wenzelm@12396: (*  Title:      HOL/Finite_Set.thy
wenzelm@12396:     ID:         $Id$
wenzelm@12396:     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
avigad@16775:                 with contributions by Jeremy Avigad
wenzelm@12396: *)
wenzelm@12396: 
wenzelm@12396: header {* Finite sets *}
wenzelm@12396: 
nipkow@15131: theory Finite_Set
haftmann@24728: imports Divides
nipkow@15131: begin
wenzelm@12396: 
nipkow@15392: subsection {* Definition and basic properties *}
wenzelm@12396: 
berghofe@23736: inductive finite :: "'a set => bool"
berghofe@22262:   where
berghofe@22262:     emptyI [simp, intro!]: "finite {}"
berghofe@22262:   | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
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: 
berghofe@22262: lemma finite_induct [case_names empty insert, induct set: finite]:
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@23389:     show "P {}" by fact
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: 
nipkow@15484: lemma finite_ne_induct[case_names singleton insert, consumes 2]:
nipkow@15484: assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
nipkow@15484:  \<lbrakk> \<And>x. P{x};
nipkow@15484:    \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
nipkow@15484:  \<Longrightarrow> P F"
nipkow@15484: using fin
nipkow@15484: proof induct
nipkow@15484:   case empty thus ?case by simp
nipkow@15484: next
nipkow@15484:   case (insert x F)
nipkow@15484:   show ?case
nipkow@15484:   proof cases
wenzelm@23389:     assume "F = {}"
wenzelm@23389:     thus ?thesis using `P {x}` by simp
nipkow@15484:   next
wenzelm@23389:     assume "F \<noteq> {}"
wenzelm@23389:     thus ?thesis using insert by blast
nipkow@15484:   qed
nipkow@15484: qed
nipkow@15484: 
wenzelm@12396: lemma finite_subset_induct [consumes 2, case_names empty insert]:
wenzelm@23389:   assumes "finite F" and "F \<subseteq> A"
wenzelm@23389:     and empty: "P {}"
wenzelm@23389:     and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
wenzelm@23389:   shows "P F"
wenzelm@12396: proof -
wenzelm@23389:   from `finite F` and `F \<subseteq> A`
wenzelm@23389:   show ?thesis
wenzelm@12396:   proof induct
wenzelm@23389:     show "P {}" by fact
wenzelm@23389:   next
wenzelm@23389:     fix x F
wenzelm@23389:     assume "finite F" and "x \<notin> F" and
wenzelm@23389:       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@23389:       show "finite F" by fact
wenzelm@23389:       show "x \<notin> F" by fact
wenzelm@12396:     qed
wenzelm@12396:   qed
wenzelm@12396: qed
wenzelm@12396: 
haftmann@23878: 
nipkow@15392: text{* Finite sets are the images of initial segments of natural numbers: *}
nipkow@15392: 
paulson@15510: lemma finite_imp_nat_seg_image_inj_on:
paulson@15510:   assumes fin: "finite A" 
paulson@15510:   shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
nipkow@15392: using fin
nipkow@15392: proof induct
nipkow@15392:   case empty
paulson@15510:   show ?case  
paulson@15510:   proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp 
paulson@15510:   qed
nipkow@15392: next
nipkow@15392:   case (insert a A)
wenzelm@23389:   have notinA: "a \<notin> A" by fact
paulson@15510:   from insert.hyps obtain n f
paulson@15510:     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
paulson@15510:   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
paulson@15510:         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
paulson@15510:     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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})"
paulson@15510: by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
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. *}
berghofe@22262:   by (induct set: finite) 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@23389:     have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
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@23389:       also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
wenzelm@12396:       finally show ?thesis .
wenzelm@12396:     next
wenzelm@23389:       show "A \<subseteq> F ==> ?thesis" by fact
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: 
nipkow@18423: lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
nipkow@17761: using finite_subset[of "{x \<in> A. P x}" "A"] by blast
nipkow@17761: 
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@23389:   assumes "finite A"
wenzelm@23389:     and "P A"
wenzelm@23389:     and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
wenzelm@23389:   shows "P {}"
wenzelm@12396: proof -
wenzelm@12396:   have "P (A - A)"
wenzelm@12396:   proof -
wenzelm@23389:     {
wenzelm@23389:       fix c b :: "'a set"
wenzelm@23389:       assume c: "finite c" and b: "finite b"
wenzelm@23389: 	and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
wenzelm@23389:       have "c \<subseteq> b ==> P (b - c)"
wenzelm@23389: 	using c
wenzelm@23389:       proof induct
wenzelm@23389: 	case empty
wenzelm@23389: 	from P1 show ?case by simp
wenzelm@23389:       next
wenzelm@23389: 	case (insert x F)
wenzelm@23389: 	have "P (b - F - {x})"
wenzelm@23389: 	proof (rule P2)
wenzelm@23389:           from _ b show "finite (b - F)" by (rule finite_subset) blast
wenzelm@23389:           from insert show "x \<in> b - F" by simp
wenzelm@23389:           from insert show "P (b - F)" by simp
wenzelm@23389: 	qed
wenzelm@23389: 	also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
wenzelm@23389: 	finally show ?case .
wenzelm@12396:       qed
wenzelm@23389:     }
wenzelm@23389:     then show ?thesis by this (simp_all add: assms)
wenzelm@12396:   qed
wenzelm@23389:   then show ?thesis 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: 
paulson@19870: lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"
paulson@19870:   by simp
paulson@19870: 
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. *}
berghofe@22262:   by (induct set: finite) 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. *}
berghofe@22262:   apply (induct set: finite)
wenzelm@21575:    apply 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)"
berghofe@22262:   by (induct set: finite) 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@17022: lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
nipkow@17022: by (simp add: Plus_def)
nipkow@17022: 
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: 
paulson@15409: lemma finite_cartesian_productD1:
paulson@15409:      "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
paulson@15409: apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409: apply (drule_tac x=n in spec) 
paulson@15409: apply (drule_tac x="fst o f" in spec) 
paulson@15409: apply (auto simp add: o_def) 
paulson@15409:  prefer 2 apply (force dest!: equalityD2) 
paulson@15409: apply (drule equalityD1) 
paulson@15409: apply (rename_tac y x)
paulson@15409: apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409:  prefer 2 apply force
paulson@15409: apply clarify
paulson@15409: apply (rule_tac x=k in image_eqI, auto)
paulson@15409: done
paulson@15409: 
paulson@15409: lemma finite_cartesian_productD2:
paulson@15409:      "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
paulson@15409: apply (auto simp add: finite_conv_nat_seg_image) 
paulson@15409: apply (drule_tac x=n in spec) 
paulson@15409: apply (drule_tac x="snd o f" in spec) 
paulson@15409: apply (auto simp add: o_def) 
paulson@15409:  prefer 2 apply (force dest!: equalityD2) 
paulson@15409: apply (drule equalityD1)
paulson@15409: apply (rename_tac x y)
paulson@15409: apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") 
paulson@15409:  prefer 2 apply force
paulson@15409: apply clarify
paulson@15409: apply (rule_tac x=k in image_eqI, auto)
paulson@15409: done
paulson@15409: 
paulson@15409: 
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"}. *}
berghofe@22262:   apply (induct set: finite)
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: 
haftmann@26041: subsection {* Class @{text finite} and code generation *}
haftmann@26041: 
haftmann@26041: lemma finite_code [code func]:
haftmann@26041:   "finite {} \<longleftrightarrow> True"
haftmann@26041:   "finite (insert a A) \<longleftrightarrow> finite A"
haftmann@26041:   by auto
haftmann@26041: 
haftmann@26041: setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
haftmann@26041: class finite (attach UNIV) = type +
haftmann@26041:   fixes itself :: "'a itself"
haftmann@26041:   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
haftmann@26041: setup {* Sign.parent_path *}
haftmann@26041: hide const finite
haftmann@26041: 
haftmann@26041: lemma finite [simp]: "finite (A \<Colon> 'a\<Colon>finite set)"
haftmann@26041:   by (rule finite_subset [OF subset_UNIV finite_UNIV])
haftmann@26041: 
haftmann@26041: lemma univ_unit [noatp]:
haftmann@26041:   "UNIV = {()}" by auto
haftmann@26041: 
haftmann@26041: instantiation unit :: finite
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: definition
haftmann@26041:   "itself = TYPE(unit)"
haftmann@26041: 
haftmann@26041: instance proof
haftmann@26041:   have "finite {()}" by simp
haftmann@26041:   also note univ_unit [symmetric]
haftmann@26041:   finally show "finite (UNIV :: unit set)" .
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemmas [code func] = univ_unit
haftmann@26041: 
haftmann@26041: lemma univ_bool [noatp]:
haftmann@26041:   "UNIV = {False, True}" by auto
haftmann@26041: 
haftmann@26041: instantiation bool :: finite
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: definition
haftmann@26041:   "itself = TYPE(bool)"
haftmann@26041: 
haftmann@26041: instance proof
haftmann@26041:   have "finite {False, True}" by simp
haftmann@26041:   also note univ_bool [symmetric]
haftmann@26041:   finally show "finite (UNIV :: bool set)" .
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemmas [code func] = univ_bool
haftmann@26041: 
haftmann@26041: instantiation * :: (finite, finite) finite
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: definition
haftmann@26041:   "itself = TYPE('a \<times> 'b)"
haftmann@26041: 
haftmann@26041: instance proof
haftmann@26041:   show "finite (UNIV :: ('a \<times> 'b) set)"
haftmann@26041:   proof (rule finite_Prod_UNIV)
haftmann@26041:     show "finite (UNIV :: 'a set)" by (rule finite)
haftmann@26041:     show "finite (UNIV :: 'b set)" by (rule finite)
haftmann@26041:   qed
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemma univ_prod [noatp, code func]:
haftmann@26041:   "UNIV = (UNIV \<Colon> 'a\<Colon>finite set) \<times> (UNIV \<Colon> 'b\<Colon>finite set)"
haftmann@26041:   unfolding UNIV_Times_UNIV ..
haftmann@26041: 
haftmann@26041: instantiation "+" :: (finite, finite) finite
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: definition
haftmann@26041:   "itself = TYPE('a + 'b)"
haftmann@26041: 
haftmann@26041: instance proof
haftmann@26041:   have a: "finite (UNIV :: 'a set)" by (rule finite)
haftmann@26041:   have b: "finite (UNIV :: 'b set)" by (rule finite)
haftmann@26041:   from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"
haftmann@26041:     by (rule finite_Plus)
haftmann@26041:   thus "finite (UNIV :: ('a + 'b) set)" by simp
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemma univ_sum [noatp, code func]:
haftmann@26041:   "UNIV = (UNIV \<Colon> 'a\<Colon>finite set) <+> (UNIV \<Colon> 'b\<Colon>finite set)"
haftmann@26041:   unfolding UNIV_Plus_UNIV ..
