src/HOL/Finite_Set.thy
 changeset 35719 99b6152aedf5 parent 35577 43b93e294522 child 35722 69419a09a7ff
--- a/src/HOL/Finite_Set.thy	Wed Mar 10 08:04:50 2010 +0100
+++ b/src/HOL/Finite_Set.thy	Wed Mar 10 16:53:27 2010 +0100
@@ -6,7 +6,7 @@

theory Finite_Set
-imports Power Product_Type Sum_Type
+imports Power Option
begin

subsection {* Definition and basic properties *}
@@ -527,17 +527,24 @@
lemma UNIV_unit [noatp]:
"UNIV = {()}" by auto

-instance unit :: finite
-  by default (simp add: UNIV_unit)
+instance unit :: finite proof

lemma UNIV_bool [noatp]:
"UNIV = {False, True}" by auto

-instance bool :: finite
-  by default (simp add: UNIV_bool)
+instance bool :: finite proof
+
+instance * :: (finite, finite) finite proof
+qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)

-instance * :: (finite, finite) finite
-  by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
+lemma finite_option_UNIV [simp]:
+  "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
+  by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
+
+instance option :: (finite) finite proof

lemma inj_graph: "inj (%f. {(x, y). y = f x})"
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
@@ -556,8 +563,8 @@
qed
qed

-instance "+" :: (finite, finite) finite
-  by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
+instance "+" :: (finite, finite) finite proof
+qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)

