wenzelm@12396: (*  Title:      HOL/Finite_Set.thy
wenzelm@12396:     ID:         $Id$
wenzelm@12396:     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
paulson@14430:                 Additions by Jeremy Avigad in Feb 2004
nipkow@15376: 
nipkow@15402: Get rid of a couple of superfluous finiteness assumptions in lemmas
nipkow@15402: about setsum and card - see FIXME.
nipkow@15402: NB: the analogous lemmas for setprod should also be simplified!
wenzelm@12396: *)
wenzelm@12396: 
wenzelm@12396: header {* Finite sets *}
wenzelm@12396: 
nipkow@15131: theory Finite_Set
nipkow@15140: imports Divides Power Inductive
nipkow@15131: begin
wenzelm@12396: 
nipkow@15392: subsection {* Definition and basic properties *}
wenzelm@12396: 
wenzelm@12396: consts Finites :: "'a set set"
nipkow@13737: syntax
nipkow@13737:   finite :: "'a set => bool"
nipkow@13737: translations
nipkow@13737:   "finite A" == "A : Finites"
wenzelm@12396: 
wenzelm@12396: inductive Finites
wenzelm@12396:   intros
wenzelm@12396:     emptyI [simp, intro!]: "{} : Finites"
wenzelm@12396:     insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
wenzelm@12396: 
wenzelm@12396: axclass finite \<subseteq> type
wenzelm@12396:   finite: "finite UNIV"
wenzelm@12396: 
nipkow@13737: lemma ex_new_if_finite: -- "does not depend on def of finite at all"
wenzelm@14661:   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
wenzelm@14661:   shows "\<exists>a::'a. a \<notin> A"
wenzelm@14661: proof -
wenzelm@14661:   from prems have "A \<noteq> UNIV" by blast
wenzelm@14661:   thus ?thesis by blast
wenzelm@14661: qed
wenzelm@12396: 
wenzelm@12396: lemma finite_induct [case_names empty insert, induct set: Finites]:
wenzelm@12396:   "finite F ==>
nipkow@15327:     P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
wenzelm@12396:   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
wenzelm@12396: proof -
wenzelm@13421:   assume "P {}" and
nipkow@15327:     insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
wenzelm@12396:   assume "finite F"
wenzelm@12396:   thus "P F"
wenzelm@12396:   proof induct
wenzelm@12396:     show "P {}" .
nipkow@15327:     fix x F assume F: "finite F" and P: "P F"
wenzelm@12396:     show "P (insert x F)"
wenzelm@12396:     proof cases
wenzelm@12396:       assume "x \<in> F"
wenzelm@12396:       hence "insert x F = F" by (rule insert_absorb)
wenzelm@12396:       with P show ?thesis by (simp only:)
wenzelm@12396:     next
wenzelm@12396:       assume "x \<notin> F"
wenzelm@12396:       from F this P show ?thesis by (rule insert)
wenzelm@12396:     qed
wenzelm@12396:   qed
wenzelm@12396: qed
wenzelm@12396: 
wenzelm@12396: lemma finite_subset_induct [consumes 2, case_names empty insert]:
wenzelm@12396:   "finite F ==> F \<subseteq> A ==>
nipkow@15327:     P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
wenzelm@12396:     P F"
wenzelm@12396: proof -
wenzelm@13421:   assume "P {}" and insert:
nipkow@15327:     "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
wenzelm@12396:   assume "finite F"
wenzelm@12396:   thus "F \<subseteq> A ==> P F"
wenzelm@12396:   proof induct
wenzelm@12396:     show "P {}" .
nipkow@15327:     fix x F assume "finite F" and "x \<notin> F"
wenzelm@12396:       and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
wenzelm@12396:     show "P (insert x F)"
wenzelm@12396:     proof (rule insert)
wenzelm@12396:       from i show "x \<in> A" by blast
wenzelm@12396:       from i have "F \<subseteq> A" by blast
wenzelm@12396:       with P show "P F" .
wenzelm@12396:     qed
wenzelm@12396:   qed
wenzelm@12396: qed
wenzelm@12396: 
nipkow@15392: text{* Finite sets are the images of initial segments of natural numbers: *}
nipkow@15392: 
nipkow@15392: lemma finite_imp_nat_seg_image:
nipkow@15392: assumes fin: "finite A" shows "\<exists> (n::nat) f. A = f ` {i::nat. i<n}"
nipkow@15392: using fin
nipkow@15392: proof induct
nipkow@15392:   case empty
nipkow@15392:   show ?case
nipkow@15392:   proof show "\<exists>f. {} = f ` {i::nat. i < 0}" by(simp add:image_def) qed
nipkow@15392: next
nipkow@15392:   case (insert a A)
nipkow@15392:   from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" by blast
nipkow@15392:   hence "insert a A = (%i. if i<n then f i else a) ` {i. i < n+1}"
nipkow@15392:     by (auto simp add:image_def Ball_def)
nipkow@15392:   thus ?case by blast
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: lemma nat_seg_image_imp_finite:
nipkow@15392:   "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
nipkow@15392: proof (induct n)
nipkow@15392:   case 0 thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (Suc n)
nipkow@15392:   let ?B = "f ` {i. i < n}"
nipkow@15392:   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
nipkow@15392:   show ?case
nipkow@15392:   proof cases
nipkow@15392:     assume "\<exists>k<n. f n = f k"
nipkow@15392:     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
nipkow@15392:     thus ?thesis using finB by simp
nipkow@15392:   next
nipkow@15392:     assume "\<not>(\<exists> k<n. f n = f k)"
nipkow@15392:     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
nipkow@15392:     thus ?thesis using finB by simp
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: lemma finite_conv_nat_seg_image:
nipkow@15392:   "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
nipkow@15392: by(blast intro: finite_imp_nat_seg_image nat_seg_image_imp_finite)
nipkow@15392: 
nipkow@15392: subsubsection{* Finiteness and set theoretic constructions *}
nipkow@15392: 
wenzelm@12396: lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
wenzelm@12396:   -- {* The union of two finite sets is finite. *}
wenzelm@12396:   by (induct set: Finites) simp_all
wenzelm@12396: 
wenzelm@12396: lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
wenzelm@12396:   -- {* Every subset of a finite set is finite. *}
wenzelm@12396: proof -
wenzelm@12396:   assume "finite B"
wenzelm@12396:   thus "!!A. A \<subseteq> B ==> finite A"
wenzelm@12396:   proof induct
wenzelm@12396:     case empty
wenzelm@12396:     thus ?case by simp
wenzelm@12396:   next
nipkow@15327:     case (insert x F A)
wenzelm@12396:     have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
wenzelm@12396:     show "finite A"
wenzelm@12396:     proof cases
wenzelm@12396:       assume x: "x \<in> A"
wenzelm@12396:       with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
wenzelm@12396:       with r have "finite (A - {x})" .
wenzelm@12396:       hence "finite (insert x (A - {x}))" ..
wenzelm@12396:       also have "insert x (A - {x}) = A" by (rule insert_Diff)
wenzelm@12396:       finally show ?thesis .
wenzelm@12396:     next
wenzelm@12396:       show "A \<subseteq> F ==> ?thesis" .
wenzelm@12396:       assume "x \<notin> A"
wenzelm@12396:       with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
wenzelm@12396:     qed
wenzelm@12396:   qed
wenzelm@12396: qed
wenzelm@12396: 
wenzelm@12396: lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
wenzelm@12396:   by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
wenzelm@12396: 
wenzelm@12396: lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
wenzelm@12396:   -- {* The converse obviously fails. *}
wenzelm@12396:   by (blast intro: finite_subset)
wenzelm@12396: 
wenzelm@12396: lemma finite_insert [simp]: "finite (insert a A) = finite A"
wenzelm@12396:   apply (subst insert_is_Un)
paulson@14208:   apply (simp only: finite_Un, blast)
wenzelm@12396:   done
wenzelm@12396: 
nipkow@15281: lemma finite_Union[simp, intro]:
nipkow@15281:  "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
nipkow@15281: by (induct rule:finite_induct) simp_all
nipkow@15281: 
wenzelm@12396: lemma finite_empty_induct:
wenzelm@12396:   "finite A ==>
wenzelm@12396:   P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
wenzelm@12396: proof -
wenzelm@12396:   assume "finite A"
wenzelm@12396:     and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
wenzelm@12396:   have "P (A - A)"
wenzelm@12396:   proof -
wenzelm@12396:     fix c b :: "'a set"
wenzelm@12396:     presume c: "finite c" and b: "finite b"
wenzelm@12396:       and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
wenzelm@12396:     from c show "c \<subseteq> b ==> P (b - c)"
wenzelm@12396:     proof induct
wenzelm@12396:       case empty
wenzelm@12396:       from P1 show ?case by simp
wenzelm@12396:     next
nipkow@15327:       case (insert x F)
wenzelm@12396:       have "P (b - F - {x})"
wenzelm@12396:       proof (rule P2)
wenzelm@12396:         from _ b show "finite (b - F)" by (rule finite_subset) blast
wenzelm@12396:         from insert show "x \<in> b - F" by simp
wenzelm@12396:         from insert show "P (b - F)" by simp
wenzelm@12396:       qed
wenzelm@12396:       also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
wenzelm@12396:       finally show ?case .
wenzelm@12396:     qed
wenzelm@12396:   next
wenzelm@12396:     show "A \<subseteq> A" ..