haftmann@26041: 
haftmann@26041: instantiation set :: (finite) finite
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: definition
haftmann@26041:   "itself = TYPE('a set)"
haftmann@26041: 
haftmann@26041: instance proof
haftmann@26041:   have "finite (UNIV :: 'a set)" by (rule finite)
haftmann@26041:   hence "finite (Pow (UNIV :: 'a set))"
haftmann@26041:     by (rule finite_Pow_iff [THEN iffD2])
haftmann@26041:   thus "finite (UNIV :: 'a set set)" by simp
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemma univ_set [noatp, code func]:
haftmann@26041:   "UNIV = Pow (UNIV \<Colon> 'a\<Colon>finite set)" unfolding Pow_UNIV ..
haftmann@26041: 
haftmann@26041: lemma inj_graph: "inj (%f. {(x, y). y = f x})"
haftmann@26041:   by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
haftmann@26041: 
haftmann@26041: instantiation "fun" :: (finite, finite) finite
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: definition
haftmann@26041:   "itself \<equiv> TYPE('a \<Rightarrow> 'b)"
haftmann@26041: 
haftmann@26041: instance proof
haftmann@26041:   show "finite (UNIV :: ('a => 'b) set)"
haftmann@26041:   proof (rule finite_imageD)
haftmann@26041:     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
haftmann@26041:     show "finite (range ?graph)" by (rule finite)
haftmann@26041:     show "inj ?graph" by (rule inj_graph)
haftmann@26041:   qed
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: hide (open) const itself
haftmann@26041: 
haftmann@26041: subsection {* Equality and order on functions *}
haftmann@26041: 
haftmann@26041: instance "fun" :: (finite, eq) eq ..
haftmann@26041: 
haftmann@26041: lemma eq_fun [code func]:
haftmann@26041:   fixes f g :: "'a\<Colon>finite \<Rightarrow> 'b\<Colon>eq"
haftmann@26041:   shows "f = g \<longleftrightarrow> (\<forall>x\<in>UNIV. f x = g x)"
haftmann@26041:   unfolding expand_fun_eq by auto
haftmann@26041: 
haftmann@26041: lemma order_fun [code func]:
haftmann@26041:   fixes f g :: "'a\<Colon>finite \<Rightarrow> 'b\<Colon>order"
haftmann@26041:   shows "f \<le> g \<longleftrightarrow> (\<forall>x\<in>UNIV. f x \<le> g x)"
haftmann@26041:     and "f < g \<longleftrightarrow> f \<le> g \<and> (\<exists>x\<in>UNIV. f x \<noteq> g x)"
haftmann@26041:   by (auto simp add: expand_fun_eq le_fun_def less_fun_def order_less_le)
haftmann@26041: 
haftmann@26041: 
nipkow@15392: subsection {* A fold functional for finite sets *}
nipkow@15392: 
nipkow@15392: text {* The intended behaviour is
nipkow@15480: @{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<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: 
berghofe@23736: inductive
berghofe@22262:   foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool"
berghofe@22262:   for f ::  "'a => 'a => 'a"
berghofe@22262:   and g :: "'b => 'a"
berghofe@22262:   and z :: 'a
berghofe@22262: where
berghofe@22262:   emptyI [intro]: "foldSet f g z {} z"
berghofe@22262: | insertI [intro]:
berghofe@22262:      "\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk>
berghofe@22262:       \<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)"
berghofe@22262: 
berghofe@23736: inductive_cases empty_foldSetE [elim!]: "foldSet f g z {} x"
nipkow@15392: 
nipkow@15392: constdefs
nipkow@21733:   fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
berghofe@22262:   "fold f g z A == THE x. foldSet f g z A x"
nipkow@15392: 
paulson@15498: text{*A tempting alternative for the definiens is
berghofe@22262: @{term "if finite A then THE x. foldSet f g e A x else e"}.
paulson@15498: It allows the removal of finiteness assumptions from the theorems
paulson@15498: @{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
paulson@15498: The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
paulson@15498: 
paulson@15498: 
nipkow@15392: lemma Diff1_foldSet:
berghofe@22262:   "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)"
nipkow@15392: by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
nipkow@15392: 
berghofe@22262: lemma foldSet_imp_finite: "foldSet f g z A x==> finite A"
nipkow@15392:   by (induct set: foldSet) auto
nipkow@15392: 
berghofe@22262: lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x"
berghofe@22262:   by (induct set: finite) auto
nipkow@15392: 
nipkow@15392: 
nipkow@15392: subsubsection{*From @{term foldSet} to @{term fold}*}
nipkow@15392: 
paulson@15510: lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
wenzelm@19868:   by (auto simp add: less_Suc_eq) 
paulson@15510: 
paulson@15510: lemma insert_image_inj_on_eq:
paulson@15510:      "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; 
paulson@15510:         inj_on h {i. i < Suc m}|] 
paulson@15510:       ==> A = h ` {i. i < m}"
paulson@15510: apply (auto simp add: image_less_Suc inj_on_def)
paulson@15510: apply (blast intro: less_trans) 
paulson@15510: done
paulson@15510: 
paulson@15510: lemma insert_inj_onE:
paulson@15510:   assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" 
paulson@15510:       and inj_on: "inj_on h {i::nat. i<n}"
paulson@15510:   shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
paulson@15510: proof (cases n)
paulson@15510:   case 0 thus ?thesis using aA by auto
paulson@15510: next
paulson@15510:   case (Suc m)
wenzelm@23389:   have nSuc: "n = Suc m" by fact
paulson@15510:   have mlessn: "m<n" by (simp add: nSuc)
paulson@15532:   from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
paulson@15520:   let ?hm = "swap k m h"
paulson@15520:   have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn 
paulson@15520:     by (simp add: inj_on_swap_iff inj_on)
paulson@15510:   show ?thesis
paulson@15520:   proof (intro exI conjI)
paulson@15520:     show "inj_on ?hm {i. i < m}" using inj_hm
paulson@15510:       by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
paulson@15520:     show "m<n" by (rule mlessn)
paulson@15520:     show "A = ?hm ` {i. i < m}" 
paulson@15520:     proof (rule insert_image_inj_on_eq)
paulson@15520:       show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
paulson@15520:       show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) 
paulson@15520:       show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
paulson@15520: 	using aA hkeq nSuc klessn
paulson@15520: 	by (auto simp add: swap_def image_less_Suc fun_upd_image 
paulson@15520: 			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
nipkow@15479:     qed
nipkow@15479:   qed
nipkow@15479: qed
nipkow@15479: 
haftmann@26041: context ab_semigroup_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma foldSet_determ_aux:
paulson@15510:   "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; 
haftmann@26041:                 foldSet times g z A x; foldSet times g z A x' \<rbrakk>
nipkow@15392:    \<Longrightarrow> x' = x"
paulson@15510: proof (induct n rule: less_induct)
paulson@15510:   case (less n)
paulson@15510:     have IH: "!!m h A x x'. 
paulson@15510:                \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; 
haftmann@26041:                 foldSet times g z A x; foldSet times g z A x'\<rbrakk> \<Longrightarrow> x' = x" by fact
haftmann@26041:     have Afoldx: "foldSet times g z A x" and Afoldx': "foldSet times g z A x'"
wenzelm@23389:      and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
paulson@15510:     show ?case
paulson@15510:     proof (rule foldSet.cases [OF Afoldx])
berghofe@22262:       assume "A = {}" and "x = z"
paulson@15510:       with Afoldx' show "x' = x" by blast
nipkow@15392:     next
paulson@15510:       fix B b u
haftmann@26041:       assume AbB: "A = insert b B" and x: "x = g b * u"
haftmann@26041:          and notinB: "b \<notin> B" and Bu: "foldSet times g z B u"
paulson@15510:       show "x'=x" 
paulson@15510:       proof (rule foldSet.cases [OF Afoldx'])
berghofe@22262:         assume "A = {}" and "x' = z"
paulson@15510:         with AbB show "x' = x" by blast
nipkow@15392:       next
paulson@15510: 	fix C c v
haftmann@26041: 	assume AcC: "A = insert c C" and x': "x' = g c * v"
haftmann@26041:            and notinC: "c \<notin> C" and Cv: "foldSet times g z C v"
paulson@15510: 	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
paulson@15510:         from insert_inj_onE [OF Beq notinB injh]
paulson@15510:         obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" 
paulson@15510:                      and Beq: "B = hB ` {i. i < mB}"
paulson@15510:                      and lessB: "mB < n" by auto 
paulson@15510: 	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
paulson@15510:         from insert_inj_onE [OF Ceq notinC injh]
paulson@15510:         obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
paulson@15510:                        and Ceq: "C = hC ` {i. i < mC}"
paulson@15510:                        and lessC: "mC < n" by auto 
paulson@15510: 	show "x'=x"
nipkow@15392: 	proof cases
paulson@15510:           assume "b=c"
paulson@15510: 	  then moreover have "B = C" using AbB AcC notinB notinC by auto
paulson@15510: 	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
paulson@15510:             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"
paulson@15510: 	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
nipkow@15402: 	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
paulson@15510: 	  with AbB have "finite ?D" by simp
haftmann@26041: 	  then obtain d where Dfoldd: "foldSet times g z ?D d"
nipkow@17589: 	    using finite_imp_foldSet by iprover
paulson@15506: 	  moreover have cinB: "c \<in> B" using B by auto
haftmann@26041: 	  ultimately have "foldSet times g z B (g c * d)"
nipkow@15392: 	    by(rule Diff1_foldSet)
haftmann@26041: 	  then have "g c * d = u" by (rule IH [OF lessB Beq inj_onB Bu]) 
haftmann@26041:           then have "u = g c * d" ..
haftmann@26041:           moreover have "v = g b * d"
haftmann@26041: 	  proof (rule sym, rule IH [OF lessC Ceq inj_onC Cv])
haftmann@26041: 	    show "foldSet times g z C (g b * d)" using C notinB Dfoldd
nipkow@15392: 	      by fastsimp
nipkow@15392: 	  qed
haftmann@26041: 	  ultimately show ?thesis using x x'
haftmann@26041: 	    by (simp add: mult_left_commute)
nipkow@15392: 	qed
nipkow@15392:       qed
nipkow@15392:     qed
nipkow@15392:   qed
nipkow@15392: 
haftmann@26041: lemma foldSet_determ:
haftmann@26041:   "foldSet times g z A x ==> foldSet times g z A y ==> y = x"
paulson@15510: apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) 
paulson@15510: apply (blast intro: foldSet_determ_aux [rule_format])
nipkow@15392: done
nipkow@15392: 
haftmann@26041: lemma fold_equality: "foldSet times g z A y ==> fold times g z 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: 
haftmann@26041: lemma (in -) fold_empty [simp]: "fold f g z {} = z"
nipkow@15392:   by (unfold fold_def) blast
nipkow@15392: 
haftmann@26041: lemma fold_insert_aux: "x \<notin> A ==>
haftmann@26041:     (foldSet times g z (insert x A) v) =
haftmann@26041:     (EX y. foldSet times g z A y & v = g x * y)"
nipkow@15392:   apply auto
haftmann@26041:   apply (rule_tac A1 = A and f1 = times 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: 
haftmann@26041: lemma fold_insert [simp]:
haftmann@26041:     "finite A ==> x \<notin> A ==> fold times g z (insert x A) = g x * fold times g z 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: 
haftmann@26041: lemma fold_rec:
nipkow@15535: assumes fin: "finite A" and a: "a:A"
haftmann@26041: shows "fold times g z A = g a * fold times g z (A - {a})"
nipkow@15535: proof-
nipkow@15535:   have A: "A = insert a (A - {a})" using a by blast
haftmann@26041:   hence "fold times g z A = fold times g z (insert a (A - {a}))" by simp
haftmann@26041:   also have "\<dots> = g a * fold times g z (A - {a})"
nipkow@15535:     by(rule fold_insert) (simp add:fin)+
nipkow@15535:   finally show ?thesis .