subsection {* A fold functional for finite sets *}
@@ -1053,1470 +1060,6 @@

end

-subsection {* Generalized summation over a set *}
-
-
-definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
-where "setsum f A == if finite A then fold_image (op +) f 0 A else 0"
-
-abbreviation
-  Setsum  ("\<Sum>_" [1000] 999) where
-  "\<Sum>A == setsum (%x. x) A"
-
-text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
-written @{text"\<Sum>x\<in>A. e"}. *}
-
-syntax
-  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
-syntax (xsymbols)
-  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
-syntax (HTML output)
-  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
-
-translations -- {* Beware of argument permutation! *}
-  "SUM i:A. b" == "CONST setsum (%i. b) A"
-  "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
-
-text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
- @{text"\<Sum>x|P. e"}. *}
-
-syntax
-  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
-syntax (xsymbols)
-  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
-syntax (HTML output)
-  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
-
-translations
-  "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
-  "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
-
-print_translation {*
-let
-  fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
-        if x <> y then raise Match
-        else
-          let
-            val x' = Syntax.mark_bound x;
-            val t' = subst_bound (x', t);
-            val P' = subst_bound (x', P);
-          in Syntax.const @{syntax_const "_qsetsum"} \$ Syntax.mark_bound x \$ P' \$ t' end
-    | setsum_tr' _ = raise Match;
-in [(@{const_syntax setsum}, setsum_tr')] end
-*}
-
-
-lemma setsum_empty [simp]: "setsum f {} = 0"
-
-lemma setsum_insert [simp]:
-  "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
-
-lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
-
-lemma setsum_reindex:
-     "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
-
-lemma setsum_reindex_id:
-     "inj_on f B ==> setsum f B = setsum id (f ` B)"
-
-lemma setsum_reindex_nonzero:
-  assumes fS: "finite S"
-  and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
-  shows "setsum h (f ` S) = setsum (h o f) S"
-using nz
-proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case by simp
-next
-  case (2 x F)
-  {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
-    then obtain y where y: "y \<in> F" "f x = f y" by auto
-    from "2.hyps" y have xy: "x \<noteq> y" by auto
-
-    from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
-    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
-    also have "\<dots> = setsum (h o f) (insert x F)"
-      unfolding setsum_insert[OF `finite F` `x\<notin>F`]
-      using h0
-      apply simp
-      apply (rule "2.hyps"(3))
-      apply (rule_tac y="y" in  "2.prems")
-      apply simp_all
-      done
-    finally have ?case .}
-  moreover
-  {assume fxF: "f x \<notin> f ` F"
-    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
-      using fxF "2.hyps" by simp
-    also have "\<dots> = setsum (h o f) (insert x F)"
-      unfolding setsum_insert[OF `finite F` `x\<notin>F`]
-      apply simp
-      apply (rule cong[OF refl[of "op + (h (f x))"]])
-      apply (rule "2.hyps"(3))
-      apply (rule_tac y="y" in  "2.prems")
-      apply simp_all
-      done
-    finally have ?case .}
-  ultimately show ?case by blast
-qed
-
-lemma setsum_cong:
-  "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
-
-lemma strong_setsum_cong[cong]:
-  "A = B ==> (!!x. x:B =simp=> f x = g x)
-   ==> setsum (%x. f x) A = setsum (%x. g x) B"
-by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong)
-
-lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"
-by (rule setsum_cong[OF refl], auto)
-
-lemma setsum_reindex_cong:
-   "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
-    ==> setsum h B = setsum g A"
-by (simp add: setsum_reindex cong: setsum_cong)
-
-
-lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
-apply (clarsimp simp: setsum_def)
-apply (erule finite_induct, auto)
-done
-
-lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
-
-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! *}
-
-lemma setsum_Un_disjoint: "finite A ==> finite B
-  ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
-by (subst setsum_Un_Int [symmetric], auto)
-
-lemma setsum_mono_zero_left:
-  assumes fT: "finite T" and ST: "S \<subseteq> T"
-  and z: "\<forall>i \<in> T - S. f i = 0"
-  shows "setsum f S = setsum f T"
-proof-
-  have eq: "T = S \<union> (T - S)" using ST by blast
-  have d: "S \<inter> (T - S) = {}" using ST by blast
-  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
-  show ?thesis
-  by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
-qed
-
-lemma setsum_mono_zero_right:
-  "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. f i = 0 \<Longrightarrow> setsum f T = setsum f S"
-by(blast intro!: setsum_mono_zero_left[symmetric])
-
-lemma setsum_mono_zero_cong_left:
-  assumes fT: "finite T" and ST: "S \<subseteq> T"
-  and z: "\<forall>i \<in> T - S. g i = 0"
-  and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
-  shows "setsum f S = setsum g T"
-proof-
-  have eq: "T = S \<union> (T - S)" using ST by blast
-  have d: "S \<inter> (T - S) = {}" using ST by blast
-  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
-  show ?thesis
-    using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
-qed
-
-lemma setsum_mono_zero_cong_right:
-  assumes fT: "finite T" and ST: "S \<subseteq> T"
-  and z: "\<forall>i \<in> T - S. f i = 0"
-  and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
-  shows "setsum f T = setsum g S"
-using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto
-
-lemma setsum_delta:
-  assumes fS: "finite S"
-  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else 0)"
-  {assume a: "a \<notin> S"
-    hence "\<forall> k\<in> S. ?f k = 0" by simp
-    hence ?thesis  using a by simp}
-  moreover
-  {assume a: "a \<in> S"
-    let ?A = "S - {a}"
-    let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto
-    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
-      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
-      by simp
-    then have ?thesis  using a by simp}
-  ultimately show ?thesis by blast
-qed
-lemma setsum_delta':
-  assumes fS: "finite S" shows
-  "setsum (\<lambda>k. if a = k then b k else 0) S =
-     (if a\<in> S then b a else 0)"
-  using setsum_delta[OF fS, of a b, symmetric]
-  by (auto intro: setsum_cong)
-
-lemma setsum_restrict_set:
-  assumes fA: "finite A"
-  shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
-proof-
-  from fA have fab: "finite (A \<inter> B)" by auto
-  have aba: "A \<inter> B \<subseteq> A" by blast
-  let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
-  from setsum_mono_zero_left[OF fA aba, of ?g]
-  show ?thesis by simp
-qed
-
-lemma setsum_cases:
-  assumes fA: "finite A"
-  shows "setsum (\<lambda>x. if P x then f x else g x) A =
-         setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
-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 f x else g x"
-  from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
-  show ?thesis by simp
-qed
-
-
-(*But we can't get rid of finite I. If infinite, although the rhs is 0,
-  the lhs need not be, since UNION I A could still be finite.*)
-lemma setsum_UN_disjoint:
-    "finite I ==> (ALL i:I. finite (A i)) ==>
-        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
-      setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
-
-text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
-directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
-lemma setsum_Union_disjoint:
-  "[| (ALL A:C. finite A);
-      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
-   ==> setsum f (Union C) = setsum (setsum f) C"
-apply (cases "finite C")
- prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
-  apply (frule setsum_UN_disjoint [of C id f])
- apply (unfold Union_def id_def, assumption+)
-done
-
-(*But we can't get rid of finite A. If infinite, although the lhs is 0,
-  the rhs need not be, since SIGMA A B could still be finite.*)
-lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
-    (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
-
-text{*Here we can eliminate the finiteness assumptions, by cases.*}
-lemma setsum_cartesian_product:
-   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
-apply (cases "finite A")
- apply (cases "finite B")
- apply (cases "A={}", simp)
- apply (simp)
-            dest: finite_cartesian_productD1 finite_cartesian_productD2)
-done
-
-lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
-
-
-subsubsection {* Properties in more restricted classes of structures *}
-
-lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
-apply (case_tac "finite A")
- prefer 2 apply (simp add: setsum_def)
-apply (erule rev_mp)
-apply (erule finite_induct, auto)
-done
-
-lemma setsum_eq_0_iff [simp]:
-    "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
-by (induct set: finite) auto
-
-lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
-  (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
-apply(erule finite_induct)
-done
-
-lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
-
-lemma setsum_Un_nat: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
-  -- {* For the natural numbers, we have subtraction. *}
-by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
-
-lemma setsum_Un: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: 'a :: ab_group_add) =
-   setsum f A + setsum f B - setsum f (A Int B)"
-by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
-