wenzelm@12396:   qed
wenzelm@12396:   thus "P {}" by simp
wenzelm@12396: qed
wenzelm@12396: 
wenzelm@12396: lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
wenzelm@12396:   by (rule Diff_subset [THEN finite_subset])
wenzelm@12396: 
wenzelm@12396: lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
wenzelm@12396:   apply (subst Diff_insert)
wenzelm@12396:   apply (case_tac "a : A - B")
wenzelm@12396:    apply (rule finite_insert [symmetric, THEN trans])
paulson@14208:    apply (subst insert_Diff, simp_all)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: 
nipkow@15392: text {* Image and Inverse Image over Finite Sets *}
paulson@13825: 
paulson@13825: lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
paulson@13825:   -- {* The image of a finite set is finite. *}
paulson@13825:   by (induct set: Finites) simp_all
paulson@13825: 
paulson@14430: lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
paulson@14430:   apply (frule finite_imageI)
paulson@14430:   apply (erule finite_subset, assumption)
paulson@14430:   done
paulson@14430: 
paulson@13825: lemma finite_range_imageI:
paulson@13825:     "finite (range g) ==> finite (range (%x. f (g x)))"
paulson@14208:   apply (drule finite_imageI, simp)
paulson@13825:   done
paulson@13825: 
wenzelm@12396: lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
wenzelm@12396: proof -
wenzelm@12396:   have aux: "!!A. finite (A - {}) = finite A" by simp
wenzelm@12396:   fix B :: "'a set"
wenzelm@12396:   assume "finite B"
wenzelm@12396:   thus "!!A. f`A = B ==> inj_on f A ==> finite A"
wenzelm@12396:     apply induct
wenzelm@12396:      apply simp
wenzelm@12396:     apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
wenzelm@12396:      apply clarify
wenzelm@12396:      apply (simp (no_asm_use) add: inj_on_def)
paulson@14208:      apply (blast dest!: aux [THEN iffD1], atomize)
wenzelm@12396:     apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
paulson@14208:     apply (frule subsetD [OF equalityD2 insertI1], clarify)
wenzelm@12396:     apply (rule_tac x = xa in bexI)
wenzelm@12396:      apply (simp_all add: inj_on_image_set_diff)
wenzelm@12396:     done
wenzelm@12396: qed (rule refl)
wenzelm@12396: 
wenzelm@12396: 
paulson@13825: lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
paulson@13825:   -- {* The inverse image of a singleton under an injective function
paulson@13825:          is included in a singleton. *}
paulson@14430:   apply (auto simp add: inj_on_def)
paulson@14430:   apply (blast intro: the_equality [symmetric])
paulson@13825:   done
paulson@13825: 
paulson@13825: lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
paulson@13825:   -- {* The inverse image of a finite set under an injective function
paulson@13825:          is finite. *}
paulson@14430:   apply (induct set: Finites, simp_all)
paulson@14430:   apply (subst vimage_insert)
paulson@14430:   apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
paulson@13825:   done
paulson@13825: 
paulson@13825: 
nipkow@15392: text {* The finite UNION of finite sets *}
wenzelm@12396: 
wenzelm@12396: lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
wenzelm@12396:   by (induct set: Finites) simp_all
wenzelm@12396: 
wenzelm@12396: text {*
wenzelm@12396:   Strengthen RHS to
paulson@14430:   @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
wenzelm@12396: 
wenzelm@12396:   We'd need to prove
paulson@14430:   @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
wenzelm@12396:   by induction. *}
wenzelm@12396: 
wenzelm@12396: lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
wenzelm@12396:   by (blast intro: finite_UN_I finite_subset)
wenzelm@12396: 
wenzelm@12396: 
nipkow@15392: text {* Sigma of finite sets *}
wenzelm@12396: 
wenzelm@12396: lemma finite_SigmaI [simp]:
wenzelm@12396:     "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
wenzelm@12396:   by (unfold Sigma_def) (blast intro!: finite_UN_I)
wenzelm@12396: 
nipkow@15402: lemma finite_cartesian_product: "[| finite A; finite B |] ==>
nipkow@15402:     finite (A <*> B)"
nipkow@15402:   by (rule finite_SigmaI)
nipkow@15402: 
wenzelm@12396: lemma finite_Prod_UNIV:
wenzelm@12396:     "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
wenzelm@12396:   apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
wenzelm@12396:    apply (erule ssubst)
paulson@14208:    apply (erule finite_SigmaI, auto)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: instance unit :: finite
wenzelm@12396: proof
wenzelm@12396:   have "finite {()}" by simp
wenzelm@12396:   also have "{()} = UNIV" by auto
wenzelm@12396:   finally show "finite (UNIV :: unit set)" .
wenzelm@12396: qed
wenzelm@12396: 
wenzelm@12396: instance * :: (finite, finite) finite
wenzelm@12396: proof
wenzelm@12396:   show "finite (UNIV :: ('a \<times> 'b) set)"
wenzelm@12396:   proof (rule finite_Prod_UNIV)
wenzelm@12396:     show "finite (UNIV :: 'a set)" by (rule finite)
wenzelm@12396:     show "finite (UNIV :: 'b set)" by (rule finite)
wenzelm@12396:   qed
wenzelm@12396: qed
wenzelm@12396: 
wenzelm@12396: 
nipkow@15392: text {* The powerset of a finite set *}
wenzelm@12396: 
wenzelm@12396: lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
wenzelm@12396: proof
wenzelm@12396:   assume "finite (Pow A)"
wenzelm@12396:   with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
wenzelm@12396:   thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
wenzelm@12396: next
wenzelm@12396:   assume "finite A"
wenzelm@12396:   thus "finite (Pow A)"
wenzelm@12396:     by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
wenzelm@12396: qed
wenzelm@12396: 
nipkow@15392: 
nipkow@15392: lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
nipkow@15392: by(blast intro: finite_subset[OF subset_Pow_Union])
nipkow@15392: 
nipkow@15392: 
wenzelm@12396: lemma finite_converse [iff]: "finite (r^-1) = finite r"
wenzelm@12396:   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
wenzelm@12396:    apply simp
wenzelm@12396:    apply (rule iffI)
wenzelm@12396:     apply (erule finite_imageD [unfolded inj_on_def])
wenzelm@12396:     apply (simp split add: split_split)
wenzelm@12396:    apply (erule finite_imageI)
paulson@14208:   apply (simp add: converse_def image_def, auto)
wenzelm@12396:   apply (rule bexI)
wenzelm@12396:    prefer 2 apply assumption
wenzelm@12396:   apply simp
wenzelm@12396:   done
wenzelm@12396: 
paulson@14430: 
nipkow@15392: text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
nipkow@15392: Ehmety) *}
wenzelm@12396: 
wenzelm@12396: lemma finite_Field: "finite r ==> finite (Field r)"
wenzelm@12396:   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
wenzelm@12396:   apply (induct set: Finites)
wenzelm@12396:    apply (auto simp add: Field_def Domain_insert Range_insert)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
wenzelm@12396:   apply clarify
wenzelm@12396:   apply (erule trancl_induct)
wenzelm@12396:    apply (auto simp add: Field_def)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma finite_trancl: "finite (r^+) = finite r"
wenzelm@12396:   apply auto
wenzelm@12396:    prefer 2
wenzelm@12396:    apply (rule trancl_subset_Field2 [THEN finite_subset])
wenzelm@12396:    apply (rule finite_SigmaI)
wenzelm@12396:     prefer 3
berghofe@13704:     apply (blast intro: r_into_trancl' finite_subset)
wenzelm@12396:    apply (auto simp add: finite_Field)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: 
nipkow@15392: subsection {* A fold functional for finite sets *}
nipkow@15392: 
nipkow@15392: text {* The intended behaviour is
nipkow@15392: @{text "fold f g e {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) e)\<dots>)"}
nipkow@15392: if @{text f} is associative-commutative. For an application of @{text fold}
nipkow@15392: se the definitions of sums and products over finite sets.
nipkow@15392: *}
nipkow@15392: 
nipkow@15392: consts
nipkow@15392:   foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
nipkow@15392: 
nipkow@15392: inductive "foldSet f g e"
nipkow@15392: intros
nipkow@15392: emptyI [intro]: "({}, e) : foldSet f g e"
nipkow@15392: insertI [intro]: "\<lbrakk> x \<notin> A; (A, y) : foldSet f g e \<rbrakk>
nipkow@15392:  \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g e"
nipkow@15392: 
nipkow@15392: inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g e"
nipkow@15392: 
nipkow@15392: constdefs
nipkow@15392:   fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
nipkow@15392:   "fold f g e A == THE x. (A, x) : foldSet f g e"
nipkow@15392: 
nipkow@15392: lemma Diff1_foldSet:
nipkow@15392:   "(A - {x}, y) : foldSet f g e ==> x: A ==> (A, f (g x) y) : foldSet f g e"
nipkow@15392: by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
nipkow@15392: 
nipkow@15392: lemma foldSet_imp_finite: "(A, x) : foldSet f g e ==> finite A"
nipkow@15392:   by (induct set: foldSet) auto
nipkow@15392: 
nipkow@15392: lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g e"
nipkow@15392:   by (induct set: Finites) auto
nipkow@15392: 
nipkow@15392: 
nipkow@15392: subsubsection {* Commutative monoids *}
nipkow@15392: 
nipkow@15392: locale ACf =
nipkow@15392:   fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
nipkow@15392:   assumes commute: "x \<cdot> y = y \<cdot> x"
nipkow@15392:     and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
nipkow@15392: 
nipkow@15392: locale ACe = ACf +
nipkow@15392:   fixes e :: 'a
nipkow@15392:   assumes ident [simp]: "x \<cdot> e = x"
nipkow@15392: 
nipkow@15392: lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
nipkow@15392: proof -
nipkow@15392:   have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
nipkow@15392:   also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
nipkow@15392:   also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
nipkow@15392:   finally show ?thesis .
