+
+lemma setsum_constant [simp]:
+  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
+apply (cases "finite A")
+apply (erule finite_induct)
+apply (auto simp add: algebra_simps)
+done
+
+lemma setsum_bounded:
+  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
+  shows "setsum f A \<le> of_nat (card A) * K"
+proof (cases "finite A")
+  case True
+  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
+next
+  case False thus ?thesis by simp
+qed
+
+lemma card_UN_disjoint:
+  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
+    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
+  shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
+proof -
+  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
+  with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
+qed
+
+lemma card_Union_disjoint:
+  "finite C ==> (ALL A:C. finite A) ==>
+   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
+   ==> card (Union C) = setsum card C"
+apply (frule card_UN_disjoint [of C id])
+apply (simp_all add: SUP_def id_def)
+done
+
+
+subsubsection {* Cardinality of products *}
+
+lemma card_SigmaI [simp]:
+  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
+  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
+by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
+
+(*
+lemma SigmaI_insert: "y \<notin> A ==>
+  (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
+  by auto
+*)
+
+lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
+  by (cases "finite A \<and> finite B")
+    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
+
+lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
+by (simp add: card_cartesian_product)
+
+
+subsection {* Generalized product over a set *}
+
+context comm_monoid_mult
+begin
+
+definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+where
+  "setprod = comm_monoid_set.F times 1"
+
+sublocale setprod!: comm_monoid_set times 1
+where
+  "comm_monoid_set.F times 1 = setprod"
+proof -
+  show "comm_monoid_set times 1" ..
+  then interpret setprod!: comm_monoid_set times 1 .
+  from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule
+qed
+
+abbreviation
+  Setprod ("\<Prod>_" [1000] 999) where
+  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
+
+end
+
+syntax
+  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
+syntax (xsymbols)
+  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
+syntax (HTML output)
+  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
+
+translations -- {* Beware of argument permutation! *}
+  "PROD i:A. b" == "CONST setprod (%i. b) A" 
+  "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 
+
+text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
+ @{text"\<Prod>x|P. e"}. *}
+
+syntax
+  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
+syntax (xsymbols)
+  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
+syntax (HTML output)
+  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
+
+translations
+  "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
+  "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
+
+text {* TODO These are candidates for generalization *}
+
+context comm_monoid_mult
+begin
+
+lemma setprod_reindex_id:
+  "inj_on f B ==> setprod f B = setprod id (f ` B)"
+  by (auto simp add: setprod.reindex)
+
+lemma setprod_reindex_cong:
+  "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
+  by (frule setprod.reindex, simp)
+
+lemma strong_setprod_reindex_cong:
+  assumes i: "inj_on f A"
+  and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
+  shows "setprod h B = setprod g A"
+proof-
+  have "setprod h B = setprod (h o f) A"
+    by (simp add: B setprod.reindex [OF i, of h])
+  then show ?thesis apply simp
+    apply (rule setprod.cong)
+    apply simp
+    by (simp add: eq)
+qed
+
+lemma setprod_Union_disjoint:
+  assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 
+  shows "setprod f (Union C) = setprod (setprod f) C"
+  using assms by (fact setprod.Union_disjoint)
+
+text{*Here we can eliminate the finiteness assumptions, by cases.*}
+lemma setprod_cartesian_product:
+  "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
+  by (fact setprod.cartesian_product)
+
+lemma setprod_Un2:
+  assumes "finite (A \<union> B)"
+  shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
+proof -
+  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
+    by auto
+  with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
+qed
+
+end
+
+text {* TODO These are legacy *}
+
+lemma setprod_empty: "setprod f {} = 1"
+  by (fact setprod.empty)
+
+lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
+    setprod f (insert a A) = f a * setprod f A"
+  by (fact setprod.insert)
+
+lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
+  by (fact setprod.infinite)
+
+lemma setprod_reindex:
+  "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
+  by (fact setprod.reindex)
+
+lemma setprod_cong:
+  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
+  by (fact setprod.cong)
+
+lemma strong_setprod_cong:
+  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
+  by (fact setprod.strong_cong)
+
+lemma setprod_Un_one:
+  "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
+  \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
+  by (fact setprod.union_inter_neutral)
+
+lemmas setprod_1 = setprod.neutral_const
+lemmas setprod_1' = setprod.neutral
+
+lemma setprod_Un_Int: "finite A ==> finite B
+    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
+  by (fact setprod.union_inter)
+
+lemma setprod_Un_disjoint: "finite A ==> finite B