-lemma (in comm_monoid_mult) fold_image_1: "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
-  apply (induct set: finite)
-  apply simp by auto
-
-lemma (in comm_monoid_mult) fold_image_Un_one:
-  assumes fS: "finite S" and fT: "finite T"
-  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
-  shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
-proof-
-  have "fold_image op * f 1 (S \<inter> T) = 1"
-    apply (rule fold_image_1)
-    using fS fT I0 by auto
-  with fold_image_Un_Int[OF fS fT] show ?thesis by simp
-qed
-
-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"
-  apply (simp add: setsum_def fS fT)
-  apply (erule kh)
-  apply (erule hk)
-  done
-
-
-
-lemma setsum_Un_zero:
-  assumes fS: "finite S" and fT: "finite T"
-  and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
-  shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
-  using fS fT
-  using I0 by auto
-
-
-lemma setsum_UNION_zero:
-  assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
-  and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
-  shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
-  using fSS f0
-proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case by simp
-next
-  case (2 T F)
-  then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
-    and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
-  from fTF have fUF: "finite (\<Union>F)" by auto
-  from "2.prems" TF fTF
-  show ?case
-    by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
-qed
-
-
-lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
-  (if a:A then setsum f A - f a else setsum f A)"
-apply (case_tac "finite A")
- prefer 2 apply (simp add: setsum_def)
-apply (erule finite_induct)
- apply (auto simp add: insert_Diff_if)
-apply (drule_tac a = a in mk_disjoint_insert, auto)
-done
-
-lemma setsum_diff1: "finite A \<Longrightarrow>
-  (setsum f (A - {a}) :: ('a::ab_group_add)) =
-  (if a:A then setsum f A - f a else setsum f A)"
-by (erule finite_induct) (auto simp add: insert_Diff_if)
-
-lemma setsum_diff1'[rule_format]:
-  "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
-apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
-done
-
-lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
-  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
-unfolding setsum_diff1'[OF assms] by auto
-
-(* By Jeremy Siek: *)
-
-lemma setsum_diff_nat:
-assumes "finite B" and "B \<subseteq> A"
-shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
-using assms
-proof induct
-  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
-next
-  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
-    and xFinA: "insert x F \<subseteq> A"
-    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
-  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
-  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
-  from xFinA have "F \<subseteq> A" by simp
-  with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
-  with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
-    by simp
-  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
-  with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
-    by simp
-  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
-  with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
-    by simp
-  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
-qed
-
-lemma setsum_diff:
-  assumes le: "finite A" "B \<subseteq> A"
-  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
-proof -
-  from le have finiteB: "finite B" using finite_subset by auto
-  show ?thesis using finiteB le
-  proof induct
-    case empty
-    thus ?case by auto
-  next
-    case (insert x F)
-    thus ?case using le finiteB
-      by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
-  qed
-qed
-
-lemma setsum_mono:
-  assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
-  shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
-proof (cases "finite K")
-  case True
-  thus ?thesis using le
-  proof induct
-    case empty
-    thus ?case by simp
-  next
-    case insert
-    thus ?case using add_mono by fastsimp
-  qed
-next
-  case False
-  thus ?thesis
-qed
-
-lemma setsum_strict_mono:
-  assumes "finite A"  "A \<noteq> {}"
-    and "!!x. x:A \<Longrightarrow> f x < g x"
-  shows "setsum f A < setsum g A"
-  using prems
-proof (induct rule: finite_ne_induct)
-  case singleton thus ?case by simp
-next
-  case insert thus ?case by (auto simp: add_strict_mono)
-qed
-
-lemma setsum_negf:
-  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
-proof (cases "finite A")
-  case True thus ?thesis by (induct set: finite) auto
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_subtractf:
-  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
-    setsum f A - setsum g A"
-proof (cases "finite A")
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_nonneg:
-  shows "0 \<le> setsum f A"
-proof (cases "finite A")
-  case True thus ?thesis using nn
-  proof induct
-    case empty then show ?case by simp
-  next
-    case (insert x F)
-    then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
-    with insert show ?case by simp
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_nonpos:
-  shows "setsum f A \<le> 0"
-proof (cases "finite A")
-  case True thus ?thesis using np
-  proof induct
-    case empty then show ?case by simp
-  next
-    case (insert x F)
-    then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
-    with insert show ?case by simp
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_mono2:
-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)"
-  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
-  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.
-        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 (erule setsum_nonneg)
-  apply (subst setsum_Un_disjoint [THEN sym])
-  apply (erule finite_subset, assumption)
-  apply (rule finite_subset)
-  prefer 2
-  apply assumption
-  apply (auto simp add: sup_absorb2)
-done
-
-lemma setsum_right_distrib:
-  fixes f :: "'a => ('b::semiring_0)"
-  shows "r * setsum f A = setsum (%n. r * f n) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: right_distrib)
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_left_distrib:
-  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
-proof (cases "finite A")
-  case True
-  then show ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: left_distrib)
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_divide_distrib:
-  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
-proof (cases "finite A")
-  case True
-  then show ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_abs[iff]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A)
-    thus ?case by (auto intro: abs_triangle_ineq order_trans)
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_abs_ge_zero[iff]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "0 \<le> setsum (%i. abs(f i)) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma abs_setsum_abs[simp]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert a A)
-    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
-    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
-    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
-      by (simp del: abs_of_nonneg)
-    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
-    finally show ?case .
-  qed
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-
-lemma setsum_Plus:
-  fixes A :: "'a set" and B :: "'b set"
-  assumes fin: "finite A" "finite B"
-  shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
-proof -
-  have "A <+> B = Inl ` A \<union> Inr ` B" by auto
-  moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
-    by(auto intro: finite_imageI)
-  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
-  moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
-  ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
-qed
-
-
-text {* Commuting outer and inner summation *}
-
-lemma swap_inj_on:
-  "inj_on (%(i, j). (j, i)) (A \<times> B)"
-  by (unfold inj_on_def) fast
-
-lemma swap_product:
-  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
-  by (simp add: split_def image_def) blast
-
-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)"
-  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
-    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
-    (is "?s = _")
-    apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
-    done
-  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
-    (is "_ = ?t")
-    done
-  finally show "?s = ?t" .
-qed
-
-lemma setsum_product:
-  fixes f :: "'a => ('b::semiring_0)"
-  shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
-  by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
-
-lemma setsum_mult_setsum_if_inj:
-fixes f :: "'a => ('b::semiring_0)"
-shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
-  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
-by(auto simp: setsum_product setsum_cartesian_product
-        intro!:  setsum_reindex_cong[symmetric])
-
-
-subsection {* Generalized product over a set *}
-
-definition setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
-where "setprod f A == if finite A then fold_image (op *) f 1 A else 1"
-
-abbreviation
-  Setprod  ("\<Prod>_" [1000] 999) where
-  "\<Prod>A == setprod (%x. x) A"
-
-syntax
-  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
-syntax (xsymbols)
-  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
-syntax (HTML output)
-  "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
-
-translations -- {* Beware of argument permutation! *}
-  "PROD i:A. b" == "CONST setprod (%i. b) A"
-  "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
-
-text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
- @{text"\<Prod>x|P. e"}. *}
-
-syntax