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: lemmas (in ACf) AC = assoc commute left_commute
nipkow@15392: 
nipkow@15392: lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
nipkow@15392: proof -
nipkow@15392:   have "x \<cdot> e = x" by (rule ident)
nipkow@15392:   thus ?thesis by (subst commute)
nipkow@15392: qed
nipkow@15392: 
nipkow@15402: text{* Instantiation of locales: *}
nipkow@15402: 
nipkow@15402: lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15402: by(fastsimp intro: ACf.intro add_assoc add_commute)
nipkow@15402: 
nipkow@15402: lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
nipkow@15402: by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
nipkow@15402: 
nipkow@15402: 
nipkow@15402: lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15402: by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
nipkow@15402: 
nipkow@15402: lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
nipkow@15402: by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
nipkow@15402: 
nipkow@15402: 
nipkow@15392: subsubsection{*From @{term foldSet} to @{term fold}*}
nipkow@15392: 
nipkow@15392: lemma (in ACf) foldSet_determ_aux:
nipkow@15392:   "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
nipkow@15392:    \<Longrightarrow> x' = x"
nipkow@15392: proof (induct n)
nipkow@15392:   case 0 thus ?case by auto
nipkow@15392: next
nipkow@15392:   case (Suc n)
nipkow@15392:   have IH: "!!A x x' h. \<lbrakk>A = h`{i::nat. i<n}; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
nipkow@15392:            \<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
nipkow@15392:   and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
nipkow@15392:   show ?case
nipkow@15392:   proof cases
nipkow@15392:     assume "EX k<n. h n = h k"
nipkow@15392:     hence card': "A = h ` {i. i < n}"
nipkow@15392:       using card by (auto simp:image_def less_Suc_eq)
nipkow@15392:     show ?thesis by(rule IH[OF card' Afoldx Afoldy])
nipkow@15392:   next
nipkow@15392:     assume new: "\<not>(EX k<n. h n = h k)"
nipkow@15392:     show ?thesis
nipkow@15392:     proof (rule foldSet.cases[OF Afoldx])
nipkow@15392:       assume "(A, x) = ({}, e)"
nipkow@15392:       thus "x' = x" using Afoldy by (auto)
nipkow@15392:     next
nipkow@15392:       fix B b y
nipkow@15392:       assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
nipkow@15392: 	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
nipkow@15392:       hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
nipkow@15392:       show ?thesis
nipkow@15392:       proof (rule foldSet.cases[OF Afoldy])
nipkow@15392: 	assume "(A,x') = ({}, e)"
nipkow@15392: 	thus ?thesis using A1 by auto
nipkow@15392:       next
nipkow@15392: 	fix C c z
nipkow@15392: 	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
nipkow@15392: 	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
nipkow@15392: 	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
nipkow@15392: 	let ?h = "%i. if h i = b then h n else h i"
nipkow@15392: 	have less: "B = ?h`{i. i<n}" (is "_ = ?r")
nipkow@15392: 	proof
nipkow@15392: 	  show "B \<subseteq> ?r"
nipkow@15392: 	  proof
nipkow@15392: 	    fix u assume "u \<in> B"
nipkow@15392: 	    hence uinA: "u \<in> A" and unotb: "u \<noteq> b" using A1 notinB by blast+
nipkow@15392: 	    then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
nipkow@15392: 	      using card by(auto simp:image_def)
nipkow@15392: 	    show "u \<in> ?r"
nipkow@15392: 	    proof cases
nipkow@15392: 	      assume "i\<^isub>u < n"
nipkow@15392: 	      thus ?thesis using unotb by(fastsimp)
nipkow@15392: 	    next
nipkow@15392: 	      assume "\<not> i\<^isub>u < n"
nipkow@15392: 	      with below have [simp]: "i\<^isub>u = n" by arith
nipkow@15392: 	      obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "b = h i\<^isub>k"
nipkow@15392: 		using A1 card by blast
nipkow@15392: 	      have "i\<^isub>k < n"
nipkow@15392: 	      proof (rule ccontr)
nipkow@15392: 		assume "\<not> i\<^isub>k < n"
nipkow@15392: 		hence "i\<^isub>k = n" using i\<^isub>k by arith
nipkow@15392: 		thus False using unotb by simp
nipkow@15392: 	      qed
nipkow@15392: 	      thus ?thesis by(auto simp add:image_def)
nipkow@15392: 	    qed
nipkow@15392: 	  qed
nipkow@15392: 	next
nipkow@15392: 	  show "?r \<subseteq> B"
nipkow@15392: 	  proof
nipkow@15392: 	    fix u assume "u \<in> ?r"
nipkow@15392: 	    then obtain i\<^isub>u where below: "i\<^isub>u < n" and
nipkow@15392:               or: "b = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
nipkow@15392: 	      by(auto simp:image_def)
nipkow@15392: 	    from or show "u \<in> B"
nipkow@15392: 	    proof
nipkow@15392: 	      assume [simp]: "b = h i\<^isub>u \<and> u = h n"
nipkow@15392: 	      have "u \<in> A" using card by auto
nipkow@15392:               moreover have "u \<noteq> b" using new below by auto
nipkow@15392: 	      ultimately show "u \<in> B" using A1 by blast
nipkow@15392: 	    next
nipkow@15392: 	      assume "h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
nipkow@15392: 	      moreover hence "u \<in> A" using card below by auto
nipkow@15392: 	      ultimately show "u \<in> B" using A1 by blast
nipkow@15392: 	    qed
nipkow@15392: 	  qed
nipkow@15392: 	qed
nipkow@15392: 	show ?thesis
nipkow@15392: 	proof cases
nipkow@15392: 	  assume "b = c"
nipkow@15392: 	  then moreover have "B = C" using A1 A2 notinB notinC by auto
nipkow@15392: 	  ultimately show ?thesis using IH[OF less] y z x x' by auto
nipkow@15392: 	next
nipkow@15392: 	  assume diff: "b \<noteq> c"
nipkow@15392: 	  let ?D = "B - {c}"
nipkow@15392: 	  have B: "B = insert c ?D" and C: "C = insert b ?D"
nipkow@15392: 	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
nipkow@15402: 	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
nipkow@15402: 	  with A1 have "finite ?D" by simp
nipkow@15392: 	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
nipkow@15392: 	    using finite_imp_foldSet by rules
nipkow@15392: 	  moreover have cinB: "c \<in> B" using B by(auto)
nipkow@15392: 	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
nipkow@15392: 	    by(rule Diff1_foldSet)
nipkow@15392: 	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
nipkow@15392:           moreover have "g b \<cdot> d = z"
nipkow@15392: 	  proof (rule IH[OF _ z])
nipkow@15392: 	    let ?h = "%i. if h i = c then h n else h i"
nipkow@15392: 	    show "C = ?h`{i. i<n}" (is "_ = ?r")
nipkow@15392: 	    proof
nipkow@15392: 	      show "C \<subseteq> ?r"
nipkow@15392: 	      proof
nipkow@15392: 		fix u assume "u \<in> C"
nipkow@15392: 		hence uinA: "u \<in> A" and unotc: "u \<noteq> c"
nipkow@15392: 		  using A2 notinC by blast+
nipkow@15392: 		then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
nipkow@15392: 		  using card by(auto simp:image_def)
nipkow@15392: 		show "u \<in> ?r"
nipkow@15392: 		proof cases
nipkow@15392: 		  assume "i\<^isub>u < n"
nipkow@15392: 		  thus ?thesis using unotc by(fastsimp)
nipkow@15392: 		next
nipkow@15392: 		  assume "\<not> i\<^isub>u < n"
nipkow@15392: 		  with below have [simp]: "i\<^isub>u = n" by arith
nipkow@15392: 		  obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "c = h i\<^isub>k"
nipkow@15392: 		    using A2 card by blast
nipkow@15392: 		  have "i\<^isub>k < n"
nipkow@15392: 		  proof (rule ccontr)
nipkow@15392: 		    assume "\<not> i\<^isub>k < n"
nipkow@15392: 		    hence "i\<^isub>k = n" using i\<^isub>k by arith
nipkow@15392: 		    thus False using unotc by simp
nipkow@15392: 		  qed
nipkow@15392: 		  thus ?thesis by(auto simp add:image_def)
nipkow@15392: 		qed
nipkow@15392: 	      qed
nipkow@15392: 	    next
nipkow@15392: 	      show "?r \<subseteq> C"
nipkow@15392: 	      proof
nipkow@15392: 		fix u assume "u \<in> ?r"
nipkow@15392: 		then obtain i\<^isub>u where below: "i\<^isub>u < n" and
nipkow@15392: 		  or: "c = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> c \<and> h i\<^isub>u = u"
nipkow@15392: 		  by(auto simp:image_def)
nipkow@15392: 		from or show "u \<in> C"
nipkow@15392: 		proof
nipkow@15392: 		  assume [simp]: "c = h i\<^isub>u \<and> u = h n"
nipkow@15392: 		  have "u \<in> A" using card by auto
nipkow@15392: 		  moreover have "u \<noteq> c" using new below by auto
nipkow@15392: 		  ultimately show "u \<in> C" using A2 by blast
nipkow@15392: 		next
nipkow@15392: 		  assume "h i\<^isub>u \<noteq> c \<and> h i\<^isub>u = u"
nipkow@15392: 		  moreover hence "u \<in> A" using card below by auto
nipkow@15392: 		  ultimately show "u \<in> C" using A2 by blast
nipkow@15392: 		qed
nipkow@15392: 	      qed
nipkow@15392: 	    qed
nipkow@15392: 	  next
nipkow@15392: 	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
nipkow@15392: 	      by fastsimp
nipkow@15392: 	  qed
nipkow@15392: 	  ultimately show ?thesis using x x' by(auto simp:AC)
nipkow@15392: 	qed
nipkow@15392:       qed
nipkow@15392:     qed
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: (* The same proof, but using card 
nipkow@15392: lemma (in ACf) foldSet_determ_aux:
nipkow@15392:   "!!A x x'. \<lbrakk> card A < n; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
nipkow@15392:    \<Longrightarrow> x' = x"
nipkow@15392: proof (induct n)
nipkow@15392:   case 0 thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (Suc n)
nipkow@15392:   have IH: "!!A x x'. \<lbrakk>card A < n; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
nipkow@15392:            \<Longrightarrow> x' = x" and card: "card A < Suc n"
nipkow@15392:   and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
nipkow@15392:   from card have "card A < n \<or> card A = n" by arith
nipkow@15392:   thus ?case
nipkow@15392:   proof
nipkow@15392:     assume less: "card A < n"
nipkow@15392:     show ?thesis by(rule IH[OF less Afoldx Afoldy])
nipkow@15392:   next
nipkow@15392:     assume cardA: "card A = n"
nipkow@15392:     show ?thesis
nipkow@15392:     proof (rule foldSet.cases[OF Afoldx])
nipkow@15392:       assume "(A, x) = ({}, e)"
nipkow@15392:       thus "x' = x" using Afoldy by (auto)
nipkow@15392:     next
nipkow@15392:       fix B b y
nipkow@15392:       assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
nipkow@15392: 	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
nipkow@15392:       hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
nipkow@15392:       show ?thesis
nipkow@15392:       proof (rule foldSet.cases[OF Afoldy])
nipkow@15392: 	assume "(A,x') = ({}, e)"
nipkow@15392: 	thus ?thesis using A1 by auto
nipkow@15392:       next
nipkow@15392: 	fix C c z
nipkow@15392: 	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
nipkow@15392: 	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
nipkow@15392: 	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
nipkow@15392: 	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
nipkow@15392: 	with cardA A1 notinB have less: "card B < n" by simp
nipkow@15392: 	show ?thesis
nipkow@15392: 	proof cases
nipkow@15392: 	  assume "b = c"
nipkow@15392: 	  then moreover have "B = C" using A1 A2 notinB notinC by auto
nipkow@15392: 	  ultimately show ?thesis using IH[OF less] y z x x' by auto
nipkow@15392: 	next
nipkow@15392: 	  assume diff: "b \<noteq> c"
nipkow@15392: 	  let ?D = "B - {c}"
nipkow@15392: 	  have B: "B = insert c ?D" and C: "C = insert b ?D"
nipkow@15392: 	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
nipkow@15392: 	  have "finite ?D" using finA A1 by simp
nipkow@15392: 	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
nipkow@15392: 	    using finite_imp_foldSet by rules
nipkow@15392: 	  moreover have cinB: "c \<in> B" using B by(auto)
nipkow@15392: 	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
nipkow@15392: 	    by(rule Diff1_foldSet)
nipkow@15392: 	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
nipkow@15392:           moreover have "g b \<cdot> d = z"
nipkow@15392: 	  proof (rule IH[OF _ z])
nipkow@15392: 	    show "card C < n" using C cardA A1 notinB finA cinB
nipkow@15392: 	      by(auto simp:card_Diff1_less)
nipkow@15392: 	  next
nipkow@15392: 	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
nipkow@15392: 	      by fastsimp
nipkow@15392: 	  qed
nipkow@15392: 	  ultimately show ?thesis using x x' by(auto simp:AC)
nipkow@15392: 	qed
nipkow@15392:       qed
nipkow@15392:     qed
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: *)
nipkow@15392: 
nipkow@15392: lemma (in ACf) foldSet_determ:
nipkow@15392:   "(A, x) : foldSet f g e ==> (A, y) : foldSet f g e ==> y = x"
nipkow@15392: apply(frule foldSet_imp_finite)
nipkow@15392: apply(simp add:finite_conv_nat_seg_image)
nipkow@15392: apply(blast intro: foldSet_determ_aux [rule_format])
nipkow@15392: done
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_equality: "(A, y) : foldSet f g e ==> fold f g e A = y"
nipkow@15392:   by (unfold fold_def) (blast intro: foldSet_determ)
nipkow@15392: 
nipkow@15392: text{* The base case for @{text fold}: *}
nipkow@15392: 
nipkow@15392: lemma fold_empty [simp]: "fold f g e {} = e"
nipkow@15392:   by (unfold fold_def) blast
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
nipkow@15392:     ((insert x A, v) : foldSet f g e) =
nipkow@15392:     (EX y. (A, y) : foldSet f g e & v = f (g x) y)"
nipkow@15392:   apply auto
nipkow@15392:   apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
nipkow@15392:    apply (fastsimp dest: foldSet_imp_finite)
nipkow@15392:   apply (blast intro: foldSet_determ)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: text{* The recursion equation for @{text fold}: *}
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_insert[simp]:
nipkow@15392:     "finite A ==> x \<notin> A ==> fold f g e (insert x A) = f (g x) (fold f g e A)"
nipkow@15392:   apply (unfold fold_def)
nipkow@15392:   apply (simp add: fold_insert_aux)
nipkow@15392:   apply (rule the_equality)
nipkow@15392:   apply (auto intro: finite_imp_foldSet
nipkow@15392:     cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: declare
nipkow@15392:   empty_foldSetE [rule del]  foldSet.intros [rule del]
nipkow@15392:   -- {* Delete rules to do with @{text foldSet} relation. *}
nipkow@15392: 
nipkow@15392: subsubsection{*Lemmas about @{text fold}*}
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_commute:
nipkow@15392:   "finite A ==> (!!e. f (g x) (fold f g e A) = fold f g (f (g x) e) A)"
nipkow@15392:   apply (induct set: Finites, simp)
nipkow@15392:   apply (simp add: left_commute)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_nest_Un_Int:
nipkow@15392:   "finite A ==> finite B
nipkow@15392:     ==> fold f g (fold f g e B) A = fold f g (fold f g e (A Int B)) (A Un B)"