nipkow@15535: qed
nipkow@15535: 
haftmann@26041: end
nipkow@15392: 
nipkow@15480: text{* A simplified version for idempotent functions: *}
nipkow@15480: 
haftmann@26041: context ab_semigroup_idem_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma fold_insert_idem:
nipkow@15480: assumes finA: "finite A"
haftmann@26041: shows "fold times g z (insert a A) = g a * fold times g z A"
nipkow@15480: proof cases
nipkow@15480:   assume "a \<in> A"
nipkow@15480:   then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
nipkow@15480:     by(blast dest: mk_disjoint_insert)
nipkow@15480:   show ?thesis
nipkow@15480:   proof -
nipkow@15480:     from finA A have finB: "finite B" by(blast intro: finite_subset)
haftmann@26041:     have "fold times g z (insert a A) = fold times g z (insert a B)" using A by simp
haftmann@26041:     also have "\<dots> = g a * fold times g z B"
paulson@15506:       using finB disj by simp
haftmann@26041:     also have "\<dots> = g a * fold times g z A"
haftmann@26041:       using A finB disj
haftmann@26041: 	by (simp add: mult_idem mult_assoc [symmetric])
nipkow@15480:     finally show ?thesis .
nipkow@15480:   qed
nipkow@15480: next
nipkow@15480:   assume "a \<notin> A"
nipkow@15480:   with finA show ?thesis by simp
nipkow@15480: qed
nipkow@15480: 
haftmann@26041: lemma foldI_conv_id:
haftmann@26041:   "finite A \<Longrightarrow> fold times g z A = fold times id z (g ` A)"
paulson@15509: by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
nipkow@15484: 
haftmann@26041: end
haftmann@26041: 
nipkow@15392: subsubsection{*Lemmas about @{text fold}*}
nipkow@15392: 
haftmann@26041: context ab_semigroup_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma fold_commute:
haftmann@26041:   "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
berghofe@22262:   apply (induct set: finite)
wenzelm@21575:    apply simp
haftmann@26041:   apply (simp add: mult_left_commute [of x])
nipkow@15392:   done
nipkow@15392: 
haftmann@26041: lemma fold_nest_Un_Int:
nipkow@15392:   "finite A ==> finite B
haftmann@26041:     ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
berghofe@22262:   apply (induct set: finite)
wenzelm@21575:    apply simp
nipkow@15392:   apply (simp add: fold_commute Int_insert_left insert_absorb)
nipkow@15392:   done
nipkow@15392: 
haftmann@26041: lemma fold_nest_Un_disjoint:
nipkow@15392:   "finite A ==> finite B ==> A Int B = {}
haftmann@26041:     ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
nipkow@15392:   by (simp add: fold_nest_Un_Int)
nipkow@15392: 
haftmann@26041: lemma fold_reindex:
paulson@15487: assumes fin: "finite A"
haftmann@26041: shows "inj_on h A \<Longrightarrow> fold times g z (h ` A) = fold times (g \<circ> h) z A"
paulson@15506: using fin apply induct
nipkow@15392:  apply simp
nipkow@15392: apply simp
nipkow@15392: done
nipkow@15392: 
haftmann@26041: text{*
haftmann@26041:   Fusion theorem, as described in Graham Hutton's paper,
haftmann@26041:   A Tutorial on the Universality and Expressiveness of Fold,
haftmann@26041:   JFP 9:4 (355-372), 1999.
haftmann@26041: *}
haftmann@26041: 
haftmann@26041: lemma fold_fusion:
haftmann@26041:   includes ab_semigroup_mult g
haftmann@26041:   assumes fin: "finite A"
haftmann@26041:     and hyp: "\<And>x y. h (g x y) = times x (h y)"
haftmann@26041:   shows "h (fold g j w A) = fold times j (h w) A"
haftmann@26041:   using fin hyp by (induct set: finite) simp_all
haftmann@26041: 
haftmann@26041: lemma fold_cong:
haftmann@26041:   "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold times g z A = fold times h z A"
haftmann@26041:   apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold times g z C = fold times h z 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: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: context comm_monoid_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma fold_Un_Int:
haftmann@26041:   "finite A ==> finite B ==>
haftmann@26041:     fold times g 1 A * fold times g 1 B =
haftmann@26041:     fold times g 1 (A Un B) * fold times g 1 (A Int B)"
haftmann@26041:   by (induct set: finite) 
haftmann@26041:     (auto simp add: mult_ac insert_absorb Int_insert_left)
haftmann@26041: 
haftmann@26041: corollary fold_Un_disjoint:
haftmann@26041:   "finite A ==> finite B ==> A Int B = {} ==>
haftmann@26041:     fold times g 1 (A Un B) = fold times g 1 A * fold times g 1 B"
haftmann@26041:   by (simp add: fold_Un_Int)
haftmann@26041: 
haftmann@26041: lemma fold_UN_disjoint:
haftmann@26041:   "\<lbrakk> finite I; ALL i:I. finite (A i);
haftmann@26041:      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
haftmann@26041:    \<Longrightarrow> fold times g 1 (UNION I A) =
haftmann@26041:        fold times (%i. fold times g 1 (A i)) 1 I"
haftmann@26041:   apply (induct set: finite, simp, atomize)
haftmann@26041:   apply (subgoal_tac "ALL i:F. x \<noteq> i")
haftmann@26041:    prefer 2 apply blast
haftmann@26041:   apply (subgoal_tac "A x Int UNION F A = {}")
haftmann@26041:    prefer 2 apply blast
haftmann@26041:   apply (simp add: fold_Un_disjoint)
haftmann@26041:   done
haftmann@26041: 
haftmann@26041: lemma fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
haftmann@26041:   fold times (%x. fold times (g x) 1 (B x)) 1 A =
haftmann@26041:   fold times (split g) 1 (SIGMA x:A. B x)"
nipkow@15392: apply (subst Sigma_def)
paulson@15506: apply (subst fold_UN_disjoint, assumption, simp)
nipkow@15392:  apply blast
nipkow@15392: apply (erule fold_cong)
paulson@15506: apply (subst fold_UN_disjoint, simp, simp)
nipkow@15392:  apply blast
paulson@15506: apply simp
nipkow@15392: done
nipkow@15392: 
haftmann@26041: lemma fold_distrib: "finite A \<Longrightarrow>
haftmann@26041:    fold times (%x. g x * h x) 1 A = fold times g 1 A *  fold times h 1 A"
haftmann@26041:   by (erule finite_induct) (simp_all add: mult_ac)
haftmann@26041: 
haftmann@26041: end
haftmann@22917: 
haftmann@22917: 
nipkow@15402: subsection {* Generalized summation over a set *}
nipkow@15402: 
haftmann@26041: interpretation comm_monoid_add: comm_monoid_mult ["0::'a::comm_monoid_add" "op +"]
haftmann@26041:   by unfold_locales (auto intro: add_assoc add_commute)
haftmann@26041: 
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: 
wenzelm@19535: abbreviation
wenzelm@21404:   Setsum  ("\<Sum>_" [1000] 999) where
wenzelm@19535:   "\<Sum>A == setsum (%x. x) A"
wenzelm@19535: 
nipkow@15402: text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
nipkow@15402: written @{text"\<Sum>x\<in>A. e"}. *}
nipkow@15402: 
nipkow@15402: syntax
paulson@17189:   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (xsymbols)
paulson@17189:   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (HTML output)
paulson@17189:   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: 
nipkow@15402: translations -- {* Beware of argument permutation! *}
nipkow@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
paulson@17189:   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
nipkow@15402: syntax (xsymbols)
paulson@17189:   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402: syntax (HTML output)
paulson@17189:   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
nipkow@15402: 
nipkow@15402: translations
nipkow@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: print_translation {*
nipkow@15402: let
wenzelm@19535:   fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 
wenzelm@19535:     if x<>y then raise Match
wenzelm@19535:     else let val x' = Syntax.mark_bound x
wenzelm@19535:              val t' = subst_bound(x',t)
wenzelm@19535:              val P' = subst_bound(x',P)
wenzelm@19535:          in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
wenzelm@19535: in [("setsum", setsum_tr')] end
nipkow@15402: *}
nipkow@15402: 
wenzelm@19535: 
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"
ballarin@15765:   by (simp add: setsum_def)
nipkow@15402: 
paulson@15409: lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
paulson@15409:   by (simp add: setsum_def)
paulson@15409: 
nipkow@15402: lemma setsum_reindex:
nipkow@15402:      "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
haftmann@26041: by(auto simp add: setsum_def comm_monoid_add.fold_reindex 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"
haftmann@26041: by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_cong)
nipkow@15402: 
nipkow@16733: lemma strong_setsum_cong[cong]:
nipkow@16733:   "A = B ==> (!!x. x:B =simp=> f x = g x)
nipkow@16733:    ==> setsum (%x. f x) A = setsum (%x. g x) B"
haftmann@26041: by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_cong)
berghofe@16632: 
nipkow@15554: lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
nipkow@15554:   by (rule setsum_cong[OF refl], auto);
nipkow@15554: 
nipkow@15402: lemma setsum_reindex_cong:
nipkow@15554:      "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> 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@15542: lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
nipkow@15402: apply (clarsimp simp: setsum_def)
ballarin@15765: apply (erule finite_induct, auto)
nipkow@15402: done
nipkow@15402: 
nipkow@15543: lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
nipkow@15543: by(simp add:setsum_cong)
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! *}
haftmann@26041: by(simp add: setsum_def comm_monoid_add.fold_Un_Int [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: 
paulson@15409: (*But we can't get rid of finite I. If infinite, although the rhs is 0, 
paulson@15409:   the lhs need not be, since UNION I A could still be finite.*)
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))"
haftmann@26041: by(simp add: setsum_def comm_monoid_add.fold_UN_disjoint cong: setsum_cong)
nipkow@15402: 
paulson@15409: text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
paulson@15409: directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
nipkow@15402: lemma setsum_Union_disjoint:
paulson@15409:   "[| (ALL A:C. finite A);
paulson@15409:       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
paulson@15409:    ==> setsum f (Union C) = setsum (setsum f) C"
paulson@15409: apply (cases "finite C") 
paulson@15409:  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
nipkow@15402:   apply (frule setsum_UN_disjoint [of C id f])
paulson@15409:  apply (unfold Union_def id_def, assumption+)
paulson@15409: done
nipkow@15402: 
paulson@15409: (*But we can't get rid of finite A. If infinite, although the lhs is 0, 
paulson@15409:   the rhs need not be, since SIGMA A B could still be finite.*)
nipkow@15402: lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
paulson@17189:     (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
haftmann@26041: by(simp add:setsum_def comm_monoid_add.fold_Sigma split_def cong:setsum_cong)
nipkow@15402: 
paulson@15409: text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409: lemma setsum_cartesian_product: 
paulson@17189:    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
paulson@15409: apply (cases "finite A") 
paulson@15409:  apply (cases "finite B") 
paulson@15409:   apply (simp add: setsum_Sigma)
paulson@15409:  apply (cases "A={}", simp)
nipkow@15543:  apply (simp) 
paulson@15409: apply (auto simp add: setsum_def
paulson@15409:             dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409: done
nipkow@15402: 
nipkow@15402: lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
haftmann@26041: by(simp add:setsum_def comm_monoid_add.fold_distrib)
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))"