+  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
+  by (fact setprod.union_disjoint)
+
+lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
+    setprod f A = setprod f (A - B) * setprod f B"
+  by (fact setprod.subset_diff)
+
+lemma setprod_mono_one_left:
+  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
+  by (fact setprod.mono_neutral_left)
+
+lemmas setprod_mono_one_right = setprod.mono_neutral_right
+
+lemma setprod_mono_one_cong_left: 
+  "\<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>
+  \<Longrightarrow> setprod f S = setprod g T"
+  by (fact setprod.mono_neutral_cong_left)
+
+lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
+
+lemma setprod_delta: "finite S \<Longrightarrow>
+  setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
+  by (fact setprod.delta)
+
+lemma setprod_delta': "finite S \<Longrightarrow>
+  setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
+  by (fact setprod.delta')
+
+lemma setprod_UN_disjoint:
+    "finite I ==> (ALL i:I. finite (A i)) ==>
+        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
+      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
+  by (fact setprod.UNION_disjoint)
+
+lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
+    (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
+    (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
+  by (fact setprod.Sigma)
+
+lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
+  by (fact setprod.distrib)
+
+
+subsubsection {* Properties in more restricted classes of structures *}
+
+lemma setprod_zero:
+     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
+apply (induct set: finite, force, clarsimp)
+apply (erule disjE, auto)
+done
+
+lemma setprod_zero_iff[simp]: "finite A ==> 
+  (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
+  (EX x: A. f x = 0)"
+by (erule finite_induct, auto simp:no_zero_divisors)
+
+lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
+  (setprod f (A Un B) :: 'a ::{field})
+   = setprod f A * setprod f B / setprod f (A Int B)"
+by (subst setprod_Un_Int [symmetric], auto)
+
+lemma setprod_nonneg [rule_format]:
+   "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
+by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
+
+lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
+  --> 0 < setprod f A"
+by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
+
+lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
+  (setprod f (A - {a}) :: 'a :: {field}) =
+  (if a:A then setprod f A / f a else setprod f A)"
+  by (erule finite_induct) (auto simp add: insert_Diff_if)
+
+lemma setprod_inversef: 
+  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
+  shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
+by (erule finite_induct) auto
+
+lemma setprod_dividef:
+  fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
+  shows "finite A
+    ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
+apply (subgoal_tac
+         "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
+apply (erule ssubst)
+apply (subst divide_inverse)
+apply (subst setprod_timesf)
+apply (subst setprod_inversef, assumption+, rule refl)
+apply (rule setprod_cong, rule refl)
+apply (subst divide_inverse, auto)
+done
+
+lemma setprod_dvd_setprod [rule_format]: 
+    "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
+  apply (cases "finite A")
+  apply (induct set: finite)
+  apply (auto simp add: dvd_def)
+  apply (rule_tac x = "k * ka" in exI)
+  apply (simp add: algebra_simps)
+done
+
+lemma setprod_dvd_setprod_subset:
+  "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
+  apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
+  apply (unfold dvd_def, blast)
+  apply (subst setprod_Un_disjoint [symmetric])
+  apply (auto elim: finite_subset intro: setprod_cong)
+done
+
+lemma setprod_dvd_setprod_subset2:
+  "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 
+      setprod f A dvd setprod g B"
+  apply (rule dvd_trans)
+  apply (rule setprod_dvd_setprod, erule (1) bspec)
+  apply (erule (1) setprod_dvd_setprod_subset)
+done
+
+lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 
+    (f i ::'a::comm_semiring_1) dvd setprod f A"
+by (induct set: finite) (auto intro: dvd_mult)
+
+lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 
+    (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
+  apply (cases "finite A")
+  apply (induct set: finite)
+  apply auto
+done
+
+lemma setprod_mono:
+  fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
+  assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
+  shows "setprod f A \<le> setprod g A"
+proof (cases "finite A")
+  case True
+  hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
+  proof (induct A rule: finite_subset_induct)
+    case (insert a F)
+    thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
+      unfolding setprod_insert[OF insert(1,3)]
+      using assms[rule_format,OF insert(2)] insert
+      by (auto intro: mult_mono mult_nonneg_nonneg)
+  qed auto
+  thus ?thesis by simp
+qed auto
+
+lemma abs_setprod:
+  fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
+  shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
+proof (cases "finite A")
+  case True thus ?thesis
+    by induct (auto simp add: field_simps abs_mult)
+qed auto
+
+lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
+apply (erule finite_induct)
+apply auto
+done
+
+lemma setprod_gen_delta:
+  assumes fS: "finite S"
+  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
+proof-
+  let ?f = "(\<lambda>k. if k=a then b k else c)"