nipkow@15392:   apply (induct set: Finites, simp)
nipkow@15392:   apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_nest_Un_disjoint:
nipkow@15392:   "finite A ==> finite B ==> A Int B = {}
nipkow@15392:     ==> fold f g e (A Un B) = fold f g (fold f g e B) A"
nipkow@15392:   by (simp add: fold_nest_Un_Int)
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_reindex:
nipkow@15392: assumes fin: "finite B"
nipkow@15392: shows "inj_on h B \<Longrightarrow> fold f g e (h ` B) = fold f (g \<circ> h) e B"
nipkow@15392: using fin apply (induct)
nipkow@15392:  apply simp
nipkow@15392: apply simp
nipkow@15392: done
nipkow@15392: 
nipkow@15392: lemma (in ACe) fold_Un_Int:
nipkow@15392:   "finite A ==> finite B ==>
nipkow@15392:     fold f g e A \<cdot> fold f g e B =
nipkow@15392:     fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
nipkow@15392:   apply (induct set: Finites, simp)
nipkow@15392:   apply (simp add: AC insert_absorb Int_insert_left)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: corollary (in ACe) fold_Un_disjoint:
nipkow@15392:   "finite A ==> finite B ==> A Int B = {} ==>
nipkow@15392:     fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
nipkow@15392:   by (simp add: fold_Un_Int)
nipkow@15392: 
nipkow@15392: lemma (in ACe) fold_UN_disjoint:
nipkow@15392:   "\<lbrakk> finite I; ALL i:I. finite (A i);
nipkow@15392:      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
nipkow@15392:    \<Longrightarrow> fold f g e (UNION I A) =
nipkow@15392:        fold f (%i. fold f g e (A i)) e I"
nipkow@15392:   apply (induct set: Finites, simp, atomize)
nipkow@15392:   apply (subgoal_tac "ALL i:F. x \<noteq> i")
nipkow@15392:    prefer 2 apply blast
nipkow@15392:   apply (subgoal_tac "A x Int UNION F A = {}")
nipkow@15392:    prefer 2 apply blast
nipkow@15392:   apply (simp add: fold_Un_disjoint)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold_cong:
nipkow@15392:   "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g a A = fold f h a A"
nipkow@15392:   apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g a C = fold f h a C")
nipkow@15392:    apply simp
nipkow@15392:   apply (erule finite_induct, simp)
nipkow@15392:   apply (simp add: subset_insert_iff, clarify)
nipkow@15392:   apply (subgoal_tac "finite C")
nipkow@15392:    prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
nipkow@15392:   apply (subgoal_tac "C = insert x (C - {x})")
nipkow@15392:    prefer 2 apply blast
nipkow@15392:   apply (erule ssubst)
nipkow@15392:   apply (drule spec)
nipkow@15392:   apply (erule (1) notE impE)
nipkow@15392:   apply (simp add: Ball_def del: insert_Diff_single)
nipkow@15392:   done
nipkow@15392: 
nipkow@15392: lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15392:   fold f (%x. fold f (g x) e (B x)) e A =
nipkow@15392:   fold f (split g) e (SIGMA x:A. B x)"
nipkow@15392: apply (subst Sigma_def)
nipkow@15392: apply (subst fold_UN_disjoint)
nipkow@15392:    apply assumption
nipkow@15392:   apply simp
nipkow@15392:  apply blast
nipkow@15392: apply (erule fold_cong)
nipkow@15392: apply (subst fold_UN_disjoint)
nipkow@15392:    apply simp
nipkow@15392:   apply simp
nipkow@15392:  apply blast
nipkow@15392: apply (simp)
nipkow@15392: done
nipkow@15392: 
nipkow@15392: lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
nipkow@15392:    fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
nipkow@15392: apply (erule finite_induct)
nipkow@15392:  apply simp
nipkow@15392: apply (simp add:AC)
nipkow@15392: done
nipkow@15392: 
nipkow@15392: 
nipkow@15402: subsection {* Generalized summation over a set *}
nipkow@15402: 
nipkow@15402: constdefs
nipkow@15402:   setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
nipkow@15402:   "setsum f A == if finite A then fold (op +) f 0 A else 0"
nipkow@15402: 
nipkow@15402: text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402: written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402: 
nipkow@15402: syntax
nipkow@15402:   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (xsymbols)
nipkow@15402:   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (HTML output)
nipkow@15402:   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: 
nipkow@15402: translations -- {* Beware of argument permutation! *}
nipkow@15402:   "SUM i:A. b" == "setsum (%i. b) A"
nipkow@15402:   "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
nipkow@15402: 
nipkow@15402: text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
nipkow@15402:  @{text"\<Sum>x|P. e"}. *}
nipkow@15402: 
nipkow@15402: syntax
nipkow@15402:   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402: syntax (xsymbols)
nipkow@15402:   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402: syntax (HTML output)
nipkow@15402:   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402: 
nipkow@15402: translations
nipkow@15402:   "SUM x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402:   "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
nipkow@15402: 
nipkow@15402: text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
nipkow@15402: 
nipkow@15402: syntax
nipkow@15402:   "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
nipkow@15402: 
nipkow@15402: parse_translation {*
nipkow@15402:   let
nipkow@15402:     fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
nipkow@15402:   in [("_Setsum", Setsum_tr)] end;
nipkow@15402: *}
nipkow@15402: 
nipkow@15402: print_translation {*
nipkow@15402: let
nipkow@15402:   fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
nipkow@15402:     | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
nipkow@15402:        if x<>y then raise Match
nipkow@15402:        else let val x' = Syntax.mark_bound x
nipkow@15402:                 val t' = subst_bound(x',t)
nipkow@15402:                 val P' = subst_bound(x',P)
nipkow@15402:             in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
nipkow@15402: in
nipkow@15402: [("setsum", setsum_tr')]
nipkow@15402: end
nipkow@15402: *}
nipkow@15402: 
nipkow@15402: lemma setsum_empty [simp]: "setsum f {} = 0"
nipkow@15402:   by (simp add: setsum_def)
nipkow@15402: 
nipkow@15402: lemma setsum_insert [simp]:
nipkow@15402:     "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
nipkow@15402:   by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
nipkow@15402: 
nipkow@15402: lemma setsum_reindex:
nipkow@15402:      "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
nipkow@15402: by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
nipkow@15402: 
nipkow@15402: lemma setsum_reindex_id:
nipkow@15402:      "inj_on f B ==> setsum f B = setsum id (f ` B)"
nipkow@15402: by (auto simp add: setsum_reindex)
nipkow@15402: 
nipkow@15402: lemma setsum_cong:
nipkow@15402:   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
nipkow@15402: by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
nipkow@15402: 
nipkow@15402: lemma setsum_reindex_cong:
nipkow@15402:      "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|] 
nipkow@15402:       ==> setsum h B = setsum g A"
nipkow@15402:   by (simp add: setsum_reindex cong: setsum_cong)
nipkow@15402: 
nipkow@15402: lemma setsum_0: "setsum (%i. 0) A = 0"
nipkow@15402: apply (clarsimp simp: setsum_def)
nipkow@15402: apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
nipkow@15402: done
nipkow@15402: 
nipkow@15402: lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
nipkow@15402:   apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
nipkow@15402:   apply (erule ssubst, rule setsum_0)
nipkow@15402:   apply (rule setsum_cong, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setsum_Un_Int: "finite A ==> finite B ==>
nipkow@15402:   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
nipkow@15402:   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
nipkow@15402: by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
nipkow@15402: 
nipkow@15402: lemma setsum_Un_disjoint: "finite A ==> finite B
nipkow@15402:   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
nipkow@15402: by (subst setsum_Un_Int [symmetric], auto)
nipkow@15402: 
nipkow@15402: (* FIXME get rid of finite I. If infinite, rhs is directly 0, and UNION I A
nipkow@15402: is also infinite and hence also 0 *)
nipkow@15402: lemma setsum_UN_disjoint:
nipkow@15402:     "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402:         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402:       setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
nipkow@15402: by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
nipkow@15402: 
nipkow@15402: 
nipkow@15402: (* FIXME get rid of finite C. If infinite, rhs is directly 0, and Union C
nipkow@15402: is also infinite and hence also 0 *)
nipkow@15402: lemma setsum_Union_disjoint:
nipkow@15402:   "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402:         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402:       setsum f (Union C) = setsum (setsum f) C"
nipkow@15402:   apply (frule setsum_UN_disjoint [of C id f])
nipkow@15402:   apply (unfold Union_def id_def, assumption+)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: (* FIXME get rid of finite A. If infinite, lhs is directly 0, and SIGMA A B
nipkow@15402: is either infinite or empty, and in both cases the rhs is also 0 *)
nipkow@15402: lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15402:     (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
nipkow@15402:     (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
nipkow@15402: by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
nipkow@15402: 
nipkow@15402: lemma setsum_cartesian_product: "finite A ==> finite B ==>
nipkow@15402:     (\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) =
nipkow@15402:     (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
nipkow@15402:   by (erule setsum_Sigma, auto)
nipkow@15402: 
nipkow@15402: lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
nipkow@15402: by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402: 
nipkow@15402: lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
nipkow@15402:   apply (case_tac "finite A")
nipkow@15402:    prefer 2 apply (simp add: setsum_def)
nipkow@15402:   apply (erule rev_mp)
nipkow@15402:   apply (erule finite_induct, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setsum_eq_0_iff [simp]:
nipkow@15402:     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
nipkow@15402:   by (induct set: Finites) auto
nipkow@15402: 
nipkow@15402: lemma setsum_Un_nat: "finite A ==> finite B ==>
nipkow@15402:     (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402:   -- {* For the natural numbers, we have subtraction. *}
nipkow@15402:   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402: 
nipkow@15402: lemma setsum_Un: "finite A ==> finite B ==>
nipkow@15402:     (setsum f (A Un B) :: 'a :: ab_group_add) =
nipkow@15402:       setsum f A + setsum f B - setsum f (A Int B)"
nipkow@15402:   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
nipkow@15402: 
nipkow@15402: lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
nipkow@15402:     (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402:   apply (case_tac "finite A")
nipkow@15402:    prefer 2 apply (simp add: setsum_def)
nipkow@15402:   apply (erule finite_induct)
nipkow@15402:    apply (auto simp add: insert_Diff_if)
nipkow@15402:   apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setsum_diff1: "finite A \<Longrightarrow>
nipkow@15402:   (setsum f (A - {a}) :: ('a::ab_group_add)) =
nipkow@15402:   (if a:A then setsum f A - f a else setsum f A)"
nipkow@15402:   by (erule finite_induct) (auto simp add: insert_Diff_if)
nipkow@15402: 
nipkow@15402: (* By Jeremy Siek: *)
nipkow@15402: 
nipkow@15402: lemma setsum_diff_nat: 
nipkow@15402:   assumes finB: "finite B"
nipkow@15402:   shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
nipkow@15402: using finB
nipkow@15402: proof (induct)
nipkow@15402:   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402: next
nipkow@15402:   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402:     and xFinA: "insert x F \<subseteq> A"
nipkow@15402:     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402:   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402:   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402:     by (simp add: setsum_diff1_nat)
nipkow@15402:   from xFinA have "F \<subseteq> A" by simp
nipkow@15402:   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402:   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402:     by simp
nipkow@15402:   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402:   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402:     by simp
nipkow@15402:   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402:   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402:     by simp
nipkow@15402:   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_diff:
nipkow@15402:   assumes le: "finite A" "B \<subseteq> A"
nipkow@15402:   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402: proof -
nipkow@15402:   from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402:   show ?thesis using finiteB le
nipkow@15402:     proof (induct)
nipkow@15402:       case empty
nipkow@15402:       thus ?case by auto
nipkow@15402:     next
nipkow@15402:       case (insert x F)
nipkow@15402:       thus ?case using le finiteB 
nipkow@15402: 	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402:     qed
nipkow@15402:   qed
nipkow@15402: 
nipkow@15402: lemma setsum_mono:
nipkow@15402:   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
nipkow@15402:   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
nipkow@15402: proof (cases "finite K")
nipkow@15402:   case True
nipkow@15402:   thus ?thesis using le
nipkow@15402:   proof (induct)
nipkow@15402:     case empty
nipkow@15402:     thus ?case by simp
nipkow@15402:   next
nipkow@15402:     case insert
nipkow@15402:     thus ?case using add_mono 
nipkow@15402:       by force
nipkow@15402:   qed
nipkow@15402: next
nipkow@15402:   case False
nipkow@15402:   thus ?thesis
nipkow@15402:     by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_mono2_nat:
nipkow@15402:   assumes fin: "finite B" and sub: "A \<subseteq> B"
nipkow@15402: shows "setsum f A \<le> (setsum f B :: nat)"
nipkow@15402: proof -
nipkow@15402:   have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
nipkow@15402:   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15402:     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15402:   also have "A \<union> (B-A) = B" using sub by blast
nipkow@15402:   finally show ?thesis .