berghofe@22262:   by (induct set: finite) 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@23477:   by (subst setsum_Un_Int [symmetric], auto simp add: ring_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@23477:   by (subst setsum_Un_Int [symmetric], auto simp add: ring_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: 
obua@15552: lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
obua@15552:   apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
obua@15552:   apply (auto simp add: insert_Diff_if add_ac)
obua@15552:   done
obua@15552: 
nipkow@15402: (* By Jeremy Siek: *)
nipkow@15402: 
nipkow@15402: lemma setsum_diff_nat: 
wenzelm@19535:   assumes "finite B"
wenzelm@19535:     and "B \<subseteq> A"
wenzelm@19535:   shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
wenzelm@19535:   using prems
wenzelm@19535: proof induct
nipkow@15402:   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
nipkow@15402: next
nipkow@15402:   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
nipkow@15402:     and xFinA: "insert x F \<subseteq> A"
nipkow@15402:     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
nipkow@15402:   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
nipkow@15402:   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
nipkow@15402:     by (simp add: setsum_diff1_nat)
nipkow@15402:   from xFinA have "F \<subseteq> A" by simp
nipkow@15402:   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
nipkow@15402:   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
nipkow@15402:     by simp
nipkow@15402:   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
nipkow@15402:   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
nipkow@15402:     by simp
nipkow@15402:   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
nipkow@15402:   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
nipkow@15402:     by simp
nipkow@15402:   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
nipkow@15402: qed
nipkow@15402: 
nipkow@15402: lemma setsum_diff:
nipkow@15402:   assumes le: "finite A" "B \<subseteq> A"
nipkow@15402:   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
nipkow@15402: proof -
nipkow@15402:   from le have finiteB: "finite B" using finite_subset by auto
nipkow@15402:   show ?thesis using finiteB le
wenzelm@21575:   proof induct
wenzelm@19535:     case empty
wenzelm@19535:     thus ?case by auto
wenzelm@19535:   next
wenzelm@19535:     case (insert x F)
wenzelm@19535:     thus ?case using le finiteB 
wenzelm@19535:       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
nipkow@15402:   qed
wenzelm@19535: qed
nipkow@15402: 
nipkow@15402: lemma setsum_mono:
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
wenzelm@19535:   proof induct
nipkow@15402:     case empty
nipkow@15402:     thus ?case by simp
nipkow@15402:   next
nipkow@15402:     case insert
wenzelm@19535:     thus ?case using add_mono by fastsimp
nipkow@15402:   qed
nipkow@15402: next
nipkow@15402:   case False
nipkow@15402:   thus ?thesis
nipkow@15402:     by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15554: lemma setsum_strict_mono:
wenzelm@19535:   fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
wenzelm@19535:   assumes "finite A"  "A \<noteq> {}"
wenzelm@19535:     and "!!x. x:A \<Longrightarrow> f x < g x"
wenzelm@19535:   shows "setsum f A < setsum g A"
wenzelm@19535:   using prems
nipkow@15554: proof (induct rule: finite_ne_induct)
nipkow@15554:   case singleton thus ?case by simp
nipkow@15554: next
nipkow@15554:   case insert thus ?case by (auto simp: add_strict_mono)
nipkow@15554: qed
nipkow@15554: 
nipkow@15535: lemma setsum_negf:
wenzelm@19535:   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
nipkow@15535: proof (cases "finite A")
berghofe@22262:   case True thus ?thesis by (induct set: finite) auto
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15535: lemma setsum_subtractf:
wenzelm@19535:   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
wenzelm@19535:     setsum f A - setsum g A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15535: lemma setsum_nonneg:
wenzelm@19535:   assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
wenzelm@19535:   shows "0 \<le> setsum f A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True thus ?thesis using nn
wenzelm@21575:   proof induct
wenzelm@19535:     case empty then show ?case by simp
wenzelm@19535:   next
wenzelm@19535:     case (insert x F)
wenzelm@19535:     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
wenzelm@19535:     with insert show ?case by simp
wenzelm@19535:   qed
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15535: lemma setsum_nonpos:
wenzelm@19535:   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
wenzelm@19535:   shows "setsum f A \<le> 0"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True thus ?thesis using np
wenzelm@21575:   proof induct
wenzelm@19535:     case empty then show ?case by simp
wenzelm@19535:   next
wenzelm@19535:     case (insert x F)
wenzelm@19535:     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
wenzelm@19535:     with insert show ?case by simp
wenzelm@19535:   qed
nipkow@15535: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15535: qed
nipkow@15402: 
nipkow@15539: lemma setsum_mono2:
nipkow@15539: fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
nipkow@15539: assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
nipkow@15539: shows "setsum f A \<le> setsum f B"
nipkow@15539: proof -
nipkow@15539:   have "setsum f A \<le> setsum f A + setsum f (B-A)"
nipkow@15539:     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
nipkow@15539:   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
nipkow@15539:     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
nipkow@15539:   also have "A \<union> (B-A) = B" using sub by blast
nipkow@15539:   finally show ?thesis .
nipkow@15539: qed
nipkow@15542: 
avigad@16775: lemma setsum_mono3: "finite B ==> A <= B ==> 
avigad@16775:     ALL x: B - A. 
avigad@16775:       0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
avigad@16775:         setsum f A <= setsum f B"
avigad@16775:   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
avigad@16775:   apply (erule ssubst)
avigad@16775:   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
avigad@16775:   apply simp
avigad@16775:   apply (rule add_left_mono)
avigad@16775:   apply (erule setsum_nonneg)
avigad@16775:   apply (subst setsum_Un_disjoint [THEN sym])
avigad@16775:   apply (erule finite_subset, assumption)
avigad@16775:   apply (rule finite_subset)
avigad@16775:   prefer 2
avigad@16775:   apply assumption
avigad@16775:   apply auto
avigad@16775:   apply (rule setsum_cong)
avigad@16775:   apply auto
avigad@16775: done
avigad@16775: 
ballarin@19279: lemma setsum_right_distrib: 
huffman@22934:   fixes f :: "'a => ('b::semiring_0)"
nipkow@15402:   shows "r * setsum f A = setsum (%n. r * f n) A"
nipkow@15402: proof (cases "finite A")
nipkow@15402:   case True
nipkow@15402:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15402:     case empty thus ?case by simp
nipkow@15402:   next
nipkow@15402:     case (insert x A) thus ?case by (simp add: right_distrib)
nipkow@15402:   qed
nipkow@15402: next
nipkow@15402:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
ballarin@17149: lemma setsum_left_distrib:
huffman@22934:   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
ballarin@17149: proof (cases "finite A")
ballarin@17149:   case True
ballarin@17149:   then show ?thesis
ballarin@17149:   proof induct
ballarin@17149:     case empty thus ?case by simp
ballarin@17149:   next
ballarin@17149:     case (insert x A) thus ?case by (simp add: left_distrib)
ballarin@17149:   qed
ballarin@17149: next
ballarin@17149:   case False thus ?thesis by (simp add: setsum_def)
ballarin@17149: qed
ballarin@17149: 
ballarin@17149: lemma setsum_divide_distrib:
ballarin@17149:   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
ballarin@17149: proof (cases "finite A")
ballarin@17149:   case True
ballarin@17149:   then show ?thesis
ballarin@17149:   proof induct
ballarin@17149:     case empty thus ?case by simp
ballarin@17149:   next
ballarin@17149:     case (insert x A) thus ?case by (simp add: add_divide_distrib)
ballarin@17149:   qed
ballarin@17149: next
ballarin@17149:   case False thus ?thesis by (simp add: setsum_def)
ballarin@17149: qed
ballarin@17149: 
nipkow@15535: lemma setsum_abs[iff]: 
haftmann@25303:   fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
nipkow@15402:   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True
nipkow@15535:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15535:     case empty thus ?case by simp
nipkow@15535:   next
nipkow@15535:     case (insert x A)
nipkow@15535:     thus ?case by (auto intro: abs_triangle_ineq order_trans)
nipkow@15535:   qed
nipkow@15402: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15535: lemma setsum_abs_ge_zero[iff]: 
haftmann@25303:   fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
nipkow@15402:   shows "0 \<le> setsum (%i. abs(f i)) A"
nipkow@15535: proof (cases "finite A")
nipkow@15535:   case True
nipkow@15535:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15535:     case empty thus ?case by simp
nipkow@15535:   next
nipkow@21733:     case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
nipkow@15535:   qed
nipkow@15402: next
nipkow@15535:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15402: qed
nipkow@15402: 
nipkow@15539: lemma abs_setsum_abs[simp]: 
haftmann@25303:   fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
nipkow@15539:   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
nipkow@15539: proof (cases "finite A")
nipkow@15539:   case True
nipkow@15539:   thus ?thesis
wenzelm@21575:   proof induct
nipkow@15539:     case empty thus ?case by simp
nipkow@15539:   next
nipkow@15539:     case (insert a A)
nipkow@15539:     hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
nipkow@15539:     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
avigad@16775:     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
avigad@16775:       by (simp del: abs_of_nonneg)
nipkow@15539:     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
nipkow@15539:     finally show ?case .
nipkow@15539:   qed
nipkow@15539: next
nipkow@15539:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15539: qed
nipkow@15539: 
nipkow@15402: 
ballarin@17149: text {* Commuting outer and inner summation *}
ballarin@17149: 
ballarin@17149: lemma swap_inj_on:
ballarin@17149:   "inj_on (%(i, j). (j, i)) (A \<times> B)"
ballarin@17149:   by (unfold inj_on_def) fast
ballarin@17149: 
ballarin@17149: lemma swap_product:
ballarin@17149:   "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
ballarin@17149:   by (simp add: split_def image_def) blast
ballarin@17149: 
ballarin@17149: lemma setsum_commute:
ballarin@17149:   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
ballarin@17149: proof (simp add: setsum_cartesian_product)
paulson@17189:   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
paulson@17189:     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
ballarin@17149:     (is "?s = _")
ballarin@17149:     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
ballarin@17149:     apply (simp add: split_def)
ballarin@17149:     done
paulson@17189:   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
ballarin@17149:     (is "_ = ?t")
ballarin@17149:     apply (simp add: swap_product)
ballarin@17149:     done
ballarin@17149:   finally show "?s = ?t" .