+  {assume a: "a \<notin> S"
+    hence "\<forall> k\<in> S. ?f k = c" by simp
+    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
+  moreover 
+  {assume a: "a \<in> S"
+    let ?A = "S - {a}"
+    let ?B = "{a}"
+    have eq: "S = ?A \<union> ?B" using a by blast 
+    have dj: "?A \<inter> ?B = {}" by simp
+    from fS have fAB: "finite ?A" "finite ?B" by auto  
+    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
+      apply (rule setprod_cong) by auto
+    have cA: "card ?A = card S - 1" using fS a by auto
+    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
+    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
+      using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
+      by simp
+    then have ?thesis using a cA
+      by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
+  ultimately show ?thesis by blast
+qed
+
+lemma setprod_eq_1_iff [simp]:
+  "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
+  by (induct set: finite) auto
+
+lemma setprod_pos_nat:
+  "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
+using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
+
+lemma setprod_pos_nat_iff[simp]:
+  "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
+using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
+
+end

--- a/src/HOL/Hilbert_Choice.thy	Sat Dec 14 20:46:36 2013 +0100
+++ b/src/HOL/Hilbert_Choice.thy	Sun Dec 15 15:10:14 2013 +0100
@@ -6,7 +6,7 @@
 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
 
 theory Hilbert_Choice
-imports Nat Wellfounded Big_Operators
+imports Nat Wellfounded Lattices_Big Metis
 keywords "specification" "ax_specification" :: thy_goal
 begin
 

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Lattices_Big.thy	Sun Dec 15 15:10:14 2013 +0100
@@ -0,0 +1,833 @@
+(*  Title:      HOL/Lattices_Big.thy
+    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
+                with contributions by Jeremy Avigad
+*)
+
+header {* Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets *}
+
+theory Lattices_Big
+imports Finite_Set
+begin
+
+subsection {* Generic lattice operations over a set *}
+
+no_notation times (infixl "*" 70)
+no_notation Groups.one ("1")
+
+
+subsubsection {* Without neutral element *}
+
+locale semilattice_set = semilattice
+begin
+
+interpretation comp_fun_idem f
+  by default (simp_all add: fun_eq_iff left_commute)
+
+definition F :: "'a set \<Rightarrow> 'a"
+where
+  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)"
+
+lemma eq_fold:
+  assumes "finite A"
+  shows "F (insert x A) = Finite_Set.fold f x A"
+proof (rule sym)
+  let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
+  interpret comp_fun_idem "?f"
+    by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
+  from assms show "Finite_Set.fold f x A = F (insert x A)"
+  proof induct
+    case empty then show ?case by (simp add: eq_fold')
+  next
+    case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
+  qed
+qed
+
+lemma singleton [simp]:
+  "F {x} = x"
+  by (simp add: eq_fold)
+
+lemma insert_not_elem:
+  assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
+  shows "F (insert x A) = x * F A"
+proof -
+  from `A \<noteq> {}` obtain b where "b \<in> A" by blast
+  then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
+  with `finite A` and `x \<notin> A`
+    have "finite (insert x B)" and "b \<notin> insert x B" by auto
+  then have "F (insert b (insert x B)) = x * F (insert b B)"
+    by (simp add: eq_fold)
+  then show ?thesis by (simp add: * insert_commute)
+qed
+
+lemma in_idem:
+  assumes "finite A" and "x \<in> A"
+  shows "x * F A = F A"
+proof -
+  from assms have "A \<noteq> {}" by auto
+  with `finite A` show ?thesis using `x \<in> A`
+    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
+qed
+
+lemma insert [simp]:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "F (insert x A) = x * F A"
+  using assms by (cases "x \<in> A") (simp_all add: insert_absorb in_idem insert_not_elem)
+
+lemma union:
+  assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
+  shows "F (A \<union> B) = F A * F B"
+  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
+
+lemma remove:
+  assumes "finite A" and "x \<in> A"
+  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
+proof -
+  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
+  with assms show ?thesis by simp
+qed
+
+lemma insert_remove:
+  assumes "finite A"
+  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
+  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
+
+lemma subset:
+  assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
+  shows "F B * F A = F A"
+proof -
+  from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
+  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
+qed
+
+lemma closed:
+  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
+  shows "F A \<in> A"
+using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
+  case singleton then show ?case by simp
+next
+  case insert with elem show ?case by force
+qed
+
+lemma hom_commute:
+  assumes hom: "\<And>x y. h (x * y) = h x * h y"
+  and N: "finite N" "N \<noteq> {}"
+  shows "h (F N) = F (h ` N)"
+using N proof (induct rule: finite_ne_induct)
+  case singleton thus ?case by simp
+next
+  case (insert n N)
+  then have "h (F (insert n N)) = h (n * F N)" by simp
+  also have "\<dots> = h n * h (F N)" by (rule hom)
+  also have "h (F N) = F (h ` N)" by (rule insert)
+  also have "h n * \<dots> = F (insert (h n) (h ` N))"
+    using insert by simp
+  also have "insert (h n) (h ` N) = h ` insert n N" by simp
+  finally show ?case .