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
nipkow@15402:   - setsum f A"
nipkow@15402:   by (induct set: Finites, auto)
nipkow@15402: 
nipkow@15402: lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
nipkow@15402:   setsum f A - setsum g A"
nipkow@15402:   by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15402: 
nipkow@15402: lemma setsum_nonneg: "[| finite A;
nipkow@15402:     \<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
nipkow@15402:     0 \<le> setsum f A";
nipkow@15402:   apply (induct set: Finites, auto)
nipkow@15402:   apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
nipkow@15402:   apply (blast intro: add_mono)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setsum_nonpos: "[| finite A;
nipkow@15402:     \<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
nipkow@15402:     setsum f A \<le> 0";
nipkow@15402:   apply (induct set: Finites, auto)
nipkow@15402:   apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
nipkow@15402:   apply (blast intro: add_mono)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setsum_mult: 
nipkow@15402:   fixes f :: "'a => ('b::semiring_0_cancel)"
nipkow@15402:   shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402: proof (cases "finite A")
nipkow@15402:   case True
nipkow@15402:   thus ?thesis
nipkow@15402:   proof (induct)
nipkow@15402:     case empty thus ?case by simp
nipkow@15402:   next
nipkow@15402:     case (insert x A) thus ?case by (simp add: right_distrib)
nipkow@15402:   qed
nipkow@15402: next
nipkow@15402:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_abs: 
nipkow@15402:   fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402:   assumes fin: "finite A" 
nipkow@15402:   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15402: using fin 
nipkow@15402: proof (induct) 
nipkow@15402:   case empty thus ?case by simp
nipkow@15402: next
nipkow@15402:   case (insert x A)
nipkow@15402:   thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_abs_ge_zero: 
nipkow@15402:   fixes f :: "'a => ('b::lordered_ab_group_abs)"
nipkow@15402:   assumes fin: "finite A" 
nipkow@15402:   shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15402: using fin 
nipkow@15402: proof (induct) 
nipkow@15402:   case empty thus ?case by simp
nipkow@15402: next
nipkow@15402:   case (insert x A) thus ?case by (auto intro: order_trans)
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsection {* Generalized product over a set *}
nipkow@15402: 
nipkow@15402: constdefs
nipkow@15402:   setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
nipkow@15402:   "setprod f A == if finite A then fold (op *) f 1 A else 1"
nipkow@15402: 
nipkow@15402: syntax
nipkow@15402:   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
nipkow@15402: 
nipkow@15402: syntax (xsymbols)
nipkow@15402:   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (HTML output)
nipkow@15402:   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: translations
nipkow@15402:   "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
nipkow@15402: 
nipkow@15402: syntax
nipkow@15402:   "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
nipkow@15402: 
nipkow@15402: parse_translation {*
nipkow@15402:   let
nipkow@15402:     fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
nipkow@15402:   in [("_Setprod", Setprod_tr)] end;
nipkow@15402: *}
nipkow@15402: print_translation {*
nipkow@15402: let fun setprod_tr' [Abs(x,Tx,t), A] =
nipkow@15402:     if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
nipkow@15402: in
nipkow@15402: [("setprod", setprod_tr')]
nipkow@15402: end
nipkow@15402: *}
nipkow@15402: 
nipkow@15402: 
nipkow@15402: lemma setprod_empty [simp]: "setprod f {} = 1"
nipkow@15402:   by (auto simp add: setprod_def)
nipkow@15402: 
nipkow@15402: lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
nipkow@15402:     setprod f (insert a A) = f a * setprod f A"
nipkow@15402: by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
nipkow@15402: 
nipkow@15402: lemma setprod_reindex:
nipkow@15402:      "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
nipkow@15402: by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
nipkow@15402: 
nipkow@15402: lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
nipkow@15402: by (auto simp add: setprod_reindex)
nipkow@15402: 
nipkow@15402: lemma setprod_cong:
nipkow@15402:   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
nipkow@15402: by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
nipkow@15402: 
nipkow@15402: lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402:     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@15402:   by (frule setprod_reindex, simp)
nipkow@15402: 
nipkow@15402: 
nipkow@15402: lemma setprod_1: "setprod (%i. 1) A = 1"
nipkow@15402:   apply (case_tac "finite A")
nipkow@15402:   apply (erule finite_induct, auto simp add: mult_ac)
nipkow@15402:   apply (simp add: setprod_def)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
nipkow@15402:   apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
nipkow@15402:   apply (erule ssubst, rule setprod_1)
nipkow@15402:   apply (rule setprod_cong, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_Un_Int: "finite A ==> finite B
nipkow@15402:     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
nipkow@15402: by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
nipkow@15402: 
nipkow@15402: lemma setprod_Un_disjoint: "finite A ==> finite B
nipkow@15402:   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
nipkow@15402: by (subst setprod_Un_Int [symmetric], auto)
nipkow@15402: 
nipkow@15402: lemma setprod_UN_disjoint:
nipkow@15402:     "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402:         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402:       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
nipkow@15402: by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
nipkow@15402: 
nipkow@15402: lemma setprod_Union_disjoint:
nipkow@15402:   "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402:         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402:       setprod f (Union C) = setprod (setprod f) C"
nipkow@15402:   apply (frule setprod_UN_disjoint [of C id f])
nipkow@15402:   apply (unfold Union_def id_def, assumption+)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@15402:     (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
nipkow@15402:     (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
nipkow@15402: by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
nipkow@15402: 
nipkow@15402: lemma setprod_cartesian_product: "finite A ==> finite B ==>
nipkow@15402:     (\<Prod>x:A. (\<Prod>y: B. f x y)) =
nipkow@15402:     (\<Prod>z:(A <*> B). f (fst z) (snd z))"
nipkow@15402:   by (erule setprod_Sigma, auto)
nipkow@15402: 
nipkow@15402: lemma setprod_timesf:
nipkow@15402:   "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
nipkow@15402: by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsubsection {* Properties in more restricted classes of structures *}
nipkow@15402: 
nipkow@15402: lemma setprod_eq_1_iff [simp]:
nipkow@15402:     "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
nipkow@15402:   by (induct set: Finites) auto
nipkow@15402: 
nipkow@15402: lemma setprod_zero:
nipkow@15402:      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
nipkow@15402:   apply (induct set: Finites, force, clarsimp)
nipkow@15402:   apply (erule disjE, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_nonneg [rule_format]:
nipkow@15402:      "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
nipkow@15402:   apply (case_tac "finite A")
nipkow@15402:   apply (induct set: Finites, force, clarsimp)
nipkow@15402:   apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
nipkow@15402:   apply (rule mult_mono, assumption+)
nipkow@15402:   apply (auto simp add: setprod_def)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
nipkow@15402:      --> 0 < setprod f A"
nipkow@15402:   apply (case_tac "finite A")
nipkow@15402:   apply (induct set: Finites, force, clarsimp)
nipkow@15402:   apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
nipkow@15402:   apply (rule mult_strict_mono, assumption+)
nipkow@15402:   apply (auto simp add: setprod_def)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_nonzero [rule_format]:
nipkow@15402:     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402:       finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
nipkow@15402:   apply (erule finite_induct, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_zero_eq:
nipkow@15402:     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
nipkow@15402:      finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
nipkow@15402:   apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_nonzero_field:
nipkow@15402:     "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
nipkow@15402:   apply (rule setprod_nonzero, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_zero_eq_field:
nipkow@15402:     "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
nipkow@15402:   apply (rule setprod_zero_eq, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
nipkow@15402:     (setprod f (A Un B) :: 'a ::{field})
nipkow@15402:       = setprod f A * setprod f B / setprod f (A Int B)"
nipkow@15402:   apply (subst setprod_Un_Int [symmetric], auto)
nipkow@15402:   apply (subgoal_tac "finite (A Int B)")
nipkow@15402:   apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
nipkow@15402:   apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
nipkow@15402:     (setprod f (A - {a}) :: 'a :: {field}) =
nipkow@15402:       (if a:A then setprod f A / f a else setprod f A)"
nipkow@15402:   apply (erule finite_induct)
nipkow@15402:    apply (auto simp add: insert_Diff_if)
nipkow@15402:   apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
nipkow@15402:   apply (erule ssubst)
nipkow@15402:   apply (subst times_divide_eq_right [THEN sym])
nipkow@15402:   apply (auto simp add: mult_ac times_divide_eq_right divide_self)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_inversef: "finite A ==>
nipkow@15402:     ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
nipkow@15402:       setprod (inverse \<circ> f) A = inverse (setprod f A)"
nipkow@15402:   apply (erule finite_induct)
nipkow@15402:   apply (simp, simp)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_dividef:
nipkow@15402:      "[|finite A;
nipkow@15402:         \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
nipkow@15402:       ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
nipkow@15402:   apply (subgoal_tac
nipkow@15402:          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
nipkow@15402:   apply (erule ssubst)
nipkow@15402:   apply (subst divide_inverse)
nipkow@15402:   apply (subst setprod_timesf)
nipkow@15402:   apply (subst setprod_inversef, assumption+, rule refl)
nipkow@15402:   apply (rule setprod_cong, rule refl)
nipkow@15402:   apply (subst divide_inverse, auto)
nipkow@15402:   done
nipkow@15402: 