ballarin@17149: qed
ballarin@17149: 
ballarin@19279: lemma setsum_product:
huffman@22934:   fixes f :: "'a => ('b::semiring_0)"
ballarin@19279:   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
ballarin@19279:   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
ballarin@19279: 
ballarin@17149: 
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: 
wenzelm@19535: abbreviation
wenzelm@21404:   Setprod  ("\<Prod>_" [1000] 999) where
wenzelm@19535:   "\<Prod>A == setprod (%x. x) A"
wenzelm@19535: 
nipkow@15402: syntax
paulson@17189:   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (xsymbols)
paulson@17189:   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@15402: syntax (HTML output)
paulson@17189:   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
nipkow@16550: 
nipkow@16550: translations -- {* Beware of argument permutation! *}
nipkow@16550:   "PROD i:A. b" == "setprod (%i. b) A" 
nipkow@16550:   "\<Prod>i\<in>A. b" == "setprod (%i. b) A" 
nipkow@16550: 
nipkow@16550: text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
nipkow@16550:  @{text"\<Prod>x|P. e"}. *}
nipkow@16550: 
nipkow@16550: syntax
paulson@17189:   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
nipkow@16550: syntax (xsymbols)
paulson@17189:   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550: syntax (HTML output)
paulson@17189:   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
nipkow@16550: 
nipkow@15402: translations
nipkow@16550:   "PROD x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550:   "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
nipkow@16550: 
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"
ballarin@19931:   by (simp add: setprod_def)
nipkow@15402: 
paulson@15409: lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
paulson@15409:   by (simp add: setprod_def)
paulson@15409: 
nipkow@15402: lemma setprod_reindex:
nipkow@15402:      "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
haftmann@26041: by(auto simp: setprod_def fold_reindex 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"
haftmann@26041: by(fastsimp simp: setprod_def intro: fold_cong)
nipkow@15402: 
berghofe@16632: lemma strong_setprod_cong:
berghofe@16632:   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
haftmann@26041: by(fastsimp simp: simp_implies_def setprod_def intro: fold_cong)
berghofe@16632: 
nipkow@15402: lemma setprod_reindex_cong: "inj_on f A ==>
nipkow@15402:     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
nipkow@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:   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"
haftmann@26041: by(simp add: setprod_def fold_Un_Int[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"
haftmann@26041: by(simp add: setprod_def fold_UN_disjoint cong: setprod_cong)
nipkow@15402: 
nipkow@15402: lemma setprod_Union_disjoint:
paulson@15409:   "[| (ALL A:C. finite A);
paulson@15409:       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] 
paulson@15409:    ==> setprod f (Union C) = setprod (setprod f) C"
paulson@15409: apply (cases "finite C") 
paulson@15409:  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
nipkow@15402:   apply (frule setprod_UN_disjoint [of C id f])
paulson@15409:  apply (unfold Union_def id_def, assumption+)
paulson@15409: done
nipkow@15402: 
nipkow@15402: lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
nipkow@16550:     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
paulson@17189:     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
haftmann@26041: by(simp add:setprod_def fold_Sigma split_def cong:setprod_cong)
nipkow@15402: 
paulson@15409: text{*Here we can eliminate the finiteness assumptions, by cases.*}
paulson@15409: lemma setprod_cartesian_product: 
paulson@17189:      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
paulson@15409: apply (cases "finite A") 
paulson@15409:  apply (cases "finite B") 
paulson@15409:   apply (simp add: setprod_Sigma)
paulson@15409:  apply (cases "A={}", simp)
paulson@15409:  apply (simp add: setprod_1) 
paulson@15409: apply (auto simp add: setprod_def
paulson@15409:             dest: finite_cartesian_productD1 finite_cartesian_productD2) 
paulson@15409: done
nipkow@15402: 
nipkow@15402: lemma setprod_timesf:
paulson@15409:      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
haftmann@26041: by(simp add:setprod_def fold_distrib)
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))"
berghofe@22262:   by (induct set: finite) auto
nipkow@15402: 
nipkow@15402: lemma setprod_zero:
huffman@23277:      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
berghofe@22262:   apply (induct set: finite, 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")
berghofe@22262:   apply (induct set: finite, 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")
berghofe@22262:   apply (induct set: finite, 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]:
huffman@23277:     "(ALL x y. (x::'a::comm_semiring_1) * 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:
huffman@23277:     "(ALL x y. (x::'a::comm_semiring_1) * 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:
huffman@23277:     "finite A ==> (ALL x: A. f x \<noteq> (0::'a::idom)) ==> setprod f A \<noteq> 0"
nipkow@15402:   apply (rule setprod_nonzero, auto)
nipkow@15402:   done
nipkow@15402: 
nipkow@15402: lemma setprod_zero_eq_field:
huffman@23277:     "finite A ==> (setprod f A = (0::'a::idom)) = (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@23398:   apply (subst times_divide_eq_right [THEN sym], auto)
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@23413: by (erule finite_induct) (auto simp add: insert_Diff_if)
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: 
haftmann@25459: definition
haftmann@25459:   card :: "'a set \<Rightarrow> nat"
haftmann@25459: where
haftmann@25459:   [code func del]: "card A = setsum (\<lambda>x. 1) A"
wenzelm@12396: 
wenzelm@12396: lemma card_empty [simp]: "card {} = 0"
nipkow@24853: by (simp add: card_def)
nipkow@15402: 
paulson@24427: lemma card_infinite [simp]: "~ finite A ==> card A = 0"
nipkow@24853: by (simp add: card_def)
paulson@15409: 
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)"
ballarin@15765: by(simp add: card_def)
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: 
paulson@24286: lemma card_0_eq [simp,noatp]: "finite A ==> (card A = 0) = (A = {})"
wenzelm@12396:   apply auto
paulson@15506:   apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
wenzelm@12396:   done
wenzelm@12396: 
paulson@15409: lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
paulson@15409: by auto
paulson@15409: 
nipkow@24853: 
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:
nipkow@24853:   "finite A ==> x: A ==> card (A - {x}) = card A - 1"
nipkow@24853: by (simp add: card_Suc_Diff1 [symmetric])
wenzelm@12396: 
wenzelm@12396: lemma card_Diff_singleton_if:
nipkow@24853:   "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
nipkow@24853: by (simp add: card_Diff_singleton)
nipkow@24853: 
nipkow@24853: lemma card_Diff_insert[simp]:
nipkow@24853: assumes "finite A" and "a:A" and "a ~: B"
nipkow@24853: shows "card(A - insert a B) = card(A - B) - 1"
nipkow@24853: proof -
nipkow@24853:   have "A - insert a B = (A - B) - {a}" using assms by blast
nipkow@24853:   then show ?thesis using assms by(simp add:card_Diff_singleton)
nipkow@24853: qed
wenzelm@12396: 
wenzelm@12396: lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
nipkow@24853: by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
wenzelm@12396: 
haftmann@26041: lemma card_code [code func]:
haftmann@26041:   "card {} = 0"
haftmann@26041:   "card (insert a A) =
haftmann@26041:     (if finite A then Suc (card (A - {a})) else card (insert a A))"
haftmann@26041:   by (auto simp add: card_insert)
haftmann@26041: 
wenzelm@12396: lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
nipkow@24853: 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@15539: by (simp add: card_def setsum_mono2)
nipkow@15402: 
wenzelm@12396: lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
berghofe@22262:   apply (induct set: finite, 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"
nipkow@24853: apply (simp add: psubset_def linorder_not_le [symmetric])
nipkow@24853: apply (blast dest: card_seteq)
nipkow@24853: 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"
nipkow@24853: 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)
nipkow@24853:   apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
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")
nipkow@24853:    apply (simp add: card_Diff1_less del:card_Diff_insert)
wenzelm@12396:   apply (rule less_trans)
nipkow@24853:    prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
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@19793: text{* 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: 
paulson@19793: text{*The form of a finite set of given cardinality*}
paulson@19793: 
paulson@19793: lemma card_eq_SucD:
nipkow@24853: assumes "card A = Suc k"
nipkow@24853: shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
paulson@19793: proof -
nipkow@24853:   have fin: "finite A" using assms by (auto intro: ccontr)
nipkow@24853:   moreover have "card A \<noteq> 0" using assms by auto
nipkow@24853:   ultimately obtain b where b: "b \<in> A" by auto
paulson@19793:   show ?thesis
paulson@19793:   proof (intro exI conjI)
paulson@19793:     show "A = insert b (A-{b})" using b by blast
paulson@19793:     show "b \<notin> A - {b}" by blast
nipkow@24853:     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
nipkow@24853:       using assms b fin by(fastsimp dest:mk_disjoint_insert)+
paulson@19793:   qed
paulson@19793: qed
paulson@19793: 
paulson@19793: lemma card_Suc_eq:
nipkow@24853:   "(card A = Suc k) =
nipkow@24853:    (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
nipkow@24853: apply(rule iffI)
nipkow@24853:  apply(erule card_eq_SucD)
nipkow@24853: apply(auto)
nipkow@24853: apply(subst card_insert)
nipkow@24853:  apply(auto intro:ccontr)
nipkow@24853: done
paulson@19793: 
nipkow@15539: lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
nipkow@15539: apply (cases "finite A")
nipkow@15539: apply (erule finite_induct)
nipkow@23477: apply (auto simp add: ring_simps)
paulson@15409: done
nipkow@15402: 
krauss@21199: lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)"
nipkow@15402:   apply (erule finite_induct)
nipkow@15402:   apply (auto simp add: power_Suc)
nipkow@15402:   done
nipkow@15402: 
nipkow@15542: lemma setsum_bounded:
huffman@23277:   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, pordered_ab_semigroup_add})"
nipkow@15542:   shows "setsum f A \<le> of_nat(card A) * K"
nipkow@15542: proof (cases "finite A")
nipkow@15542:   case True