+qed
+
+end
+
+locale semilattice_order_set = semilattice_order + semilattice_set
+begin
+
+lemma bounded_iff:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
+  using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
+
+lemma boundedI:
+  assumes "finite A"
+  assumes "A \<noteq> {}"
+  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+  shows "x \<preceq> F A"
+  using assms by (simp add: bounded_iff)
+
+lemma boundedE:
+  assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
+  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+  using assms by (simp add: bounded_iff)
+
+lemma coboundedI:
+  assumes "finite A"
+    and "a \<in> A"
+  shows "F A \<preceq> a"
+proof -
+  from assms have "A \<noteq> {}" by auto
+  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
+  proof (induct rule: finite_ne_induct)
+    case singleton thus ?case by (simp add: refl)
+  next
+    case (insert x B)
+    from insert have "a = x \<or> a \<in> B" by simp
+    then show ?case using insert by (auto intro: coboundedI2)
+  qed
+qed
+
+lemma antimono:
+  assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
+  shows "F B \<preceq> F A"
+proof (cases "A = B")
+  case True then show ?thesis by (simp add: refl)
+next
+  case False
+  have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
+  then have "F B = F (A \<union> (B - A))" by simp
+  also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
+  also have "\<dots> \<preceq> F A" by simp
+  finally show ?thesis .
+qed
+
+end
+
+
+subsubsection {* With neutral element *}
+
+locale semilattice_neutr_set = semilattice_neutr
+begin
+
+interpretation comp_fun_idem f
+  by default (simp_all add: fun_eq_iff left_commute)
+
+definition F :: "'a set \<Rightarrow> 'a"
+where
+  eq_fold: "F A = Finite_Set.fold f 1 A"
+
+lemma infinite [simp]:
+  "\<not> finite A \<Longrightarrow> F A = 1"
+  by (simp add: eq_fold)
+
+lemma empty [simp]:
+  "F {} = 1"
+  by (simp add: eq_fold)
+
+lemma insert [simp]:
+  assumes "finite A"
+  shows "F (insert x A) = x * F A"
+  using assms by (simp add: eq_fold)
+
+lemma in_idem:
+  assumes "finite A" and "x \<in> A"
+  shows "x * F A = F A"
+proof -
+  from assms have "A \<noteq> {}" by auto
+  with `finite A` show ?thesis using `x \<in> A`
+    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
+qed
+
+lemma union:
+  assumes "finite A" and "finite B"
+  shows "F (A \<union> B) = F A * F B"
+  using assms by (induct A) (simp_all add: ac_simps)
+
+lemma remove:
+  assumes "finite A" and "x \<in> A"
+  shows "F A = x * F (A - {x})"
+proof -
+  from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
+  with assms show ?thesis by simp
+qed
+
+lemma insert_remove:
+  assumes "finite A"
+  shows "F (insert x A) = x * F (A - {x})"
+  using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
+
+lemma subset:
+  assumes "finite A" and "B \<subseteq> A"
+  shows "F B * F A = F A"
+proof -
+  from assms have "finite B" by (auto dest: finite_subset)
+  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
+qed
+
+lemma closed:
+  assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
+  shows "F A \<in> A"
+using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
+  case singleton then show ?case by simp
+next
+  case insert with elem show ?case by force
+qed
+
+end
+
+locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
+begin
+
+lemma bounded_iff:
+  assumes "finite A"
+  shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
+  using assms by (induct A) (simp_all add: bounded_iff)
+
+lemma boundedI:
+  assumes "finite A"
+  assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+  shows "x \<preceq> F A"
+  using assms by (simp add: bounded_iff)
+
+lemma boundedE:
+  assumes "finite A" and "x \<preceq> F A"
+  obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+  using assms by (simp add: bounded_iff)
+
+lemma coboundedI:
+  assumes "finite A"
+    and "a \<in> A"
+  shows "F A \<preceq> a"
+proof -
+  from assms have "A \<noteq> {}" by auto
+  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
+  proof (induct rule: finite_ne_induct)
+    case singleton thus ?case by (simp add: refl)
+  next
+    case (insert x B)
+    from insert have "a = x \<or> a \<in> B" by simp
+    then show ?case using insert by (auto intro: coboundedI2)
+  qed
+qed
+
+lemma antimono:
+  assumes "A \<subseteq> B" and "finite B"
+  shows "F B \<preceq> F A"
+proof (cases "A = B")
+  case True then show ?thesis by (simp add: refl)
+next
+  case False
+  have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
+  then have "F B = F (A \<union> (B - A))" by simp
+  also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
+  also have "\<dots> \<preceq> F A" by simp
+  finally show ?thesis .