-  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
-syntax (xsymbols)
-  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
-syntax (HTML output)
-  "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
-
-translations
-  "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
-  "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
-
-
-lemma setprod_empty [simp]: "setprod f {} = 1"
-
-lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
-    setprod f (insert a A) = f a * setprod f A"
-
-lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
-
-lemma setprod_reindex:
-   "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
-by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD)
-
-lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
-
-lemma setprod_cong:
-  "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
-by(fastsimp simp: setprod_def intro: fold_image_cong)
-
-lemma strong_setprod_cong[cong]:
-  "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
-by(fastsimp simp: simp_implies_def setprod_def intro: fold_image_cong)
-
-lemma setprod_reindex_cong: "inj_on f A ==>
-    B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
-by (frule setprod_reindex, simp)
-
-lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
-  and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
-  shows "setprod h B = setprod g A"
-proof-
-    have "setprod h B = setprod (h o f) A"
-      by (simp add: B setprod_reindex[OF i, of h])
-    then show ?thesis apply simp
-      apply (rule setprod_cong)
-      apply simp
-qed
-
-lemma setprod_Un_one:
-  assumes fS: "finite S" and fT: "finite T"
-  and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
-  shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
-  using fS fT
-  apply (rule fold_image_Un_one)
-  using I0 by auto
-
-
-lemma setprod_1: "setprod (%i. 1) A = 1"
-apply (case_tac "finite A")
-apply (erule finite_induct, auto simp add: mult_ac)
-done
-
-lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
-apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
-apply (erule ssubst, rule setprod_1)
-apply (rule setprod_cong, auto)
-done
-
-lemma setprod_Un_Int: "finite A ==> finite B
-    ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
-
-lemma setprod_Un_disjoint: "finite A ==> finite B
-  ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
-by (subst setprod_Un_Int [symmetric], auto)
-
-lemma setprod_mono_one_left:
-  assumes fT: "finite T" and ST: "S \<subseteq> T"
-  and z: "\<forall>i \<in> T - S. f i = 1"
-  shows "setprod f S = setprod f T"
-proof-
-  have eq: "T = S \<union> (T - S)" using ST by blast
-  have d: "S \<inter> (T - S) = {}" using ST by blast
-  from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
-  show ?thesis
-  by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z])
-qed
-
-lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym]
-
-lemma setprod_delta:
-  assumes fS: "finite S"
-  shows "setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else 1)"
-  {assume a: "a \<notin> S"
-    hence "\<forall> k\<in> S. ?f k = 1" by simp
-    hence ?thesis  using a by (simp add: setprod_1 cong add: setprod_cong) }
-  moreover
-  {assume a: "a \<in> S"
-    let ?A = "S - {a}"
-    let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto
-    have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto
-    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
-      using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
-      by simp
-    then have ?thesis  using a by (simp add: fA1 cong add: setprod_cong cong del: if_weak_cong)}
-  ultimately show ?thesis by blast
-qed
-
-lemma setprod_delta':
-  assumes fS: "finite S" shows
-  "setprod (\<lambda>k. if a = k then b k else 1) S =
-     (if a\<in> S then b a else 1)"
-  using setprod_delta[OF fS, of a b, symmetric]
-  by (auto intro: setprod_cong)
-
-
-lemma setprod_UN_disjoint:
-    "finite I ==> (ALL i:I. finite (A i)) ==>
-        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
-      setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
-by(simp add: setprod_def fold_image_UN_disjoint cong: setprod_cong)
-
-lemma setprod_Union_disjoint:
-  "[| (ALL A:C. finite A);
-      (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
-   ==> setprod f (Union C) = setprod (setprod f) C"
-apply (cases "finite C")
- prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
-  apply (frule setprod_UN_disjoint [of C id f])
- apply (unfold Union_def id_def, assumption+)
-done
-
-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)"
-
-text{*Here we can eliminate the finiteness assumptions, by cases.*}
-lemma setprod_cartesian_product:
-     "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
-apply (cases "finite A")
- apply (cases "finite B")
- apply (cases "A={}", simp)
-            dest: finite_cartesian_productD1 finite_cartesian_productD2)
-done
-
-lemma setprod_timesf:
-     "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
-
-
-subsubsection {* Properties in more restricted classes of structures *}
-
-lemma setprod_eq_1_iff [simp]:
-  "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
-by (induct set: finite) auto
-
-lemma setprod_zero:
-     "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
-apply (induct set: finite, force, clarsimp)
-apply (erule disjE, auto)
-done
-
-lemma setprod_nonneg [rule_format]:
-   "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
-by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
-
-lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
-  --> 0 < setprod f A"
-by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
-
-lemma setprod_zero_iff[simp]: "finite A ==>
-  (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
-  (EX x: A. f x = 0)"
-by (erule finite_induct, auto simp:no_zero_divisors)
-
-lemma setprod_pos_nat:
-  "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
-
-lemma setprod_pos_nat_iff[simp]:
-  "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
-
-lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
-  (setprod f (A Un B) :: 'a ::{field})
-   = setprod f A * setprod f B / setprod f (A Int B)"
-by (subst setprod_Un_Int [symmetric], auto)
-
-lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
-  (setprod f (A - {a}) :: 'a :: {field}) =
-  (if a:A then setprod f A / f a else setprod f A)"
-by (erule finite_induct) (auto simp add: insert_Diff_if)
-
-lemma setprod_inversef:
-  fixes f :: "'b \<Rightarrow> 'a::{field,division_by_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,division_by_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)
-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
-
-
-subsection {* Finite cardinality *}
-
-text {* This definition, although traditional, is ugly to work with:
-@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
-But now that we have @{text setsum} things are easy:
-*}
-
-definition card :: "'a set \<Rightarrow> nat" where
-  "card A = setsum (\<lambda>x. 1) A"
-
-lemmas card_eq_setsum = card_def
-
-lemma card_empty [simp]: "card {} = 0"
-
-lemma card_insert_disjoint [simp]:
-  "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
-
-lemma card_insert_if:
-  "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
-
-lemma card_infinite [simp]: "~ finite A ==> card A = 0"
-
-lemma card_ge_0_finite:
-  "card A > 0 \<Longrightarrow> finite A"
-  by (rule ccontr) simp
-
-lemma card_0_eq [simp,noatp]: "finite A ==> (card A = 0) = (A = {})"
-  apply auto
-  apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
-  done
-
-lemma finite_UNIV_card_ge_0:
-  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
-  by (rule ccontr) simp
-
-lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
-  by auto
-
-lemma card_gt_0_iff: "(0 < card A) = (A \<noteq> {} & finite A)"
-  by (simp add: neq0_conv [symmetric] card_eq_0_iff)
-
-lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
-apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
-apply(simp del:insert_Diff_single)
-done
-
-lemma card_Diff_singleton:
-  "finite A ==> x: A ==> card (A - {x}) = card A - 1"
-
-lemma card_Diff_singleton_if:
-  "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
-
-lemma card_Diff_insert[simp]:
-assumes "finite A" and "a:A" and "a ~: B"
-shows "card(A - insert a B) = card(A - B) - 1"
-proof -
-  have "A - insert a B = (A - B) - {a}" using assms by blast
-  then show ?thesis using assms by(simp add:card_Diff_singleton)
-qed
-
-lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
-by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
-
-lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
-
-lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
-
-lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
-apply (induct set: finite, simp, clarify)
-apply (subgoal_tac "finite A & A - {x} <= F")
- prefer 2 apply (blast intro: finite_subset, atomize)
-apply (drule_tac x = "A - {x}" in spec)
-apply (case_tac "card A", auto)
-done
-
-lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
-apply (simp add: psubset_eq linorder_not_le [symmetric])
-apply (blast dest: card_seteq)
-done
-
-lemma card_Un_Int: "finite A ==> finite B
-    ==> card A + card B = card (A Un B) + card (A Int B)"
-
-lemma card_Un_disjoint: "finite A ==> finite B
-    ==> A Int B = {} ==> card (A Un B) = card A + card B"
-
-lemma card_Diff_subset:
-  "finite B ==> B <= A ==> card (A - B) = card A - card B"
-
-lemma card_Diff_subset_Int:
-  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
-proof -
-  have "A - B = A - A \<inter> B" by auto
-  thus ?thesis
-    by (simp add: card_Diff_subset AB)
-qed
-
-lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
-apply (rule Suc_less_SucD)
-done
-
-lemma card_Diff2_less:
-  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