nipkow@15542:   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
nipkow@15542: next
nipkow@15542:   case False thus ?thesis by (simp add: setsum_def)
nipkow@15542: qed
nipkow@15542: 
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@15539:   apply (simp add: card_def del: setsum_constant)
nipkow@15402:   apply (subgoal_tac
nipkow@15402:            "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
nipkow@15539:   apply (simp add: setsum_UN_disjoint del: setsum_constant)
nipkow@15539:   apply (simp 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: 
paulson@15447: text{*The image of a finite set can be expressed using @{term fold}.*}
paulson@15447: lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
haftmann@26041: proof (induct rule: finite_induct)
haftmann@26041:   case empty then show ?case by simp
haftmann@26041: next
haftmann@26041:   invoke ab_semigroup_mult ["op Un"]
haftmann@26041:     by unfold_locales auto
haftmann@26041:   case insert 
haftmann@26041:   then show ?case by simp
haftmann@26041: qed
paulson@15447: 
wenzelm@12396: lemma card_image_le: "finite A ==> card (f ` A) <= card A"
berghofe@22262:   apply (induct set: finite)
wenzelm@21575:    apply 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@15539: by(simp add:card_def setsum_reindex o_def del:setsum_constant)
wenzelm@12396: 
wenzelm@12396: lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
nipkow@25162: 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"
wenzelm@21575: apply (induct rule:finite_induct)
wenzelm@21575: 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@15539: by(simp add:card_def setsum_Sigma del:setsum_constant)
nipkow@15402: 
paulson@15409: lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
paulson@15409: apply (cases "finite A") 
paulson@15409: apply (cases "finite B") 
paulson@15409: apply (auto simp add: card_eq_0_iff
nipkow@15539:             dest: finite_cartesian_productD1 finite_cartesian_productD2)
paulson@15409: done
nipkow@15402: 
nipkow@15402: lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
nipkow@15539: by (simp add: card_cartesian_product)
paulson@15409: 
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 (!?) *)
berghofe@22262:   apply (induct set: finite)
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: 
haftmann@24342: text {* Relates to equivalence classes.  Based on a theorem of F. Kammüller.  *}
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)
berghofe@22262:   apply (induct set: finite, 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@25162: subsubsection {* Relating injectivity and surjectivity *}
nipkow@25162: 
nipkow@25162: lemma finite_surj_inj: "finite(A) \<Longrightarrow> A <= f`A \<Longrightarrow> inj_on f A"
nipkow@25162: apply(rule eq_card_imp_inj_on, assumption)
nipkow@25162: apply(frule finite_imageI)
nipkow@25162: apply(drule (1) card_seteq)
nipkow@25162: apply(erule card_image_le)
nipkow@25162: apply simp
nipkow@25162: done
nipkow@25162: 
nipkow@25162: lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
nipkow@25162: shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
nipkow@25162: by (blast intro: finite_surj_inj subset_UNIV dest:surj_range)
nipkow@25162: 
nipkow@25162: lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
nipkow@25162: shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
nipkow@25162: by(fastsimp simp:surj_def dest!: endo_inj_surj)
nipkow@25162: 
nipkow@25162: corollary infinite_UNIV_nat: "~finite(UNIV::nat set)"
nipkow@25162: proof
nipkow@25162:   assume "finite(UNIV::nat set)"
nipkow@25162:   with finite_UNIV_inj_surj[of Suc]
nipkow@25162:   show False by simp (blast dest: Suc_neq_Zero surjD)
nipkow@25162: qed
nipkow@25162: 
nipkow@25162: 
nipkow@15392: subsection{* A fold functional for non-empty sets *}
nipkow@15392: 
nipkow@15392: text{* Does not require start value. *}
wenzelm@12396: 
berghofe@23736: inductive
berghofe@22262:   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
berghofe@22262:   for f :: "'a => 'a => 'a"
berghofe@22262: where
paulson@15506:   fold1Set_insertI [intro]:
berghofe@22262:    "\<lbrakk> foldSet f id a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
wenzelm@12396: 
nipkow@15392: constdefs
nipkow@15392:   fold1 :: "('a => 'a => 'a) => 'a set => 'a"
berghofe@22262:   "fold1 f A == THE x. fold1Set f A x"
paulson@15506: 
paulson@15506: lemma fold1Set_nonempty:
haftmann@22917:   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
haftmann@22917:   by(erule fold1Set.cases, simp_all) 
nipkow@15392: 
berghofe@23736: inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
berghofe@23736: 
berghofe@23736: inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
berghofe@22262: 
berghofe@22262: 
berghofe@22262: lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
paulson@15506:   by (blast intro: foldSet.intros elim: foldSet.cases)
nipkow@15392: 
haftmann@22917: lemma fold1_singleton [simp]: "fold1 f {a} = a"
paulson@15508:   by (unfold fold1_def) blast
wenzelm@12396: 
paulson@15508: lemma finite_nonempty_imp_fold1Set:
berghofe@22262:   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
paulson@15508: apply (induct A rule: finite_induct)
paulson@15508: apply (auto dest: finite_imp_foldSet [of _ f id])  
paulson@15508: done
paulson@15506: 
paulson@15506: text{*First, some lemmas about @{term foldSet}.*}
nipkow@15392: 
haftmann@26041: context ab_semigroup_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma foldSet_insert_swap:
haftmann@26041: assumes fold: "foldSet times id b A y"
haftmann@26041: shows "b \<notin> A \<Longrightarrow> foldSet times id z (insert b A) (z * y)"
paulson@15508: using fold
paulson@15508: proof (induct rule: foldSet.induct)
haftmann@26041:   case emptyI thus ?case by (force simp add: fold_insert_aux mult_commute)
paulson@15508: next
berghofe@22262:   case (insertI x A y)
haftmann@26041:     have "foldSet times (\<lambda>u. u) z (insert x (insert b A)) (x * (z * y))"
paulson@15521:       using insertI by force  --{*how does @{term id} get unfolded?*}
haftmann@26041:     thus ?case by (simp add: insert_commute mult_ac)
paulson@15508: qed
paulson@15508: 
haftmann@26041: lemma foldSet_permute_diff:
haftmann@26041: assumes fold: "foldSet times id b A x"
haftmann@26041: shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> foldSet times id a (insert b (A-{a})) x"
paulson@15508: using fold
paulson@15508: proof (induct rule: foldSet.induct)
paulson@15508:   case emptyI thus ?case by simp
paulson@15508: next
berghofe@22262:   case (insertI x A y)
paulson@15521:   have "a = x \<or> a \<in> A" using insertI by simp
paulson@15521:   thus ?case
paulson@15521:   proof
paulson@15521:     assume "a = x"
paulson@15521:     with insertI show ?thesis
paulson@15521:       by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap) 
paulson@15521:   next
paulson@15521:     assume ainA: "a \<in> A"
haftmann@26041:     hence "foldSet times id a (insert x (insert b (A - {a}))) (x * y)"
paulson@15521:       using insertI by (force simp: id_def)
paulson@15521:     moreover
paulson@15521:     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
paulson@15521:       using ainA insertI by blast
paulson@15521:     ultimately show ?thesis by (simp add: id_def)
paulson@15508:   qed
paulson@15508: qed
paulson@15508: 
haftmann@26041: lemma fold1_eq_fold:
haftmann@26041:      "[|finite A; a \<notin> A|] ==> fold1 times (insert a A) = fold times id a A"
paulson@15508: apply (simp add: fold1_def fold_def) 
paulson@15508: apply (rule the_equality)
haftmann@26041: apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ times id]) 
paulson@15508: apply (rule sym, clarify)
paulson@15508: apply (case_tac "Aa=A")
paulson@15508:  apply (best intro: the_equality foldSet_determ)  
haftmann@26041: apply (subgoal_tac "foldSet times id a A x")
paulson@15508:  apply (best intro: the_equality foldSet_determ)  
paulson@15508: apply (subgoal_tac "insert aa (Aa - {a}) = A") 
paulson@15508:  prefer 2 apply (blast elim: equalityE) 
paulson@15508: apply (auto dest: foldSet_permute_diff [where a=a]) 
paulson@15508: done
paulson@15508: 
paulson@15521: lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
paulson@15521: apply safe
paulson@15521: apply simp 
paulson@15521: apply (drule_tac x=x in spec)
paulson@15521: apply (drule_tac x="A-{x}" in spec, auto) 
paulson@15508: done
paulson@15508: 
haftmann@26041: lemma fold1_insert:
paulson@15521:   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
haftmann@26041:   shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521: proof -
paulson@15521:   from nonempty obtain a A' where "A = insert a A' & a ~: A'" 
paulson@15521:     by (auto simp add: nonempty_iff)
paulson@15521:   with A show ?thesis
paulson@15521:     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) 
paulson@15521: qed
paulson@15521: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: context ab_semigroup_idem_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma fold1_insert_idem [simp]:
paulson@15521:   assumes nonempty: "A \<noteq> {}" and A: "finite A" 
haftmann@26041:   shows "fold1 times (insert x A) = x * fold1 times A"
paulson@15521: proof -
paulson@15521:   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" 
paulson@15521:     by (auto simp add: nonempty_iff)
paulson@15521:   show ?thesis
paulson@15521:   proof cases
paulson@15521:     assume "a = x"
paulson@15521:     thus ?thesis 
paulson@15521:     proof cases
paulson@15521:       assume "A' = {}"
haftmann@26041:       with prems show ?thesis by (simp add: mult_idem) 
paulson@15521:     next
paulson@15521:       assume "A' \<noteq> {}"
paulson@15521:       with prems show ?thesis
haftmann@26041: 	by (simp add: fold1_insert mult_assoc [symmetric] mult_idem) 
paulson@15521:     qed
paulson@15521:   next
paulson@15521:     assume "a \<noteq> x"
paulson@15521:     with prems show ?thesis
paulson@15521:       by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
paulson@15521:   qed
paulson@15521: qed
paulson@15506: 
haftmann@26041: lemma hom_fold1_commute:
haftmann@26041: assumes hom: "!!x y. h (x * y) = h x * h y"
haftmann@26041: and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
haftmann@22917: using N proof (induct rule: finite_ne_induct)
haftmann@22917:   case singleton thus ?case by simp
haftmann@22917: next
haftmann@22917:   case (insert n N)
haftmann@26041:   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
haftmann@26041:   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
haftmann@26041:   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
haftmann@26041:   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
haftmann@22917:     using insert by(simp)
haftmann@22917:   also have "insert (h n) (h ` N) = h ` insert n N" by simp
haftmann@22917:   finally show ?case .