wenzelm@12396: subsection {* Finite cardinality *}
wenzelm@12396: 
nipkow@15402: text {* This definition, although traditional, is ugly to work with:
nipkow@15402: @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
nipkow@15402: But now that we have @{text setsum} things are easy:
wenzelm@12396: *}
wenzelm@12396: 
wenzelm@12396: constdefs
wenzelm@12396:   card :: "'a set => nat"
nipkow@15402:   "card A == setsum (%x. 1::nat) A"
wenzelm@12396: 
wenzelm@12396: lemma card_empty [simp]: "card {} = 0"
nipkow@15402:   by (simp add: card_def)
nipkow@15402: 
nipkow@15402: lemma card_eq_setsum: "card A = setsum (%x. 1) A"
nipkow@15402: by (simp add: card_def)
wenzelm@12396: 
wenzelm@12396: lemma card_insert_disjoint [simp]:
wenzelm@12396:   "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
nipkow@15402: by(simp add: card_def ACf.fold_insert[OF ACf_add])
nipkow@15402: 
nipkow@15402: lemma card_insert_if:
nipkow@15402:     "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
nipkow@15402:   by (simp add: insert_absorb)
wenzelm@12396: 
wenzelm@12396: lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396:   apply auto
paulson@14208:   apply (drule_tac a = x in mk_disjoint_insert, clarify)
nipkow@15402:   apply (auto)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
nipkow@14302: apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
nipkow@14302: apply(simp del:insert_Diff_single)
nipkow@14302: done
wenzelm@12396: 
wenzelm@12396: lemma card_Diff_singleton:
wenzelm@12396:     "finite A ==> x: A ==> card (A - {x}) = card A - 1"
wenzelm@12396:   by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396: 
wenzelm@12396: lemma card_Diff_singleton_if:
wenzelm@12396:     "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
wenzelm@12396:   by (simp add: card_Diff_singleton)
wenzelm@12396: 
wenzelm@12396: lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
wenzelm@12396:   by (simp add: card_insert_if card_Suc_Diff1)
wenzelm@12396: 
wenzelm@12396: lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
wenzelm@12396:   by (simp add: card_insert_if)
wenzelm@12396: 
nipkow@15402: lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
nipkow@15402: by (simp add: card_def setsum_mono2_nat)
nipkow@15402: 
wenzelm@12396: lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
paulson@14208:   apply (induct set: Finites, simp, clarify)
wenzelm@12396:   apply (subgoal_tac "finite A & A - {x} <= F")
paulson@14208:    prefer 2 apply (blast intro: finite_subset, atomize)
wenzelm@12396:   apply (drule_tac x = "A - {x}" in spec)
wenzelm@12396:   apply (simp add: card_Diff_singleton_if split add: split_if_asm)
paulson@14208:   apply (case_tac "card A", auto)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
wenzelm@12396:   apply (simp add: psubset_def linorder_not_le [symmetric])
wenzelm@12396:   apply (blast dest: card_seteq)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma card_Un_Int: "finite A ==> finite B
wenzelm@12396:     ==> card A + card B = card (A Un B) + card (A Int B)"
nipkow@15402: by(simp add:card_def setsum_Un_Int)
wenzelm@12396: 
wenzelm@12396: lemma card_Un_disjoint: "finite A ==> finite B
wenzelm@12396:     ==> A Int B = {} ==> card (A Un B) = card A + card B"
wenzelm@12396:   by (simp add: card_Un_Int)
wenzelm@12396: 
wenzelm@12396: lemma card_Diff_subset:
nipkow@15402:   "finite B ==> B <= A ==> card (A - B) = card A - card B"
nipkow@15402: by(simp add:card_def setsum_diff_nat)
wenzelm@12396: 
wenzelm@12396: lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
wenzelm@12396:   apply (rule Suc_less_SucD)
wenzelm@12396:   apply (simp add: card_Suc_Diff1)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma card_Diff2_less:
wenzelm@12396:     "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
wenzelm@12396:   apply (case_tac "x = y")
wenzelm@12396:    apply (simp add: card_Diff1_less)
wenzelm@12396:   apply (rule less_trans)
wenzelm@12396:    prefer 2 apply (auto intro!: card_Diff1_less)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
wenzelm@12396:   apply (case_tac "x : A")
wenzelm@12396:    apply (simp_all add: card_Diff1_less less_imp_le)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
paulson@14208: by (erule psubsetI, blast)
wenzelm@12396: 
paulson@14889: lemma insert_partition:
nipkow@15402:   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
nipkow@15402:   \<Longrightarrow> x \<inter> \<Union> F = {}"
paulson@14889: by auto
paulson@14889: 
paulson@14889: (* main cardinality theorem *)
paulson@14889: lemma card_partition [rule_format]:
paulson@14889:      "finite C ==>  
paulson@14889:         finite (\<Union> C) -->  
paulson@14889:         (\<forall>c\<in>C. card c = k) -->   
paulson@14889:         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->  
paulson@14889:         k * card(C) = card (\<Union> C)"
paulson@14889: apply (erule finite_induct, simp)
paulson@14889: apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 
paulson@14889:        finite_subset [of _ "\<Union> (insert x F)"])
paulson@14889: done
paulson@14889: 
wenzelm@12396: 
nipkow@15402: lemma setsum_constant_nat:
nipkow@15402:     "finite A ==> (\<Sum>x\<in>A. y) = (card A) * y"
nipkow@15402:   -- {* Generalized to any @{text comm_semiring_1_cancel} in
nipkow@15402:         @{text IntDef} as @{text setsum_constant}. *}
nipkow@15402: by (erule finite_induct, auto)
nipkow@15402: 
nipkow@15402: lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
nipkow@15402:   apply (erule finite_induct)
nipkow@15402:   apply (auto simp add: power_Suc)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsubsection {* Cardinality of unions *}
nipkow@15402: 
nipkow@15402: lemma card_UN_disjoint:
nipkow@15402:     "finite I ==> (ALL i:I. finite (A i)) ==>
nipkow@15402:         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
nipkow@15402:       card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
nipkow@15402:   apply (simp add: card_def)
nipkow@15402:   apply (subgoal_tac
nipkow@15402:            "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
nipkow@15402:   apply (simp add: setsum_UN_disjoint)
nipkow@15402:   apply (simp add: setsum_constant_nat cong: setsum_cong)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma card_Union_disjoint:
nipkow@15402:   "finite C ==> (ALL A:C. finite A) ==>
nipkow@15402:         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
nipkow@15402:       card (Union C) = setsum card C"
nipkow@15402:   apply (frule card_UN_disjoint [of C id])
nipkow@15402:   apply (unfold Union_def id_def, assumption+)
nipkow@15402:   done
nipkow@15402: 
wenzelm@12396: subsubsection {* Cardinality of image *}
wenzelm@12396: 
wenzelm@12396: lemma card_image_le: "finite A ==> card (f ` A) <= card A"
paulson@14208:   apply (induct set: Finites, simp)
wenzelm@12396:   apply (simp add: le_SucI finite_imageI card_insert_if)
wenzelm@12396:   done
wenzelm@12396: 
nipkow@15402: lemma card_image: "inj_on f A ==> card (f ` A) = card A"
nipkow@15402: by(simp add:card_def setsum_reindex o_def)
wenzelm@12396: 
wenzelm@12396: lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
wenzelm@12396:   by (simp add: card_seteq card_image)
wenzelm@12396: 
nipkow@15111: lemma eq_card_imp_inj_on:
nipkow@15111:   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
nipkow@15111: apply(induct rule:finite_induct)
nipkow@15111:  apply simp
nipkow@15111: apply(frule card_image_le[where f = f])
nipkow@15111: apply(simp add:card_insert_if split:if_splits)
nipkow@15111: done
nipkow@15111: 
nipkow@15111: lemma inj_on_iff_eq_card:
nipkow@15111:   "finite A ==> inj_on f A = (card(f ` A) = card A)"
nipkow@15111: by(blast intro: card_image eq_card_imp_inj_on)
nipkow@15111: 
wenzelm@12396: 
nipkow@15402: lemma card_inj_on_le:
nipkow@15402:     "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
nipkow@15402: apply (subgoal_tac "finite A") 
nipkow@15402:  apply (force intro: card_mono simp add: card_image [symmetric])
nipkow@15402: apply (blast intro: finite_imageD dest: finite_subset) 
nipkow@15402: done
nipkow@15402: 
nipkow@15402: lemma card_bij_eq:
nipkow@15402:     "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
nipkow@15402:        finite A; finite B |] ==> card A = card B"
nipkow@15402:   by (auto intro: le_anti_sym card_inj_on_le)
nipkow@15402: 
nipkow@15402: 
nipkow@15402: subsubsection {* Cardinality of products *}
nipkow@15402: 
nipkow@15402: (*
nipkow@15402: lemma SigmaI_insert: "y \<notin> A ==>
nipkow@15402:   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
nipkow@15402:   by auto
nipkow@15402: *)
nipkow@15402: 
nipkow@15402: lemma card_SigmaI [simp]:
nipkow@15402:   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
nipkow@15402:   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
nipkow@15402: by(simp add:card_def setsum_Sigma)
nipkow@15402: 
nipkow@15402: (* FIXME get rid of prems *)
nipkow@15402: lemma card_cartesian_product:
nipkow@15402:      "[| finite A; finite B |] ==> card (A <*> B) = card(A) * card(B)"
nipkow@15402:   by (simp add: setsum_constant_nat)
nipkow@15402: 
nipkow@15402: (* FIXME should really be a consequence of card_cartesian_product *)
nipkow@15402: lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
nipkow@15402:   apply (subgoal_tac "inj_on (%y .(x,y)) A")
nipkow@15402:   apply (frule card_image)
nipkow@15402:   apply (subgoal_tac "(Pair x ` A) = {x} <*> A")
nipkow@15402:   apply (auto simp add: inj_on_def)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: 
wenzelm@12396: subsubsection {* Cardinality of the Powerset *}
wenzelm@12396: 
wenzelm@12396: lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
wenzelm@12396:   apply (induct set: Finites)
wenzelm@12396:    apply (simp_all add: Pow_insert)
paulson@14208:   apply (subst card_Un_disjoint, blast)
paulson@14208:     apply (blast intro: finite_imageI, blast)
wenzelm@12396:   apply (subgoal_tac "inj_on (insert x) (Pow F)")
wenzelm@12396:    apply (simp add: card_image Pow_insert)
wenzelm@12396:   apply (unfold inj_on_def)
wenzelm@12396:   apply (blast elim!: equalityE)
wenzelm@12396:   done
wenzelm@12396: 
nipkow@15392: text {* Relates to equivalence classes.  Based on a theorem of
nipkow@15392: F. Kammüller's.  *}
wenzelm@12396: 
wenzelm@12396: lemma dvd_partition:
nipkow@15392:   "finite (Union C) ==>
wenzelm@12396:     ALL c : C. k dvd card c ==>
paulson@14430:     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
wenzelm@12396:   k dvd card (Union C)"
nipkow@15392: apply(frule finite_UnionD)
nipkow@15392: apply(rotate_tac -1)
paulson@14208:   apply (induct set: Finites, simp_all, clarify)
wenzelm@12396:   apply (subst card_Un_disjoint)
wenzelm@12396:   apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
wenzelm@12396:   done
wenzelm@12396: 
wenzelm@12396: 
nipkow@15392: subsubsection {* Theorems about @{text "choose"} *}
wenzelm@12396: 
wenzelm@12396: text {*
nipkow@15392:   \medskip Basic theorem about @{text "choose"}.  By Florian
nipkow@15392:   Kamm\"uller, tidied by LCP.