-apply (case_tac "x = y")
- apply (simp add: card_Diff1_less del:card_Diff_insert)
-apply (rule less_trans)
- prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
-done
-
-lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
-apply (case_tac "x : A")
- apply (simp_all add: card_Diff1_less less_imp_le)
-done
-
-lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
-by (erule psubsetI, blast)
-
-lemma insert_partition:
-  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
-  \<Longrightarrow> x \<inter> \<Union> F = {}"
-by auto
-
-lemma finite_psubset_induct[consumes 1, case_names psubset]:
-  assumes "finite A" and "!!A. finite A \<Longrightarrow> (!!B. finite B \<Longrightarrow> B \<subset> A \<Longrightarrow> P(B)) \<Longrightarrow> P(A)" shows "P A"
-using assms(1)
-proof (induct A rule: measure_induct_rule[where f=card])
-  case (less A)
-  show ?case
-  proof(rule assms(2)[OF less(2)])
-    fix B assume "finite B" "B \<subset> A"
-    show "P B" by(rule less(1)[OF psubset_card_mono[OF less(2) `B \<subset> A`] `finite B`])
-  qed
-qed
-
-text{* main cardinality theorem *}
-lemma card_partition [rule_format]:
-  "finite C ==>
-     finite (\<Union> C) -->
-     (\<forall>c\<in>C. card c = k) -->
-     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
-     k * card(C) = card (\<Union> C)"
-apply (erule finite_induct, simp)
-       finite_subset [of _ "\<Union> (insert x F)"])
-done
-
-lemma card_eq_UNIV_imp_eq_UNIV:
-  assumes fin: "finite (UNIV :: 'a set)"
-  and card: "card A = card (UNIV :: 'a set)"
-  shows "A = (UNIV :: 'a set)"
-proof
-  show "A \<subseteq> UNIV" by simp
-  show "UNIV \<subseteq> A"
-  proof
-    fix x
-    show "x \<in> A"
-    proof (rule ccontr)
-      assume "x \<notin> A"
-      then have "A \<subset> UNIV" by auto
-      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
-      with card show False by simp
-    qed
-  qed
-qed
-
-text{*The form of a finite set of given cardinality*}
-
-lemma card_eq_SucD:
-assumes "card A = Suc k"
-shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
-proof -
-  have fin: "finite A" using assms by (auto intro: ccontr)
-  moreover have "card A \<noteq> 0" using assms by auto
-  ultimately obtain b where b: "b \<in> A" by auto
-  show ?thesis
-  proof (intro exI conjI)
-    show "A = insert b (A-{b})" using b by blast
-    show "b \<notin> A - {b}" by blast
-    show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
-      using assms b fin by(fastsimp dest:mk_disjoint_insert)+
-  qed
-qed
-
-lemma card_Suc_eq:
-  "(card A = Suc k) =
-   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
-apply(rule iffI)
- apply(erule card_eq_SucD)
-apply(auto)
-apply(subst card_insert)
- apply(auto intro:ccontr)
-done
-
-lemma finite_fun_UNIVD2:
-  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
-  shows "finite (UNIV :: 'b set)"
-proof -
-  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
-    by(rule finite_imageI)
-  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
-    by(rule UNIV_eq_I) auto
-  ultimately show "finite (UNIV :: 'b set)" by simp
-qed
-
-lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
-apply (cases "finite A")
-apply (erule finite_induct)
-done
-
-lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
-apply (erule finite_induct)
-apply auto
-done
-
-lemma setprod_gen_delta:
-  assumes fS: "finite S"
-  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::{comm_monoid_mult}) * c^ (card S - 1) else c^ card S)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else c)"
-  {assume a: "a \<notin> S"
-    hence "\<forall> k\<in> S. ?f k = c" by simp
-    hence ?thesis  using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) }
-  moreover
-  {assume a: "a \<in> S"
-    let ?A = "S - {a}"
-    let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto
-    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
-      apply (rule setprod_cong) by auto
-    have cA: "card ?A = card S - 1" using fS a by auto
-    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
-    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
-      using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
-      by simp
-    then have ?thesis using a cA
-      by (simp add: fA1 ring_simps cong add: setprod_cong cong del: if_weak_cong)}
-  ultimately show ?thesis by blast
-qed
-
-
-lemma setsum_bounded:
-  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
-  shows "setsum f A \<le> of_nat(card A) * K"
-proof (cases "finite A")
-  case True
-  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
-next
-  case False thus ?thesis by (simp add: setsum_def)
-qed
-
-
-lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
-  unfolding UNIV_unit by simp
-
-
-subsubsection {* Cardinality of unions *}
-
-lemma card_UN_disjoint:
-  "finite I ==> (ALL i:I. finite (A i)) ==>
-   (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {})
-   ==> card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
-apply (simp add: card_def del: setsum_constant)
-apply (subgoal_tac
-         "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
-apply (simp add: setsum_UN_disjoint del: setsum_constant)
-apply (simp cong: setsum_cong)
-done
-
-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 (unfold Union_def id_def, assumption+)
-done
-
-
-subsubsection {* Cardinality of image *}
-
-text{*The image of a finite set can be expressed using @{term fold_image}.*}
-lemma image_eq_fold_image:
-  "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
-proof (induct rule: finite_induct)
-  case empty then show ?case by simp
-next
-  interpret ab_semigroup_mult "op Un"
-    proof qed auto
-  case insert
-  then show ?case by simp
-qed
-
-lemma card_image_le: "finite A ==> card (f ` A) <= card A"
-apply (induct set: finite)
- apply simp
-done
-
-lemma card_image: "inj_on f A ==> card (f ` A) = card A"
-
-lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
-by(auto simp: card_image bij_betw_def)
-
-lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
-
-lemma eq_card_imp_inj_on:
-  "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
-apply (induct rule:finite_induct)
-apply simp
-apply(frule card_image_le[where f = f])
-done
-
-lemma inj_on_iff_eq_card:
-  "finite A ==> inj_on f A = (card(f ` A) = card A)"
-by(blast intro: card_image eq_card_imp_inj_on)
-
-
-lemma card_inj_on_le:
-  "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
-apply (subgoal_tac "finite A")
- apply (force intro: card_mono simp add: card_image [symmetric])
-apply (blast intro: finite_imageD dest: finite_subset)
-done
-
-lemma card_bij_eq:
-  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
-     finite A; finite B |] ==> card A = card B"
-by (auto intro: le_antisym card_inj_on_le)
-
-
-subsubsection {* Cardinality of products *}
-
-(*
-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_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))"
-
-lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
-apply (cases "finite A")
-apply (cases "finite B")
-            dest: finite_cartesian_productD1 finite_cartesian_productD2)
-done
-
-lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
-
-
-subsubsection {* Cardinality of sums *}
-
-lemma card_Plus:
-  assumes "finite A" and "finite B"
-  shows "card (A <+> B) = card A + card B"
-proof -
-  have "Inl`A \<inter> Inr`B = {}" by fast
-  with assms show ?thesis
-    unfolding Plus_def
-    by (simp add: card_Un_disjoint card_image)
-qed
-
-lemma card_Plus_conv_if:
-  "card (A <+> B) = (if finite A \<and> finite B then card(A) + card(B) else 0)"
-by(auto simp: card_def setsum_Plus simp del: setsum_constant)
-
-
-subsubsection {* Cardinality of the Powerset *}
-
-lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
-apply (induct set: finite)
-apply (subst card_Un_disjoint, blast)
-  apply (blast intro: finite_imageI, blast)
-apply (subgoal_tac "inj_on (insert x) (Pow F)")
- apply (simp add: card_image Pow_insert)
-apply (unfold inj_on_def)
-apply (blast elim!: equalityE)
-done
-
-text {* Relates to equivalence classes.  Based on a theorem of F. Kammüller.  *}
-
-lemma dvd_partition:
-  "finite (Union C) ==>
-    ALL c : C. k dvd card c ==>
-    (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
-  k dvd card (Union C)"
-apply(frule finite_UnionD)
-apply(rotate_tac -1)
-apply (induct set: finite, simp_all, clarify)
-apply (subst card_Un_disjoint)
-   apply (auto simp add: disjoint_eq_subset_Compl)
-done
-
-
-subsubsection {* Relating injectivity and surjectivity *}
-
-lemma finite_surj_inj: "finite(A) \<Longrightarrow> A <= f`A \<Longrightarrow> inj_on f A"
-apply(rule eq_card_imp_inj_on, assumption)
-apply(frule finite_imageI)
-apply(drule (1) card_seteq)
- apply(erule card_image_le)
-apply simp
-done
-
-lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
-shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
-by (blast intro: finite_surj_inj subset_UNIV dest:surj_range)
-
-lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
-shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
-by(fastsimp simp:surj_def dest!: endo_inj_surj)
-
-corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
-proof
-  assume "finite(UNIV::nat set)"
-  with finite_UNIV_inj_surj[of Suc]
-  show False by simp (blast dest: Suc_neq_Zero surjD)
-qed
-
-(* Often leads to bogus ATP proofs because of reduced type information, hence noatp *)
-lemma infinite_UNIV_char_0[noatp]:
-  "\<not> finite (UNIV::'a::semiring_char_0 set)"
-proof
-  assume "finite (UNIV::'a set)"
-  with subset_UNIV have "finite (range of_nat::'a set)"
-    by (rule finite_subset)
-  moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
-  ultimately have "finite (UNIV::nat set)"
-    by (rule finite_imageD)
-  then show "False"
-    by simp
-qed