+qed
+
+end
+
+notation times (infixl "*" 70)
+notation Groups.one ("1")
+
+
+subsection {* Lattice operations on finite sets *}
+
+text {*
+  For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
+  to @{class linorder}.  This is badly designed: both should depend on a common abstract
+  distributive lattice rather than having this non-subclass dependecy between two
+  classes.  But for the moment we have to live with it.  This forces us to setup
+  this sublocale dependency simultaneously with the lattice operations on finite
+  sets, to avoid garbage.
+*}
+
+definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
+where
+  "Inf_fin = semilattice_set.F inf"
+
+definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^sub>f\<^sub>i\<^sub>n_" [900] 900)
+where
+  "Sup_fin = semilattice_set.F sup"
+
+context linorder
+begin
+
+definition Min :: "'a set \<Rightarrow> 'a"
+where
+  "Min = semilattice_set.F min"
+
+definition Max :: "'a set \<Rightarrow> 'a"
+where
+  "Max = semilattice_set.F max"
+
+sublocale Min!: semilattice_order_set min less_eq less
+  + Max!: semilattice_order_set max greater_eq greater
+where
+  "semilattice_set.F min = Min"
+  and "semilattice_set.F max = Max"
+proof -
+  show "semilattice_order_set min less_eq less" by default (auto simp add: min_def)
+  then interpret Min!: semilattice_order_set min less_eq less .
+  show "semilattice_order_set max greater_eq greater" by default (auto simp add: max_def)
+  then interpret Max!: semilattice_order_set max greater_eq greater .
+  from Min_def show "semilattice_set.F min = Min" by rule
+  from Max_def show "semilattice_set.F max = Max" by rule
+qed
+
+
+text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
+
+sublocale min_max!: distrib_lattice min less_eq less max
+where
+  "semilattice_inf.Inf_fin min = Min"
+  and "semilattice_sup.Sup_fin max = Max"
+proof -
+  show "class.distrib_lattice min less_eq less max"
+  proof
+    fix x y z
+    show "max x (min y z) = min (max x y) (max x z)"
+      by (auto simp add: min_def max_def)
+  qed (auto simp add: min_def max_def not_le less_imp_le)
+  then interpret min_max!: distrib_lattice min less_eq less max .
+  show "semilattice_inf.Inf_fin min = Min"
+    by (simp only: min_max.Inf_fin_def Min_def)
+  show "semilattice_sup.Sup_fin max = Max"
+    by (simp only: min_max.Sup_fin_def Max_def)
+qed
+
+lemmas le_maxI1 = min_max.sup_ge1
+lemmas le_maxI2 = min_max.sup_ge2
+ 
+lemmas min_ac = min_max.inf_assoc min_max.inf_commute
+  min.left_commute
+
+lemmas max_ac = min_max.sup_assoc min_max.sup_commute
+  max.left_commute
+
+end
+
+
+text {* Lattice operations proper *}
+
+sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
+where
+  "semilattice_set.F inf = Inf_fin"
+proof -
+  show "semilattice_order_set inf less_eq less" ..
+  then interpret Inf_fin!: semilattice_order_set inf less_eq less .