haftmann@22917: qed
haftmann@22917: 
haftmann@26041: end
haftmann@26041: 
paulson@15506: 
paulson@15508: text{* Now the recursion rules for definitions: *}
paulson@15508: 
haftmann@22917: lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
paulson@15508: by(simp add:fold1_singleton)
paulson@15508: 
haftmann@26041: lemma (in ab_semigroup_mult) fold1_insert_def:
haftmann@26041:   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041: by (simp add:fold1_insert)
haftmann@26041: 
haftmann@26041: lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
haftmann@26041:   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
haftmann@26041: by simp
paulson@15508: 
paulson@15508: subsubsection{* Determinacy for @{term fold1Set} *}
paulson@15508: 
paulson@15508: text{*Not actually used!!*}
wenzelm@12396: 
haftmann@26041: context ab_semigroup_mult
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma foldSet_permute:
haftmann@26041:   "[|foldSet times id b (insert a A) x; a \<notin> A; b \<notin> A|]
haftmann@26041:    ==> foldSet times id a (insert b A) x"
haftmann@26041: apply (cases "a=b") 
paulson@15506: apply (auto dest: foldSet_permute_diff) 
paulson@15506: done
nipkow@15376: 
haftmann@26041: lemma fold1Set_determ:
haftmann@26041:   "fold1Set times A x ==> fold1Set times A y ==> y = x"
paulson@15506: proof (clarify elim!: fold1Set.cases)
paulson@15506:   fix A x B y a b
haftmann@26041:   assume Ax: "foldSet times id a A x"
haftmann@26041:   assume By: "foldSet times id b B y"
paulson@15506:   assume anotA:  "a \<notin> A"
paulson@15506:   assume bnotB:  "b \<notin> B"
paulson@15506:   assume eq: "insert a A = insert b B"
paulson@15506:   show "y=x"
paulson@15506:   proof cases
paulson@15506:     assume same: "a=b"
paulson@15506:     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
paulson@15506:     thus ?thesis using Ax By same by (blast intro: foldSet_determ)
nipkow@15392:   next
paulson@15506:     assume diff: "a\<noteq>b"
paulson@15506:     let ?D = "B - {a}"
paulson@15506:     have B: "B = insert a ?D" and A: "A = insert b ?D"
paulson@15506:      and aB: "a \<in> B" and bA: "b \<in> A"
paulson@15506:       using eq anotA bnotB diff by (blast elim!:equalityE)+
paulson@15506:     with aB bnotB By
haftmann@26041:     have "foldSet times id a (insert b ?D) y" 
paulson@15506:       by (auto intro: foldSet_permute simp add: insert_absorb)
paulson@15506:     moreover
haftmann@26041:     have "foldSet times id a (insert b ?D) x"
paulson@15506:       by (simp add: A [symmetric] Ax) 
paulson@15506:     ultimately show ?thesis by (blast intro: foldSet_determ) 
nipkow@15392:   qed
wenzelm@12396: qed
wenzelm@12396: 
haftmann@26041: lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
paulson@15506:   by (unfold fold1_def) (blast intro: fold1Set_determ)
paulson@15506: 
haftmann@26041: end
haftmann@26041: 
paulson@15506: declare
paulson@15506:   empty_foldSetE [rule del]   foldSet.intros [rule del]
paulson@15506:   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
ballarin@19931:   -- {* No more proofs involve these relations. *}
nipkow@15376: 
haftmann@26041: subsubsection {* Lemmas about @{text fold1} *}
haftmann@26041: 
haftmann@26041: context ab_semigroup_mult
haftmann@22917: begin
haftmann@22917: 
haftmann@26041: lemma fold1_Un:
nipkow@15484: assumes A: "finite A" "A \<noteq> {}"
nipkow@15484: shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
haftmann@26041:        fold1 times (A Un B) = fold1 times A * fold1 times B"
haftmann@26041: using A by (induct rule: finite_ne_induct)
haftmann@26041:   (simp_all add: fold1_insert mult_assoc)
haftmann@26041: 
haftmann@26041: lemma fold1_in:
haftmann@26041:   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
haftmann@26041:   shows "fold1 times A \<in> A"
nipkow@15484: using A
nipkow@15484: proof (induct rule:finite_ne_induct)
paulson@15506:   case singleton thus ?case by simp
nipkow@15484: next
nipkow@15484:   case insert thus ?case using elem by (force simp add:fold1_insert)
nipkow@15484: qed
nipkow@15484: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemma (in ab_semigroup_idem_mult) fold1_Un2:
nipkow@15497: assumes A: "finite A" "A \<noteq> {}"
haftmann@26041: shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
haftmann@26041:        fold1 times (A Un B) = fold1 times A * fold1 times B"
nipkow@15497: using A
haftmann@26041: proof(induct rule:finite_ne_induct)
nipkow@15497:   case singleton thus ?case by simp
nipkow@15484: next
haftmann@26041:   case insert thus ?case by (simp add: mult_assoc)
nipkow@18423: qed
nipkow@18423: 
nipkow@18423: 
haftmann@22917: subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *}
haftmann@22917: 
haftmann@22917: text{*
haftmann@22917:   As an application of @{text fold1} we define infimum
haftmann@22917:   and supremum in (not necessarily complete!) lattices
haftmann@22917:   over (non-empty) sets by means of @{text fold1}.
haftmann@22917: *}
haftmann@22917: 
haftmann@26041: context lower_semilattice
haftmann@26041: begin
haftmann@26041: 
haftmann@26041: lemma ab_semigroup_idem_mult_inf:
haftmann@26041:   "ab_semigroup_idem_mult inf"
haftmann@26041:   apply unfold_locales
haftmann@26041:   apply (rule inf_assoc)
haftmann@26041:   apply (rule inf_commute)
haftmann@26041:   apply (rule inf_idem)
haftmann@26041:   done
haftmann@26041: 
haftmann@26041: lemma below_fold1_iff:
haftmann@26041:   assumes "finite A" "A \<noteq> {}"
haftmann@26041:   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [inf]
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
haftmann@26041:   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_belowI:
haftmann@26041:   assumes "finite A" "A \<noteq> {}"
haftmann@26041:     and "a \<in> A"
haftmann@26041:   shows "fold1 inf A \<le> a"
haftmann@26041: using assms proof (induct rule: finite_ne_induct)
haftmann@26041:   case singleton thus ?case by simp
haftmann@26041: next
haftmann@26041:   invoke ab_semigroup_idem_mult [inf]
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
haftmann@26041:   case (insert x F)
haftmann@26041:   from insert(5) have "a = x \<or> a \<in> F" by simp
haftmann@26041:   thus ?case
haftmann@26041:   proof
haftmann@26041:     assume "a = x" thus ?thesis using insert
haftmann@26041:       by (simp add: mult_ac_idem)
haftmann@26041:   next
haftmann@26041:     assume "a \<in> F"
haftmann@26041:     hence bel: "fold1 inf F \<le> a" by (rule insert)
haftmann@26041:     have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
haftmann@26041:       using insert by (simp add: mult_ac_idem)
haftmann@26041:     also have "inf (fold1 inf F) a = fold1 inf F"
haftmann@26041:       using bel by (auto intro: antisym)
haftmann@26041:     also have "inf x \<dots> = fold1 inf (insert x F)"
haftmann@26041:       using insert by (simp add: mult_ac_idem)
haftmann@26041:     finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
haftmann@26041:     moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
haftmann@26041:     ultimately show ?thesis by simp
haftmann@26041:   qed
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: end
haftmann@26041: 
haftmann@26041: lemma (in upper_semilattice) ab_semigroup_idem_mult_sup:
haftmann@26041:   "ab_semigroup_idem_mult sup"
haftmann@26041:   by (rule lower_semilattice.ab_semigroup_idem_mult_inf)
haftmann@26041:     (rule dual_lattice)
nipkow@15500: 
haftmann@24342: context lattice
haftmann@22917: begin
haftmann@22917: 
haftmann@22917: definition
haftmann@24342:   Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900)
haftmann@22917: where
haftmann@25062:   "Inf_fin = fold1 inf"
haftmann@22917: 
haftmann@22917: definition
haftmann@24342:   Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900)
haftmann@22917: where
haftmann@25062:   "Sup_fin = fold1 sup"
haftmann@25062: 
haftmann@25062: lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
haftmann@24342: apply(unfold Sup_fin_def Inf_fin_def)
nipkow@15500: apply(subgoal_tac "EX a. a:A")
nipkow@15500: prefer 2 apply blast
nipkow@15500: apply(erule exE)
haftmann@22388: apply(rule order_trans)
haftmann@26041: apply(erule (2) fold1_belowI)
haftmann@26041: apply(erule (2) lower_semilattice.fold1_belowI [OF dual_lattice])
nipkow@15500: done
nipkow@15500: 
haftmann@24342: lemma sup_Inf_absorb [simp]:
haftmann@25062:   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (sup a (\<Sqinter>\<^bsub>fin\<^esub>A)) = a"
nipkow@15512: apply(subst sup_commute)
haftmann@26041: apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
nipkow@15504: done
nipkow@15504: 
haftmann@24342: lemma inf_Sup_absorb [simp]:
haftmann@25062:   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (inf a (\<Squnion>\<^bsub>fin\<^esub>A)) = a"
haftmann@26041: by (simp add: Sup_fin_def inf_absorb1
haftmann@26041:   lower_semilattice.fold1_belowI [OF dual_lattice])
haftmann@24342: 
haftmann@24342: end
haftmann@24342: 
haftmann@24342: context distrib_lattice
haftmann@24342: begin
haftmann@24342: 
haftmann@24342: lemma sup_Inf1_distrib:
haftmann@26041:   assumes "finite A"
haftmann@26041:     and "A \<noteq> {}"
haftmann@26041:   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [inf]
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Inf_fin_def image_def
haftmann@26041:       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
haftmann@26041:         (rule arg_cong, blast)
haftmann@26041: qed
nipkow@18423: 
haftmann@24342: lemma sup_Inf2_distrib:
haftmann@24342:   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
haftmann@25062:   shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
haftmann@24342: using A proof (induct rule: finite_ne_induct)
nipkow@15500:   case singleton thus ?case
haftmann@24342:     by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def])
nipkow@15500: next
haftmann@26041:   invoke ab_semigroup_idem_mult [inf]
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
nipkow@15500:   case (insert x A)
haftmann@25062:   have finB: "finite {sup x b |b. b \<in> B}"
haftmann@25062:     by(rule finite_surj[where f = "sup x", OF B(1)], auto)
haftmann@25062:   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
nipkow@15500:   proof -
haftmann@25062:     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
nipkow@15500:       by blast
berghofe@15517:     thus ?thesis by(simp add: insert(1) B(1))
nipkow@15500:   qed
haftmann@25062:   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@25062:   have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
haftmann@26041:     using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def])
haftmann@25062:   also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
haftmann@25062:   also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
nipkow@15500:     using insert by(simp add:sup_Inf1_distrib[OF B])
haftmann@25062:   also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
haftmann@24342:     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
nipkow@15500:     using B insert
haftmann@26041:     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
haftmann@25062:   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@15500:     by blast
nipkow@15500:   finally show ?case .
nipkow@15500: qed
nipkow@15500: 
haftmann@24342: lemma inf_Sup1_distrib:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [sup]
haftmann@26041:     by (rule ab_semigroup_idem_mult_sup)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
haftmann@26041:       (rule arg_cong, blast)
haftmann@26041: qed
nipkow@18423: 
haftmann@24342: lemma inf_Sup2_distrib:
haftmann@24342:   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
haftmann@25062:   shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"
haftmann@24342: using A proof (induct rule: finite_ne_induct)
nipkow@18423:   case singleton thus ?case
haftmann@24342:     by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def])
nipkow@18423: next
nipkow@18423:   case (insert x A)
haftmann@25062:   have finB: "finite {inf x b |b. b \<in> B}"
haftmann@25062:     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
haftmann@25062:   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
nipkow@18423:   proof -
haftmann@25062:     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
nipkow@18423:       by blast
nipkow@18423:     thus ?thesis by(simp add: insert(1) B(1))
nipkow@18423:   qed
haftmann@25062:   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
haftmann@26041:   invoke ab_semigroup_idem_mult [sup]
haftmann@26041:     by (rule ab_semigroup_idem_mult_sup)
haftmann@25062:   have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
haftmann@26041:     using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def])
haftmann@25062:   also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
haftmann@25062:   also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
nipkow@18423:     using insert by(simp add:inf_Sup1_distrib[OF B])
haftmann@25062:   also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
haftmann@24342:     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
nipkow@18423:     using B insert
haftmann@26041:     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
haftmann@25062:   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
nipkow@18423:     by blast
nipkow@18423:   finally show ?case .
nipkow@18423: qed
nipkow@18423: 
haftmann@24342: end
haftmann@24342: 
haftmann@24342: context complete_lattice
haftmann@24342: begin
haftmann@24342: 
haftmann@22917: text {*
haftmann@24342:   Coincidence on finite sets in complete lattices:
haftmann@22917: *}
haftmann@22917: 
haftmann@24342: lemma Inf_fin_Inf:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [inf]
haftmann@26041:     by (rule ab_semigroup_idem_mult_inf)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   unfolding Inf_fin_def by (induct A set: finite)
haftmann@26041:     (simp_all add: Inf_insert_simp)
haftmann@26041: qed
haftmann@22917: 
haftmann@24342: lemma Sup_fin_Sup:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [sup]
haftmann@26041:     by (rule ab_semigroup_idem_mult_sup)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   unfolding Sup_fin_def by (induct A set: finite)
haftmann@26041:     (simp_all add: Sup_insert_simp)
haftmann@26041: qed
haftmann@22917: 
haftmann@24342: end
haftmann@24342: 
haftmann@22917: 
haftmann@22917: subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *}
haftmann@22917: 
haftmann@22917: text{*
haftmann@22917:   As an application of @{text fold1} we define minimum
haftmann@22917:   and maximum in (not necessarily complete!) linear orders
haftmann@22917:   over (non-empty) sets by means of @{text fold1}.