subsection{* A fold functional for non-empty sets *}

@@ -2811,561 +1354,6 @@
qed

-subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *}
-
-text{*
-  As an application of @{text fold1} we define infimum
-  and supremum in (not necessarily complete!) lattices
-  over (non-empty) sets by means of @{text fold1}.
-*}
-
-context semilattice_inf
-begin
-
-lemma below_fold1_iff:
-  assumes "finite A" "A \<noteq> {}"
-  shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
-proof -
-  interpret ab_semigroup_idem_mult inf
-    by (rule ab_semigroup_idem_mult_inf)
-  show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
-qed
-
-lemma fold1_belowI:
-  assumes "finite A"
-    and "a \<in> A"
-  shows "fold1 inf A \<le> a"
-proof -
-  from assms have "A \<noteq> {}" by auto
-  from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
-  proof (induct rule: finite_ne_induct)
-    case singleton thus ?case by simp
-  next
-    interpret ab_semigroup_idem_mult inf
-      by (rule ab_semigroup_idem_mult_inf)
-    case (insert x F)
-    from insert(5) have "a = x \<or> a \<in> F" by simp
-    thus ?case
-    proof
-      assume "a = x" thus ?thesis using insert
-    next
-      assume "a \<in> F"
-      hence bel: "fold1 inf F \<le> a" by (rule insert)
-      have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
-        using insert by (simp add: mult_ac)
-      also have "inf (fold1 inf F) a = fold1 inf F"
-        using bel by (auto intro: antisym)
-      also have "inf x \<dots> = fold1 inf (insert x F)"
-        using insert by (simp add: mult_ac)
-      finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
-      moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
-      ultimately show ?thesis by simp
-    qed
-  qed
-qed
-
-end
-
-context lattice
-begin
-
-definition
-  Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900)
-where
-  "Inf_fin = fold1 inf"
-
-definition
-  Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900)
-where
-  "Sup_fin = fold1 sup"
-
-lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
-apply(unfold Sup_fin_def Inf_fin_def)
-apply(subgoal_tac "EX a. a:A")
-prefer 2 apply blast
-apply(erule exE)
-apply(rule order_trans)
-apply(erule (1) fold1_belowI)
-apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
-done
-
-lemma sup_Inf_absorb [simp]:
-  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
-apply(subst sup_commute)
-done
-
-lemma inf_Sup_absorb [simp]:
-  "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
-  semilattice_inf.fold1_belowI [OF dual_semilattice])
-
-end
-
-context distrib_lattice
-begin
-
-lemma sup_Inf1_distrib:
-  assumes "finite A"
-    and "A \<noteq> {}"
-  shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
-proof -
-  interpret ab_semigroup_idem_mult inf
-    by (rule ab_semigroup_idem_mult_inf)
-  from assms show ?thesis
-    by (simp add: Inf_fin_def image_def
-      hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
-        (rule arg_cong [where f="fold1 inf"], blast)
-qed
-
-lemma sup_Inf2_distrib:
-  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
-  shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
-using A proof (induct rule: finite_ne_induct)
-  case singleton thus ?case
-    by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def])
-next
-  interpret ab_semigroup_idem_mult inf
-    by (rule ab_semigroup_idem_mult_inf)
-  case (insert x A)
-  have finB: "finite {sup x b |b. b \<in> B}"
-    by(rule finite_surj[where f = "sup x", OF B(1)], auto)
-  have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
-  proof -
-    have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
-      by blast
-    thus ?thesis by(simp add: insert(1) B(1))
-  qed
-  have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
-  have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
-    using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def])
-  also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
-  also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
-    using insert by(simp add:sup_Inf1_distrib[OF B])
-  also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
-    (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
-    using B insert
-    by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
-  also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
-    by blast
-  finally show ?case .
-qed
-
-lemma inf_Sup1_distrib:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
-proof -
-  interpret ab_semigroup_idem_mult sup
-    by (rule ab_semigroup_idem_mult_sup)
-  from assms show ?thesis
-    by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
-      (rule arg_cong [where f="fold1 sup"], blast)
-qed
-
-lemma inf_Sup2_distrib:
-  assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
-  shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"
-using A proof (induct rule: finite_ne_induct)
-  case singleton thus ?case
-    by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def])
-next
-  case (insert x A)
-  have finB: "finite {inf x b |b. b \<in> B}"
-    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
-  have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
-  proof -
-    have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
-      by blast
-    thus ?thesis by(simp add: insert(1) B(1))
-  qed
-  have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
-  interpret ab_semigroup_idem_mult sup
-    by (rule ab_semigroup_idem_mult_sup)
-  have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
-    using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def])
-  also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
-  also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
-    using insert by(simp add:inf_Sup1_distrib[OF B])
-  also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
-    (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
-    using B insert
-    by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
-  also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
-    by blast
-  finally show ?case .
-qed
-
-end
-
-
-subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *}
-
-text{*
-  As an application of @{text fold1} we define minimum
-  and maximum in (not necessarily complete!) linear orders
-  over (non-empty) sets by means of @{text fold1}.
-*}
-
-context linorder
-begin
-
-lemma ab_semigroup_idem_mult_min:
-  "ab_semigroup_idem_mult min"
-  proof qed (auto simp add: min_def)
-
-lemma ab_semigroup_idem_mult_max:
-  "ab_semigroup_idem_mult max"
-  proof qed (auto simp add: max_def)
-
-lemma max_lattice:
-  "semilattice_inf (op \<ge>) (op >) max"
-  by (fact min_max.dual_semilattice)
-
-lemma dual_max:
-  "ord.max (op \<ge>) = min"
-  by (auto simp add: ord.max_def_raw min_def expand_fun_eq)
-
-lemma dual_min:
-  "ord.min (op \<ge>) = max"
-  by (auto simp add: ord.min_def_raw max_def expand_fun_eq)
-
-lemma strict_below_fold1_iff:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms show ?thesis
-  by (induct rule: finite_ne_induct)
-qed
-
-lemma fold1_below_iff:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms show ?thesis
-  by (induct rule: finite_ne_induct)
-qed
-
-lemma fold1_strict_below_iff:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms show ?thesis
-  by (induct rule: finite_ne_induct)
-qed
-
-lemma fold1_antimono:
-  assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
-  shows "fold1 min B \<le> fold1 min A"
-proof cases
-  assume "A = B" thus ?thesis by simp
-next
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  assume "A \<noteq> B"