+  from Inf_fin_def show "semilattice_set.F inf = Inf_fin" by rule
+qed
+
+sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
+where
+  "semilattice_set.F sup = Sup_fin"
+proof -
+  show "semilattice_order_set sup greater_eq greater" ..
+  then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
+  from Sup_fin_def show "semilattice_set.F sup = Sup_fin" by rule
+qed
+
+
+text {* An aside again: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin} *}
+
+lemma Inf_fin_Min:
+  "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set \<Rightarrow> 'a)"
+  by (simp add: Inf_fin_def Min_def inf_min)
+
+lemma Sup_fin_Max:
+  "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set \<Rightarrow> 'a)"
+  by (simp add: Sup_fin_def Max_def sup_max)
+
+
+
+subsection {* Infimum and Supremum over non-empty sets *}
+
+text {*
+  After this non-regular bootstrap, things continue canonically.
+*}
+
+context lattice
+begin
+
+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"
+apply(subgoal_tac "EX a. a:A")
+prefer 2 apply blast
+apply(erule exE)
+apply(rule order_trans)
+apply(erule (1) Inf_fin.coboundedI)
+apply(erule (1) Sup_fin.coboundedI)
+done
+
+lemma sup_Inf_absorb [simp]:
+  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = a"
+apply(subst sup_commute)
+apply(simp add: sup_absorb2 Inf_fin.coboundedI)
+done
+
+lemma inf_Sup_absorb [simp]:
+  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = a"
+by (simp add: inf_absorb1 Sup_fin.coboundedI)
+
+end
+
+context distrib_lattice
+begin
+
+lemma sup_Inf1_distrib:
+  assumes "finite A"
+    and "A \<noteq> {}"
+  shows "sup x (\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA) = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n{sup x a|a. a \<in> A}"
+using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
+  (rule arg_cong [where f="Inf_fin"], blast)
+
+lemma sup_Inf2_distrib:
+  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
+  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}"
+using A proof (induct rule: finite_ne_induct)
+  case singleton then show ?case
+    by (simp add: sup_Inf1_distrib [OF B])
+next
+  case (insert x A)
+  have finB: "finite {sup x b |b. b \<in> B}"
+    by (rule finite_surj [where f = "sup x", OF B(1)], auto)
+  have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
+  proof -
+    have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
+      by blast
+    thus ?thesis by(simp add: insert(1) B(1))
+  qed
+  have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
+  have "sup (\<Sqinter>\<^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)"
+    using insert by simp
+  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)
+  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})"
+    using insert by(simp add:sup_Inf1_distrib[OF B])
+  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})"
+    (is "_ = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n?M")
+    using B insert
+    by (simp add: Inf_fin.union [OF finB _ finAB ne])
+  also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
+    by blast
+  finally show ?case .
+qed
+
+lemma inf_Sup1_distrib:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "inf x (\<Squnion>\<^sub>f\<^sub>i\<^sub>nA) = \<Squnion>\<^sub>f\<^sub>i\<^sub>n{inf x a|a. a \<in> A}"
+using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
+  (rule arg_cong [where f="Sup_fin"], blast)
+
+lemma inf_Sup2_distrib:
+  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
+  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}"
+using A proof (induct rule: finite_ne_induct)
+  case singleton thus ?case
+    by(simp add: inf_Sup1_distrib [OF B])
+next
+  case (insert x A)
+  have finB: "finite {inf x b |b. b \<in> B}"
+    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
+  have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
+  proof -
+    have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
+      by blast
+    thus ?thesis by(simp add: insert(1) B(1))
+  qed
+  have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
+  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)"
+    using insert by simp
+  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)
+  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})"
+    using insert by(simp add:inf_Sup1_distrib[OF B])
+  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})"
+    (is "_ = \<Squnion>\<^sub>f\<^sub>i\<^sub>n?M")
+    using B insert
+    by (simp add: Sup_fin.union [OF finB _ finAB ne])
+  also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
+    by blast
+  finally show ?case .