wenzelm@12396: *}
wenzelm@12396: 
nipkow@15392: lemma card_s_0_eq_empty:
nipkow@15392:     "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
nipkow@15392:   apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
nipkow@15392:   apply (simp cong add: rev_conj_cong)
nipkow@15392:   done
wenzelm@12396: 
nipkow@15392: lemma choose_deconstruct: "finite M ==> x \<notin> M
nipkow@15392:   ==> {s. s <= insert x M & card(s) = Suc k}
nipkow@15392:        = {s. s <= M & card(s) = Suc k} Un
nipkow@15392:          {s. EX t. t <= M & card(t) = k & s = insert x t}"
nipkow@15392:   apply safe
nipkow@15392:    apply (auto intro: finite_subset [THEN card_insert_disjoint])
nipkow@15392:   apply (drule_tac x = "xa - {x}" in spec)
nipkow@15392:   apply (subgoal_tac "x \<notin> xa", auto)
nipkow@15392:   apply (erule rev_mp, subst card_Diff_singleton)
nipkow@15392:   apply (auto intro: finite_subset)
wenzelm@12396:   done
wenzelm@12396: 
nipkow@15392: text{*There are as many subsets of @{term A} having cardinality @{term k}
nipkow@15392:  as there are sets obtained from the former by inserting a fixed element
nipkow@15392:  @{term x} into each.*}
nipkow@15392: lemma constr_bij:
nipkow@15392:    "[|finite A; x \<notin> A|] ==>
nipkow@15392:     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
nipkow@15392:     card {B. B <= A & card(B) = k}"
nipkow@15392:   apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
nipkow@15392:        apply (auto elim!: equalityE simp add: inj_on_def)
nipkow@15392:     apply (subst Diff_insert0, auto)
nipkow@15392:    txt {* finiteness of the two sets *}
nipkow@15392:    apply (rule_tac [2] B = "Pow (A)" in finite_subset)
nipkow@15392:    apply (rule_tac B = "Pow (insert x A)" in finite_subset)
nipkow@15392:    apply fast+
wenzelm@12396:   done
wenzelm@12396: 
nipkow@15392: text {*
nipkow@15392:   Main theorem: combinatorial statement about number of subsets of a set.
nipkow@15392: *}
wenzelm@12396: 
nipkow@15392: lemma n_sub_lemma:
nipkow@15392:   "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
nipkow@15392:   apply (induct k)
nipkow@15392:    apply (simp add: card_s_0_eq_empty, atomize)
nipkow@15392:   apply (rotate_tac -1, erule finite_induct)
nipkow@15392:    apply (simp_all (no_asm_simp) cong add: conj_cong
nipkow@15392:      add: card_s_0_eq_empty choose_deconstruct)
nipkow@15392:   apply (subst card_Un_disjoint)
nipkow@15392:      prefer 4 apply (force simp add: constr_bij)
nipkow@15392:     prefer 3 apply force
nipkow@15392:    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
nipkow@15392:      finite_subset [of _ "Pow (insert x F)", standard])
nipkow@15392:   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
wenzelm@12396:   done
wenzelm@12396: 
nipkow@15392: theorem n_subsets:
nipkow@15392:     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
nipkow@15392:   by (simp add: n_sub_lemma)
nipkow@15392: 
nipkow@15392: 
nipkow@15392: subsection{* A fold functional for non-empty sets *}
nipkow@15392: 
nipkow@15392: text{* Does not require start value. *}
wenzelm@12396: 
nipkow@15392: consts
nipkow@15392:   foldSet1 :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
nipkow@15392: 
nipkow@15392: inductive "foldSet1 f"
nipkow@15392: intros
nipkow@15392: foldSet1_singletonI [intro]: "({a}, a) : foldSet1 f"
nipkow@15392: foldSet1_insertI [intro]:
nipkow@15392:  "\<lbrakk> (A, x) : foldSet1 f; a \<notin> A; A \<noteq> {} \<rbrakk>
nipkow@15392:   \<Longrightarrow> (insert a A, f a x) : foldSet1 f"
wenzelm@12396: 
nipkow@15392: constdefs
nipkow@15392:   fold1 :: "('a => 'a => 'a) => 'a set => 'a"
nipkow@15392:   "fold1 f A == THE x. (A, x) : foldSet1 f"
nipkow@15392: 
nipkow@15392: lemma foldSet1_nonempty:
nipkow@15392:  "(A, x) : foldSet1 f \<Longrightarrow> A \<noteq> {}"
nipkow@15392: by(erule foldSet1.cases, simp_all) 
nipkow@15392: 
wenzelm@12396: 
nipkow@15392: inductive_cases empty_foldSet1E [elim!]: "({}, x) : foldSet1 f"
nipkow@15392: 
nipkow@15392: lemma foldSet1_sing[iff]: "(({a},b) : foldSet1 f) = (a = b)"
nipkow@15392: apply(rule iffI)
nipkow@15392:  prefer 2 apply fast
nipkow@15392: apply (erule foldSet1.cases)
nipkow@15392:  apply blast
nipkow@15392: apply (erule foldSet1.cases)
nipkow@15392:  apply blast
nipkow@15392: apply blast
nipkow@15376: done
wenzelm@12396: 
nipkow@15392: lemma Diff1_foldSet1:
nipkow@15392:   "(A - {x}, y) : foldSet1 f ==> x: A ==> (A, f x y) : foldSet1 f"
nipkow@15392: by (erule insert_Diff [THEN subst], rule foldSet1.intros,
nipkow@15392:     auto dest!:foldSet1_nonempty)
wenzelm@12396: 
nipkow@15392: lemma foldSet1_imp_finite: "(A, x) : foldSet1 f ==> finite A"
nipkow@15392:   by (induct set: foldSet1) auto
wenzelm@12396: 
nipkow@15392: lemma finite_nonempty_imp_foldSet1:
nipkow@15392:   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : foldSet1 f"
nipkow@15392:   by (induct set: Finites) auto
nipkow@15376: 
nipkow@15392: lemma (in ACf) foldSet1_determ_aux:
nipkow@15392:   "!!A x y. \<lbrakk> card A < n; (A, x) : foldSet1 f; (A, y) : foldSet1 f \<rbrakk> \<Longrightarrow> y = x"
nipkow@15392: proof (induct n)
nipkow@15392:   case 0 thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (Suc n)
nipkow@15392:   have IH: "!!A x y. \<lbrakk>card A < n; (A, x) \<in> foldSet1 f; (A, y) \<in> foldSet1 f\<rbrakk>
nipkow@15392:            \<Longrightarrow> y = x" and card: "card A < Suc n"
nipkow@15392:   and Afoldx: "(A, x) \<in> foldSet1 f" and Afoldy: "(A, y) \<in> foldSet1 f" .