-  have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
-  have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
-  also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
-  proof -
-    have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
-    moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
-    moreover have "(B-A) \<noteq> {}" using prems by blast
-    moreover have "A Int (B-A) = {}" using prems by blast
-    ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
-  qed
-  also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
-  finally show ?thesis .
-qed
-
-definition
-  Min :: "'a set \<Rightarrow> 'a"
-where
-  "Min = fold1 min"
-
-definition
-  Max :: "'a set \<Rightarrow> 'a"
-where
-  "Max = fold1 max"
-
-lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def]
-lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def]
-
-lemma Min_insert [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Min (insert x A) = min x (Min A)"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms show ?thesis by (rule fold1_insert_idem_def [OF Min_def])
-qed
-
-lemma Max_insert [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Max (insert x A) = max x (Max A)"
-proof -
-  interpret ab_semigroup_idem_mult max
-    by (rule ab_semigroup_idem_mult_max)
-  from assms show ?thesis by (rule fold1_insert_idem_def [OF Max_def])
-qed
-
-lemma Min_in [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Min A \<in> A"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def)
-qed
-
-lemma Max_in [simp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Max A \<in> A"
-proof -
-  interpret ab_semigroup_idem_mult max
-    by (rule ab_semigroup_idem_mult_max)
-  from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def)
-qed
-
-lemma Min_Un:
-  assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
-  shows "Min (A \<union> B) = min (Min A) (Min B)"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms show ?thesis
-    by (simp add: Min_def fold1_Un2)
-qed
-
-lemma Max_Un:
-  assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
-  shows "Max (A \<union> B) = max (Max A) (Max B)"
-proof -
-  interpret ab_semigroup_idem_mult max
-    by (rule ab_semigroup_idem_mult_max)
-  from assms show ?thesis
-    by (simp add: Max_def fold1_Un2)
-qed
-
-lemma hom_Min_commute:
-  assumes "\<And>x y. h (min x y) = min (h x) (h y)"
-    and "finite N" and "N \<noteq> {}"
-  shows "h (Min N) = Min (h ` N)"
-proof -
-  interpret ab_semigroup_idem_mult min
-    by (rule ab_semigroup_idem_mult_min)
-  from assms show ?thesis
-    by (simp add: Min_def hom_fold1_commute)
-qed
-
-lemma hom_Max_commute:
-  assumes "\<And>x y. h (max x y) = max (h x) (h y)"
-    and "finite N" and "N \<noteq> {}"
-  shows "h (Max N) = Max (h ` N)"
-proof -
-  interpret ab_semigroup_idem_mult max
-    by (rule ab_semigroup_idem_mult_max)
-  from assms show ?thesis
-    by (simp add: Max_def hom_fold1_commute [of h])
-qed
-
-lemma Min_le [simp]:
-  assumes "finite A" and "x \<in> A"
-  shows "Min A \<le> x"
-  using assms by (simp add: Min_def min_max.fold1_belowI)
-
-lemma Max_ge [simp]:
-  assumes "finite A" and "x \<in> A"
-  shows "x \<le> Max A"
-proof -
-  interpret semilattice_inf "op \<ge>" "op >" max
-    by (rule max_lattice)
-  from assms show ?thesis by (simp add: Max_def fold1_belowI)
-qed
-
-lemma Min_ge_iff [simp, noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
-  using assms by (simp add: Min_def min_max.below_fold1_iff)
-
-lemma Max_le_iff [simp, noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
-proof -
-  interpret semilattice_inf "op \<ge>" "op >" max
-    by (rule max_lattice)
-  from assms show ?thesis by (simp add: Max_def below_fold1_iff)
-qed
-
-lemma Min_gr_iff [simp, noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
-  using assms by (simp add: Min_def strict_below_fold1_iff)
-
-lemma Max_less_iff [simp, noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
-proof -
-  interpret dual: linorder "op \<ge>" "op >"
-    by (rule dual_linorder)
-  from assms show ?thesis
-    by (simp add: Max_def dual.strict_below_fold1_iff [folded dual.dual_max])
-qed
-
-lemma Min_le_iff [noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
-  using assms by (simp add: Min_def fold1_below_iff)
-
-lemma Max_ge_iff [noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
-proof -
-  interpret dual: linorder "op \<ge>" "op >"
-    by (rule dual_linorder)
-  from assms show ?thesis
-    by (simp add: Max_def dual.fold1_below_iff [folded dual.dual_max])
-qed
-
-lemma Min_less_iff [noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
-  using assms by (simp add: Min_def fold1_strict_below_iff)
-
-lemma Max_gr_iff [noatp]:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
-proof -
-  interpret dual: linorder "op \<ge>" "op >"
-    by (rule dual_linorder)
-  from assms show ?thesis
-    by (simp add: Max_def dual.fold1_strict_below_iff [folded dual.dual_max])
-qed
-
-lemma Min_eqI:
-  assumes "finite A"
-  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
-    and "x \<in> A"
-  shows "Min A = x"
-proof (rule antisym)
-  from `x \<in> A` have "A \<noteq> {}" by auto
-  with assms show "Min A \<ge> x" by simp
-next
-  from assms show "x \<ge> Min A" by simp
-qed
-
-lemma Max_eqI:
-  assumes "finite A"
-  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
-    and "x \<in> A"
-  shows "Max A = x"
-proof (rule antisym)
-  from `x \<in> A` have "A \<noteq> {}" by auto
-  with assms show "Max A \<le> x" by simp
-next
-  from assms show "x \<le> Max A" by simp
-qed
-
-lemma Min_antimono:
-  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
-  shows "Min N \<le> Min M"
-  using assms by (simp add: Min_def fold1_antimono)
-
-lemma Max_mono:
-  assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
-  shows "Max M \<le> Max N"
-proof -
-  interpret dual: linorder "op \<ge>" "op >"
-    by (rule dual_linorder)
-  from assms show ?thesis
-    by (simp add: Max_def dual.fold1_antimono [folded dual.dual_max])
-qed
-
-lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
- "finite A \<Longrightarrow> P {} \<Longrightarrow>
-  (!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))
-  \<Longrightarrow> P A"
-proof (induct rule: finite_psubset_induct)
-  fix A :: "'a set"
-  assume IH: "!! B. finite B \<Longrightarrow> B < A \<Longrightarrow> P {} \<Longrightarrow>
-                 (!!b A. finite A \<Longrightarrow> (\<forall>a\<in>A. a<b) \<Longrightarrow> P A \<Longrightarrow> P (insert b A))
-                  \<Longrightarrow> P B"
-  and "finite A" and "P {}"
-  and step: "!!b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)"
-  show "P A"
-  proof (cases "A = {}")
-    assume "A = {}" thus "P A" using `P {}` by simp
-  next
-    let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B"
-    assume "A \<noteq> {}"
-    with `finite A` have "Max A : A" by auto
-    hence A: "?A = A" using insert_Diff_single insert_absorb by auto
-    moreover have "finite ?B" using `finite A` by simp
-    ultimately 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 fastsimp
-    ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastsimp
-  qed
-qed
-
-lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
- "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
-by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
-
-end
-
-begin
-
-  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
-
-  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
-
-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
-
-
subsection {* Expressing set operations via @{const fold} *}