+qed
+
+end
+
+context complete_lattice
+begin
+
+lemma Inf_fin_Inf:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>nA = Inf A"
+proof -
+  from assms obtain b B where "A = insert b B" and "finite B" by auto
+  then show ?thesis
+    by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
+qed
+
+lemma Sup_fin_Sup:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>nA = Sup A"
+proof -
+  from assms obtain b B where "A = insert b B" and "finite B" by auto
+  then show ?thesis
+    by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
+qed
+
+end
+
+
+subsection {* Minimum and Maximum over non-empty sets *}
+
+context linorder
+begin
+
+lemma dual_min:
+  "ord.min greater_eq = max"
+  by (auto simp add: ord.min_def max_def fun_eq_iff)
+
+lemma dual_max:
+  "ord.max greater_eq = min"
+  by (auto simp add: ord.max_def min_def fun_eq_iff)
+
+lemma dual_Min:
+  "linorder.Min greater_eq = Max"
+proof -
+  interpret dual!: linorder greater_eq greater by (fact dual_linorder)
+  show ?thesis by (simp add: dual.Min_def dual_min Max_def)
+qed
+
+lemma dual_Max:
+  "linorder.Max greater_eq = Min"
+proof -
+  interpret dual!: linorder greater_eq greater by (fact dual_linorder)
+  show ?thesis by (simp add: dual.Max_def dual_max Min_def)
+qed
+
+lemmas Min_singleton = Min.singleton
+lemmas Max_singleton = Max.singleton
+lemmas Min_insert = Min.insert
+lemmas Max_insert = Max.insert
+lemmas Min_Un = Min.union
+lemmas Max_Un = Max.union
+lemmas hom_Min_commute = Min.hom_commute
+lemmas hom_Max_commute = Max.hom_commute
+
+lemma Min_in [simp]:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "Min A \<in> A"
+  using assms by (auto simp add: min_def Min.closed)
+
+lemma Max_in [simp]:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "Max A \<in> A"
+  using assms by (auto simp add: max_def Max.closed)
+
+lemma Min_le [simp]:
+  assumes "finite A" and "x \<in> A"
+  shows "Min A \<le> x"
+  using assms by (fact Min.coboundedI)
+
+lemma Max_ge [simp]:
+  assumes "finite A" and "x \<in> A"
+  shows "x \<le> Max A"
+  using assms by (fact Max.coboundedI)
+
+lemma Min_eqI:
+  assumes "finite A"
+  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
+    and "x \<in> A"
+  shows "Min A = x"
+proof (rule antisym)
+  from `x \<in> A` have "A \<noteq> {}" by auto
+  with assms show "Min A \<ge> x" by simp
+next
+  from assms show "x \<ge> Min A" by simp
+qed
+
+lemma Max_eqI:
+  assumes "finite A"
+  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
+    and "x \<in> A"
+  shows "Max A = x"
+proof (rule antisym)
+  from `x \<in> A` have "A \<noteq> {}" by auto
+  with assms show "Max A \<le> x" by simp
+next
+  from assms show "x \<le> Max A" by simp
+qed
+
+context
+  fixes A :: "'a set"
+  assumes fin_nonempty: "finite A" "A \<noteq> {}"
+begin
+
+lemma Min_ge_iff [simp]:
+  "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
+  using fin_nonempty by (fact Min.bounded_iff)
+
+lemma Max_le_iff [simp]:
+  "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
+  using fin_nonempty by (fact Max.bounded_iff)
+
+lemma Min_gr_iff [simp]:
+  "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
+  using fin_nonempty  by (induct rule: finite_ne_induct) simp_all
+
+lemma Max_less_iff [simp]:
+  "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
+  using fin_nonempty by (induct rule: finite_ne_induct) simp_all
+
+lemma Min_le_iff:
+  "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
+  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
+
+lemma Max_ge_iff:
+  "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
+  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
+
+lemma Min_less_iff:
+  "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
+  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
+
+lemma Max_gr_iff:
+  "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
+  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
+
+end
+
+lemma Min_antimono:
+  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
+  shows "Min N \<le> Min M"
+  using assms by (fact Min.antimono)
+
+lemma Max_mono:
+  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
+  shows "Max M \<le> Max N"
+  using assms by (fact Max.antimono)
+
+lemma mono_Min_commute:
+  assumes "mono f"
+  assumes "finite A" and "A \<noteq> {}"
+  shows "f (Min A) = Min (f ` A)"
+proof (rule linorder_class.Min_eqI [symmetric])
+  from `finite A` show "finite (f ` A)" by simp
+  from assms show "f (Min A) \<in> f ` A" by simp
+  fix x
+  assume "x \<in> f ` A"
+  then obtain y where "y \<in> A" and "x = f y" ..