haftmann@22917: *}
haftmann@22917: 
haftmann@24342: context linorder
haftmann@22917: begin
haftmann@22917: 
haftmann@26041: lemma ab_semigroup_idem_mult_min:
haftmann@26041:   "ab_semigroup_idem_mult min"
haftmann@26041:   by unfold_locales (auto simp add: min_def)
haftmann@26041: 
haftmann@26041: lemma ab_semigroup_idem_mult_max:
haftmann@26041:   "ab_semigroup_idem_mult max"
haftmann@26041:   by unfold_locales (auto simp add: max_def)
haftmann@26041: 
haftmann@26041: lemma min_lattice:
haftmann@26041:   "lower_semilattice (op \<le>) (op <) min"
haftmann@26041:   by unfold_locales (auto simp add: min_def)
haftmann@26041: 
haftmann@26041: lemma max_lattice:
haftmann@26041:   "lower_semilattice (op \<ge>) (op >) max"
haftmann@26041:   by unfold_locales (auto simp add: max_def)
haftmann@26041: 
haftmann@26041: lemma dual_max:
haftmann@26041:   "ord.max (op \<ge>) = min"
haftmann@26041:   by (auto simp add: ord.max_def_raw min_def_raw expand_fun_eq)
haftmann@26041: 
haftmann@26041: lemma dual_min:
haftmann@26041:   "ord.min (op \<ge>) = max"
haftmann@26041:   by (auto simp add: ord.min_def_raw max_def_raw expand_fun_eq)
haftmann@26041: 
haftmann@26041: lemma strict_below_fold1_iff:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   by (induct rule: finite_ne_induct)
haftmann@26041:     (simp_all add: fold1_insert)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_below_iff:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   by (induct rule: finite_ne_induct)
haftmann@26041:     (simp_all add: fold1_insert min_le_iff_disj)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_strict_below_iff:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:   by (induct rule: finite_ne_induct)
haftmann@26041:     (simp_all add: fold1_insert min_less_iff_disj)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma fold1_antimono:
haftmann@26041:   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
haftmann@26041:   shows "fold1 min B \<le> fold1 min A"
haftmann@26041: proof cases
haftmann@26041:   assume "A = B" thus ?thesis by simp
haftmann@26041: next
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   assume "A \<noteq> B"
haftmann@26041:   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
haftmann@26041:   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
haftmann@26041:   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
haftmann@26041:   proof -
haftmann@26041:     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
haftmann@26041:     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
haftmann@26041:     moreover have "(B-A) \<noteq> {}" using prems by blast
haftmann@26041:     moreover have "A Int (B-A) = {}" using prems by blast
haftmann@26041:     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
haftmann@26041:   qed
haftmann@26041:   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
haftmann@26041:   finally show ?thesis .
haftmann@26041: qed
haftmann@26041: 
haftmann@22917: definition
haftmann@22917:   Min :: "'a set \<Rightarrow> 'a"
haftmann@22917: where
haftmann@22917:   "Min = fold1 min"
haftmann@22917: 
haftmann@22917: definition
haftmann@22917:   Max :: "'a set \<Rightarrow> 'a"
haftmann@22917: where
haftmann@22917:   "Max = fold1 max"
haftmann@22917: 
haftmann@22917: lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def]
haftmann@22917: lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def]
haftmann@26041: 
haftmann@26041: lemma Min_insert [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min (insert x A) = min x (Min A)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis by (rule fold1_insert_idem_def [OF Min_def])
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Max_insert [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max (insert x A) = max x (Max A)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [max]
haftmann@26041:     by (rule ab_semigroup_idem_mult_max)
haftmann@26041:   from assms show ?thesis by (rule fold1_insert_idem_def [OF Max_def])
haftmann@26041: qed
nipkow@15392: 
paulson@24427: lemma Min_in [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A \<in> A"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def)
haftmann@26041: qed
nipkow@15392: 
paulson@24427: lemma Max_in [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A \<in> A"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [max]
haftmann@26041:     by (rule ab_semigroup_idem_mult_max)
haftmann@26041:   from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Min_Un:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
haftmann@26041:   shows "Min (A \<union> B) = min (Min A) (Min B)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Min_def fold1_Un2)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Max_Un:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
haftmann@26041:   shows "Max (A \<union> B) = max (Max A) (Max B)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [max]
haftmann@26041:     by (rule ab_semigroup_idem_mult_max)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Max_def fold1_Un2)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma hom_Min_commute:
haftmann@26041:   assumes "\<And>x y. h (min x y) = min (h x) (h y)"
haftmann@26041:     and "finite N" and "N \<noteq> {}"
haftmann@26041:   shows "h (Min N) = Min (h ` N)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [min]
haftmann@26041:     by (rule ab_semigroup_idem_mult_min)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Min_def hom_fold1_commute)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma hom_Max_commute:
haftmann@26041:   assumes "\<And>x y. h (max x y) = max (h x) (h y)"
haftmann@26041:     and "finite N" and "N \<noteq> {}"
haftmann@26041:   shows "h (Max N) = Max (h ` N)"
haftmann@26041: proof -
haftmann@26041:   invoke ab_semigroup_idem_mult [max]
haftmann@26041:     by (rule ab_semigroup_idem_mult_max)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Max_def hom_fold1_commute [of h])
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Min_le [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}" and "x \<in> A"
haftmann@26041:   shows "Min A \<le> x"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ min]
haftmann@26041:     by (rule min_lattice)
haftmann@26041:   from assms show ?thesis by (simp add: Min_def fold1_belowI)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Max_ge [simp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}" and "x \<in> A"
haftmann@26041:   shows "x \<le> Max A"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ max]
haftmann@26041:     by (rule max_lattice)
haftmann@26041:   from assms show ?thesis by (simp add: Max_def fold1_belowI)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Min_ge_iff [simp, noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ min]
haftmann@26041:     by (rule min_lattice)
haftmann@26041:   from assms show ?thesis by (simp add: Min_def below_fold1_iff)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Max_le_iff [simp, noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ max]
haftmann@26041:     by (rule max_lattice)
haftmann@26041:   from assms show ?thesis by (simp add: Max_def below_fold1_iff)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Min_gr_iff [simp, noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ min]
haftmann@26041:     by (rule min_lattice)
haftmann@26041:   from assms show ?thesis by (simp add: Min_def strict_below_fold1_iff)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Max_less_iff [simp, noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
haftmann@26041: proof -
haftmann@26041:   note Max = Max_def
haftmann@26041:   invoke linorder ["op \<ge>" "op >"]
haftmann@26041:     by (rule dual_linorder)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Max strict_below_fold1_iff [folded dual_max])
haftmann@26041: qed
nipkow@18493: 
paulson@24286: lemma Min_le_iff [noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ min]
haftmann@26041:     by (rule min_lattice)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Min_def fold1_below_iff)
haftmann@26041: qed
nipkow@15497: 
paulson@24286: lemma Max_ge_iff [noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
haftmann@26041: proof -
haftmann@26041:   note Max = Max_def
haftmann@26041:   invoke linorder ["op \<ge>" "op >"]
haftmann@26041:     by (rule dual_linorder)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Max fold1_below_iff [folded dual_max])
haftmann@26041: qed
haftmann@22917: 
paulson@24286: lemma Min_less_iff [noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
haftmann@26041: proof -
haftmann@26041:   invoke lower_semilattice [_ _ min]
haftmann@26041:     by (rule min_lattice)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Min_def fold1_strict_below_iff)
haftmann@26041: qed
haftmann@22917: 
paulson@24286: lemma Max_gr_iff [noatp]:
haftmann@26041:   assumes "finite A" and "A \<noteq> {}"
haftmann@26041:   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
haftmann@26041: proof -
haftmann@26041:   note Max = Max_def
haftmann@26041:   invoke linorder ["op \<ge>" "op >"]
haftmann@26041:     by (rule dual_linorder)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Max fold1_strict_below_iff [folded dual_max])
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Min_antimono:
haftmann@26041:   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@26041:   shows "Min N \<le> Min M"
haftmann@26041: proof -
haftmann@26041:   invoke distrib_lattice ["op \<le>" "op <" min max]
haftmann@26041:     by (rule distrib_lattice_min_max)
haftmann@26041:   from assms show ?thesis by (simp add: Min_def fold1_antimono)
haftmann@26041: qed
haftmann@26041: 
haftmann@26041: lemma Max_mono:
haftmann@26041:   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
haftmann@26041:   shows "Max M \<le> Max N"
haftmann@26041: proof -
haftmann@26041:   note Max = Max_def
haftmann@26041:   invoke linorder ["op \<ge>" "op >"]
haftmann@26041:     by (rule dual_linorder)
haftmann@26041:   from assms show ?thesis
haftmann@26041:     by (simp add: Max fold1_antimono [folded dual_max])
haftmann@26041: qed
haftmann@22917: 
haftmann@22917: end
haftmann@22917: 
haftmann@24380: context ordered_ab_semigroup_add
haftmann@22917: begin
haftmann@22917: 
haftmann@22917: lemma add_Min_commute:
haftmann@22917:   fixes k
haftmann@25062:   assumes "finite N" and "N \<noteq> {}"
haftmann@25062:   shows "k + Min N = Min {k + m | m. m \<in> N}"
haftmann@25062: proof -
haftmann@25062:   have "\<And>x y. k + min x y = min (k + x) (k + y)"
haftmann@25062:     by (simp add: min_def not_le)
haftmann@25062:       (blast intro: antisym less_imp_le add_left_mono)
haftmann@25062:   with assms show ?thesis
haftmann@25062:     using hom_Min_commute [of "plus k" N]
haftmann@25062:     by simp (blast intro: arg_cong [where f = Min])
haftmann@25062: qed
haftmann@22917: 
haftmann@22917: lemma add_Max_commute:
haftmann@22917:   fixes k
haftmann@25062:   assumes "finite N" and "N \<noteq> {}"
haftmann@25062:   shows "k + Max N = Max {k + m | m. m \<in> N}"
haftmann@25062: proof -
haftmann@25062:   have "\<And>x y. k + max x y = max (k + x) (k + y)"
haftmann@25062:     by (simp add: max_def not_le)
haftmann@25062:       (blast intro: antisym less_imp_le add_left_mono)
haftmann@25062:   with assms show ?thesis
haftmann@25062:     using hom_Max_commute [of "plus k" N]
haftmann@25062:     by simp (blast intro: arg_cong [where f = Max])
haftmann@25062: qed
haftmann@22917: 
haftmann@22917: end
haftmann@22917: 
haftmann@25571: end