lemma (in fun_left_comm) fun_left_comm_apply:
@@ -3445,32 +1433,6 @@
shows "Sup A = fold sup bot A"
using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)

-lemma Inf_fin_Inf:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
-proof -
-  interpret ab_semigroup_idem_mult inf
-    by (rule ab_semigroup_idem_mult_inf)
-  from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
-  moreover with `finite A` have "finite B" by simp
-  ultimately show ?thesis
-  by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
-qed
-
-lemma Sup_fin_Sup:
-  assumes "finite A" and "A \<noteq> {}"
-  shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
-proof -
-  interpret ab_semigroup_idem_mult sup
-    by (rule ab_semigroup_idem_mult_sup)
-  from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
-  moreover with `finite A` have "finite B" by simp
-  ultimately show ?thesis
-  by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
-qed
-
lemma inf_INFI_fold_inf:
assumes "finite A"
shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold")
@@ -3505,4 +1467,127 @@

end

+
+subsection {* Locales as mini-packages *}
+
+locale folding =
+  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
+  fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
+  assumes commute_comp: "f x \<circ> f y = f y \<circ> f x"
+  assumes eq_fold: "F A s = Finite_Set.fold f s A"
+begin
+
+lemma fun_left_commute:
+  "f x (f y s) = f y (f x s)"
+  using commute_comp [of x y] by (simp add: expand_fun_eq)
+
+lemma fun_left_comm:
+  "fun_left_comm f"
+proof
+qed (fact fun_left_commute)
+
+lemma empty [simp]:
+  "F {} = id"
+  by (simp add: eq_fold expand_fun_eq)
+
+lemma insert [simp]:
+  assumes "finite A" and "x \<notin> A"
+  shows "F (insert x A) = F A \<circ> f x"
+proof -
+  interpret fun_left_comm f by (fact fun_left_comm)
+  from fold_insert2 assms
+  have "\<And>s. Finite_Set.fold f s (insert x A) = Finite_Set.fold f (f x s) A" .
+  then show ?thesis by (simp add: eq_fold expand_fun_eq)
+qed
+
+lemma remove:
+  assumes "finite A" and "x \<in> A"
+  shows "F A = F (A - {x}) \<circ> f 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` this have "finite B" by simp
+  ultimately show ?thesis by simp
+qed
+
+lemma insert_remove:
+  assumes "finite A"
+  shows "F (insert x A) = F (A - {x}) \<circ> f x"
+proof (cases "x \<in> A")
+  case True with assms show ?thesis by (simp add: remove insert_absorb)
+next
+  case False with assms show ?thesis by simp
+qed
+
+lemma commute_comp':
+  assumes "finite A"
+  shows "f x \<circ> F A = F A \<circ> f x"
+proof (rule ext)
+  fix s
+  from assms show "(f x \<circ> F A) s = (F A \<circ> f x) s"
+    by (induct A arbitrary: s) (simp_all add: fun_left_commute)
+qed
+
+lemma fun_left_commute':
+  assumes "finite A"
+  shows "f x (F A s) = F A (f x s)"
+  using commute_comp' assms by (simp add: expand_fun_eq)
+
+lemma union:
+  assumes "finite A" and "finite B"
+  and "A \<inter> B = {}"
+  shows "F (A \<union> B) = F A \<circ> F B"
+using `finite A` `A \<inter> B = {}` proof (induct A)
+  case empty show ?case by simp
+next
+  case (insert x A)
+  then have "A \<inter> B = {}" by auto
+  with insert(3) have "F (A \<union> B) = F A \<circ> F B" .
+  moreover from insert have "x \<notin> B" by simp
+  moreover from `finite A` `finite B` have fin: "finite (A \<union> B)" by simp
+  moreover from `x \<notin> A` `x \<notin> B` have "x \<notin> A \<union> B" by simp
+  ultimately show ?case by (simp add: fun_left_commute')
+qed
+
end
+
+locale folding_idem = folding +
+  assumes idem_comp: "f x \<circ> f x = f x"
+begin
+
+declare insert [simp del]
+
+lemma fun_idem:
+  "f x (f x s) = f x s"
+  using idem_comp [of x] by (simp add: expand_fun_eq)
+
+lemma fun_left_comm_idem:
+  "fun_left_comm_idem f"
+proof
+qed (fact fun_left_commute fun_idem)+
+
+lemma insert_idem [simp]:
+  assumes "finite A"
+  shows "F (insert x A) = F A \<circ> f x"
+proof -
+  interpret fun_left_comm_idem f by (fact fun_left_comm_idem)
+  from fold_insert_idem2 assms
+  have "\<And>s. Finite_Set.fold f s (insert x A) = Finite_Set.fold f (f x s) A" .
+  then show ?thesis by (simp add: eq_fold expand_fun_eq)
+qed
+
+lemma union_idem:
+  assumes "finite A" and "finite B"
+  shows "F (A \<union> B) = F A \<circ> F B"
+using `finite A` proof (induct A)
+  case empty show ?case by simp
+next
+  case (insert x A)
+  from insert(3) have "F (A \<union> B) = F A \<circ> F B" .
+  moreover from `finite A` `finite B` have fin: "finite (A \<union> B)" by simp
+  ultimately show ?case by (simp add: fun_left_commute')
+qed
+
+end
+
+end