+  with assms have "Min A \<le> y" by auto
+  with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
+  with `x = f y` show "f (Min A) \<le> x" by simp
+qed
+
+lemma mono_Max_commute:
+  assumes "mono f"
+  assumes "finite A" and "A \<noteq> {}"
+  shows "f (Max A) = Max (f ` A)"
+proof (rule linorder_class.Max_eqI [symmetric])
+  from `finite A` show "finite (f ` A)" by simp
+  from assms show "f (Max A) \<in> f ` A" by simp
+  fix x
+  assume "x \<in> f ` A"
+  then obtain y where "y \<in> A" and "x = f y" ..
+  with assms have "y \<le> Max A" by auto
+  with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
+  with `x = f y` show "x \<le> f (Max A)" by simp
+qed
+
+lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
+  assumes fin: "finite A"
+  and empty: "P {}" 
+  and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
+  shows "P A"
+using fin empty insert
+proof (induct rule: finite_psubset_induct)
+  case (psubset A)
+  have IH: "\<And>B. \<lbrakk>B < A; P {}; (\<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. a<b; P A\<rbrakk> \<Longrightarrow> P (insert b A))\<rbrakk> \<Longrightarrow> P B" by fact 
+  have fin: "finite A" by fact 
+  have empty: "P {}" by fact
+  have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
+  show "P A"
+  proof (cases "A = {}")
+    assume "A = {}" 
+    then show "P A" using `P {}` by simp
+  next
+    let ?B = "A - {Max A}" 
+    let ?A = "insert (Max A) ?B"
+    have "finite ?B" using `finite A` by simp
+    assume "A \<noteq> {}"
+    with `finite A` have "Max A : A" by auto
+    then have A: "?A = A" using insert_Diff_single insert_absorb by auto
+    then have "P ?B" using `P {}` step IH [of ?B] by blast
+    moreover 
+    have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
+    ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
+  qed
+qed
+
+lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
+  "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
+  by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
+
+lemma Least_Min:
+  assumes "finite {a. P a}" and "\<exists>a. P a"
+  shows "(LEAST a. P a) = Min {a. P a}"
+proof -
+  { fix A :: "'a set"
+    assume A: "finite A" "A \<noteq> {}"
+    have "(LEAST a. a \<in> A) = Min A"
+    using A proof (induct A rule: finite_ne_induct)
+      case singleton show ?case by (rule Least_equality) simp_all
+    next
+      case (insert a A)
+      have "(LEAST b. b = a \<or> b \<in> A) = min a (LEAST a. a \<in> A)"
+        by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
+      with insert show ?case by simp
+    qed
+  } from this [of "{a. P a}"] assms show ?thesis by simp
+qed
+
+end
+
+context linordered_ab_semigroup_add
+begin
+
+lemma add_Min_commute:
+  fixes k
+  assumes "finite N" and "N \<noteq> {}"
+  shows "k + Min N = Min {k + m | m. m \<in> N}"
+proof -
+  have "\<And>x y. k + min x y = min (k + x) (k + y)"
+    by (simp add: min_def not_le)
+      (blast intro: antisym less_imp_le add_left_mono)
+  with assms show ?thesis
+    using hom_Min_commute [of "plus k" N]
+    by simp (blast intro: arg_cong [where f = Min])
+qed
+
+lemma add_Max_commute:
+  fixes k
+  assumes "finite N" and "N \<noteq> {}"
+  shows "k + Max N = Max {k + m | m. m \<in> N}"
+proof -
+  have "\<And>x y. k + max x y = max (k + x) (k + y)"
+    by (simp add: max_def not_le)
+      (blast intro: antisym less_imp_le add_left_mono)
+  with assms show ?thesis
+    using hom_Max_commute [of "plus k" N]
+    by simp (blast intro: arg_cong [where f = Max])
+qed
+
+end
+
+context linordered_ab_group_add
+begin
+
+lemma minus_Max_eq_Min [simp]:
+  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
+  by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
+
+lemma minus_Min_eq_Max [simp]:
+  "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
+  by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
+
+end
+
+context complete_linorder
+begin
+
+lemma Min_Inf:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "Min A = Inf A"
+proof -
+  from assms obtain b B where "A = insert b B" and "finite B" by auto
+  then show ?thesis
+    by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
+qed
+
+lemma Max_Sup:
+  assumes "finite A" and "A \<noteq> {}"
+  shows "Max A = Sup A"
+proof -
+  from assms obtain b B where "A = insert b B" and "finite B" by auto
+  then show ?thesis
+    by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
+qed
+
+end
+
+end