nipkow@15392:   from card have "card A < n \<or> card A = n" by arith
nipkow@15392:   thus ?case
nipkow@15392:   proof
nipkow@15392:     assume less: "card A < n"
nipkow@15392:     show ?thesis by(rule IH[OF less Afoldx Afoldy])
nipkow@15392:   next
nipkow@15392:     assume cardA: "card A = n"
nipkow@15392:     show ?thesis
nipkow@15392:     proof (rule foldSet1.cases[OF Afoldx])
nipkow@15392:       fix a assume "(A, x) = ({a}, a)"
nipkow@15392:       thus "y = x" using Afoldy by (simp add:foldSet1_sing)
nipkow@15392:     next
nipkow@15392:       fix Ax ax x'
nipkow@15392:       assume eq1: "(A, x) = (insert ax Ax, ax \<cdot> x')"
nipkow@15392: 	and x': "(Ax, x') \<in> foldSet1 f" and notinx: "ax \<notin> Ax"
nipkow@15392: 	and Axnon: "Ax \<noteq> {}"
nipkow@15392:       hence A1: "A = insert ax Ax" and x: "x = ax \<cdot> x'" by auto
nipkow@15392:       show ?thesis
nipkow@15392:       proof (rule foldSet1.cases[OF Afoldy])
nipkow@15392: 	fix ay assume "(A, y) = ({ay}, ay)"
nipkow@15392: 	thus ?thesis using eq1 x' Axnon notinx
nipkow@15392: 	  by (fastsimp simp:foldSet1_sing)
nipkow@15392:       next
nipkow@15392: 	fix Ay ay y'
nipkow@15392: 	assume eq2: "(A, y) = (insert ay Ay, ay \<cdot> y')"
nipkow@15392: 	  and y': "(Ay, y') \<in> foldSet1 f" and notiny: "ay \<notin> Ay"
nipkow@15392: 	  and Aynon: "Ay \<noteq> {}"
nipkow@15392: 	hence A2: "A = insert ay Ay" and y: "y = ay \<cdot> y'" by auto
nipkow@15392: 	have finA: "finite A" by(rule foldSet1_imp_finite[OF Afoldx])
nipkow@15392: 	with cardA A1 notinx have less: "card Ax < n" by simp
nipkow@15392: 	show ?thesis
nipkow@15392: 	proof cases
nipkow@15392: 	  assume "ax = ay"
nipkow@15392: 	  then moreover have "Ax = Ay" using A1 A2 notinx notiny by auto
nipkow@15392: 	  ultimately show ?thesis using IH[OF less x'] y' eq1 eq2 by auto
nipkow@15392: 	next
nipkow@15392: 	  assume diff: "ax \<noteq> ay"
nipkow@15392: 	  let ?B = "Ax - {ay}"
nipkow@15392: 	  have Ax: "Ax = insert ay ?B" and Ay: "Ay = insert ax ?B"
nipkow@15392: 	    using A1 A2 notinx notiny diff by(blast elim!:equalityE)+
nipkow@15392: 	  show ?thesis
nipkow@15392: 	  proof cases
nipkow@15392: 	    assume "?B = {}"
nipkow@15392: 	    with Ax Ay show ?thesis using x' y' x y by(simp add:commute)
nipkow@15392: 	  next
nipkow@15392: 	    assume Bnon: "?B \<noteq> {}"
nipkow@15392: 	    moreover have "finite ?B" using finA A1 by simp
nipkow@15392: 	    ultimately obtain b where Bfoldb: "(?B,b) \<in> foldSet1 f"
nipkow@15392: 	      using finite_nonempty_imp_foldSet1 by(blast)
nipkow@15392: 	    moreover have ayinAx: "ay \<in> Ax" using Ax by(auto)
nipkow@15392: 	    ultimately have "(Ax,ay\<cdot>b) \<in> foldSet1 f" by(rule Diff1_foldSet1)
nipkow@15392: 	    hence "ay\<cdot>b = x'" by(rule IH[OF less x'])
nipkow@15392:             moreover have "ax\<cdot>b = y'"
nipkow@15392: 	    proof (rule IH[OF _ y'])
nipkow@15392: 	      show "card Ay < n" using Ay cardA A1 notinx finA ayinAx
nipkow@15392: 		by(auto simp:card_Diff1_less)
nipkow@15392: 	    next
nipkow@15392: 	      show "(Ay,ax\<cdot>b) \<in> foldSet1 f" using Ay notinx Bfoldb Bnon
nipkow@15392: 		by fastsimp
nipkow@15392: 	    qed
nipkow@15392: 	    ultimately show ?thesis using x y by(auto simp:AC)
nipkow@15392: 	  qed
nipkow@15392: 	qed
nipkow@15392:       qed
nipkow@15392:     qed
nipkow@15392:   qed
wenzelm@12396: qed
wenzelm@12396: 
nipkow@15392: 
nipkow@15392: lemma (in ACf) foldSet1_determ:
nipkow@15392:   "(A, x) : foldSet1 f ==> (A, y) : foldSet1 f ==> y = x"
nipkow@15392: by (blast intro: foldSet1_determ_aux [rule_format])
nipkow@15392: 
nipkow@15392: lemma (in ACf) foldSet1_equality: "(A, y) : foldSet1 f ==> fold1 f A = y"
nipkow@15392:   by (unfold fold1_def) (blast intro: foldSet1_determ)
nipkow@15392: 
nipkow@15392: lemma fold1_singleton: "fold1 f {a} = a"
nipkow@15392:   by (unfold fold1_def) blast
wenzelm@12396: 
nipkow@15392: lemma (in ACf) foldSet1_insert_aux: "x \<notin> A ==> A \<noteq> {} \<Longrightarrow> 
nipkow@15392:     ((insert x A, v) : foldSet1 f) =
nipkow@15392:     (EX y. (A, y) : foldSet1 f & v = f x y)"
nipkow@15392: apply auto
nipkow@15392: apply (rule_tac A1 = A and f1 = f in finite_nonempty_imp_foldSet1 [THEN exE])
nipkow@15392:   apply (fastsimp dest: foldSet1_imp_finite)
nipkow@15392:  apply blast
nipkow@15392: apply (blast intro: foldSet1_determ)
nipkow@15392: done
nipkow@15376: 
nipkow@15392: lemma (in ACf) fold1_insert:
nipkow@15392:   "finite A ==> x \<notin> A ==> A \<noteq> {} \<Longrightarrow> fold1 f (insert x A) = f x (fold1 f A)"
nipkow@15392: apply (unfold fold1_def)
nipkow@15392: apply (simp add: foldSet1_insert_aux)
nipkow@15392: apply (rule the_equality)
nipkow@15392: apply (auto intro: finite_nonempty_imp_foldSet1
nipkow@15392:     cong add: conj_cong simp add: fold1_def [symmetric] foldSet1_equality)
nipkow@15392: done
nipkow@15376: 
nipkow@15392: locale ACIf = ACf +
nipkow@15392:   assumes idem: "x \<cdot> x = x"
wenzelm@12396: 
nipkow@15392: lemma (in ACIf) fold1_insert2:
nipkow@15392: assumes finA: "finite A" and nonA: "A \<noteq> {}"
nipkow@15392: shows "fold1 f (insert a A) = f a (fold1 f A)"
nipkow@15392: proof cases
nipkow@15392:   assume "a \<in> A"
nipkow@15392:   then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15392:     by(blast dest: mk_disjoint_insert)
nipkow@15392:   show ?thesis
nipkow@15392:   proof cases
nipkow@15392:     assume "B = {}"
nipkow@15392:     thus ?thesis using A by(simp add:idem fold1_singleton)
nipkow@15392:   next
nipkow@15392:     assume nonB: "B \<noteq> {}"
nipkow@15392:     from finA A have finB: "finite B" by(blast intro: finite_subset)
nipkow@15392:     have "fold1 f (insert a A) = fold1 f (insert a B)" using A by simp
nipkow@15392:     also have "\<dots> = f a (fold1 f B)"
nipkow@15392:       using finB nonB disj by(simp add: fold1_insert)
nipkow@15392:     also have "\<dots> = f a (fold1 f A)"
nipkow@15392:       using A finB nonB disj by(simp add:idem fold1_insert assoc[symmetric])
nipkow@15392:     finally show ?thesis .
nipkow@15392:   qed
nipkow@15392: next
nipkow@15392:   assume "a \<notin> A"
nipkow@15392:   with finA nonA show ?thesis by(simp add:fold1_insert)
nipkow@15392: qed
nipkow@15392: 
nipkow@15376: 
nipkow@15392: text{* Now the recursion rules for definitions: *}
nipkow@15392: 
nipkow@15392: lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
nipkow@15392: by(simp add:fold1_singleton)
nipkow@15392: 
nipkow@15392: lemma (in ACf) fold1_insert_def:
nipkow@15392:   "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
nipkow@15392: by(simp add:fold1_insert)
nipkow@15392: 
nipkow@15392: lemma (in ACIf) fold1_insert2_def:
nipkow@15392:   "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
nipkow@15392: by(simp add:fold1_insert2)
nipkow@15392: 
nipkow@15376: 
nipkow@15392: subsection{*Min and Max*}
nipkow@15392: 
nipkow@15392: text{* As an application of @{text fold1} we define the minimal and
nipkow@15392: maximal element of a (non-empty) set over a linear order. First we
nipkow@15392: show that @{text min} and @{text max} are ACI: *}
nipkow@15392: 
nipkow@15392: lemma ACf_min: "ACf(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392: apply(rule ACf.intro)
nipkow@15392: apply(auto simp:min_def)
nipkow@15392: done
nipkow@15392: 
nipkow@15392: lemma ACIf_min: "ACIf(min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392: apply(rule ACIf.intro[OF ACf_min])
nipkow@15392: apply(rule ACIf_axioms.intro)
nipkow@15392: apply(auto simp:min_def)
nipkow@15376: done
nipkow@15376: 
nipkow@15392: lemma ACf_max: "ACf(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392: apply(rule ACf.intro)
nipkow@15392: apply(auto simp:max_def)
nipkow@15392: done
nipkow@15392: 
nipkow@15392: lemma ACIf_max: "ACIf(max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
nipkow@15392: apply(rule ACIf.intro[OF ACf_max])
nipkow@15392: apply(rule ACIf_axioms.intro)
nipkow@15392: apply(auto simp:max_def)
nipkow@15376: done
wenzelm@12396: 
nipkow@15392: text{* Now we make the definitions: *}
nipkow@15392: 
nipkow@15392: constdefs
nipkow@15392:   Min :: "('a::linorder)set => 'a"
nipkow@15392:   "Min  ==  fold1 min"
nipkow@15392: 
nipkow@15392:   Max :: "('a::linorder)set => 'a"
nipkow@15392:   "Max  ==  fold1 max"
nipkow@15392: 
nipkow@15402: text{* Now we instantiate the recursion equations and declare them
nipkow@15392: simplification rules: *}
nipkow@15392: 
nipkow@15392: declare
nipkow@15392:   fold1_singleton_def[OF Min_def, simp]
nipkow@15392:   ACIf.fold1_insert2_def[OF ACIf_min Min_def, simp]
nipkow@15392:   fold1_singleton_def[OF Max_def, simp]
nipkow@15392:   ACIf.fold1_insert2_def[OF ACIf_max Max_def, simp]
nipkow@15392: 
nipkow@15392: text{* Now we prove some properties by induction: *}
nipkow@15392: 
nipkow@15392: lemma Min_in [simp]:
nipkow@15392:   assumes a: "finite S"
nipkow@15392:   shows "S \<noteq> {} \<Longrightarrow> Min S \<in> S"
nipkow@15392: using a
nipkow@15392: proof induct
nipkow@15392:   case empty thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (insert x S)
nipkow@15392:   show ?case
nipkow@15392:   proof cases
nipkow@15392:     assume "S = {}" thus ?thesis by simp
nipkow@15392:   next
nipkow@15392:     assume "S \<noteq> {}" thus ?thesis using insert by (simp add:min_def)
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: lemma Min_le [simp]:
nipkow@15392:   assumes a: "finite S"
nipkow@15392:   shows "\<lbrakk> S \<noteq> {}; x \<in> S \<rbrakk> \<Longrightarrow> Min S \<le> x"
nipkow@15392: using a
nipkow@15392: proof induct
nipkow@15392:   case empty thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (insert y S)
nipkow@15392:   show ?case
nipkow@15392:   proof cases
nipkow@15392:     assume "S = {}" thus ?thesis using insert by simp
nipkow@15392:   next
nipkow@15392:     assume "S \<noteq> {}" thus ?thesis using insert by (auto simp add:min_def)
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: lemma Max_in [simp]:
nipkow@15392:   assumes a: "finite S"
nipkow@15392:   shows "S \<noteq> {} \<Longrightarrow> Max S \<in> S"
nipkow@15392: using a
nipkow@15392: proof induct
nipkow@15392:   case empty thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (insert x S)
nipkow@15392:   show ?case
nipkow@15392:   proof cases
nipkow@15392:     assume "S = {}" thus ?thesis by simp
nipkow@15392:   next
nipkow@15392:     assume "S \<noteq> {}" thus ?thesis using insert by (simp add:max_def)
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: 
nipkow@15392: lemma Max_le [simp]:
nipkow@15392:   assumes a: "finite S"
nipkow@15392:   shows "\<lbrakk> S \<noteq> {}; x \<in> S \<rbrakk> \<Longrightarrow> x \<le> Max S"
nipkow@15392: using a
nipkow@15392: proof induct
nipkow@15392:   case empty thus ?case by simp
nipkow@15392: next
nipkow@15392:   case (insert y S)
nipkow@15392:   show ?case
nipkow@15392:   proof cases
nipkow@15392:     assume "S = {}" thus ?thesis using insert by simp
nipkow@15392:   next
nipkow@15392:     assume "S \<noteq> {}" thus ?thesis using insert by (auto simp add:max_def)
nipkow@15392:   qed
nipkow@15392: qed
nipkow@15392: 
wenzelm@12396: 
nipkow@15042: end