(* Title: HOL/Finite_Set.thy ID: $Id$ Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel Additions by Jeremy Avigad in Feb 2004 *) header {* Finite sets *} theory Finite_Set imports Divides Power Inductive Lattice_Locales begin subsection {* Definition and basic properties *} consts Finites :: "'a set set" syntax finite :: "'a set => bool" translations "finite A" == "A : Finites" inductive Finites intros emptyI [simp, intro!]: "{} : Finites" insertI [simp, intro!]: "A : Finites ==> insert a A : Finites" axclass finite \ type finite: "finite UNIV" lemma ex_new_if_finite: -- "does not depend on def of finite at all" assumes "\ finite (UNIV :: 'a set)" and "finite A" shows "\a::'a. a \ A" proof - from prems have "A \ UNIV" by blast thus ?thesis by blast qed lemma finite_induct [case_names empty insert, induct set: Finites]: "finite F ==> P {} ==> (!!x F. finite F ==> x \ F ==> P F ==> P (insert x F)) ==> P F" -- {* Discharging @{text "x \ F"} entails extra work. *} proof - assume "P {}" and insert: "!!x F. finite F ==> x \ F ==> P F ==> P (insert x F)" assume "finite F" thus "P F" proof induct show "P {}" . fix x F assume F: "finite F" and P: "P F" show "P (insert x F)" proof cases assume "x \ F" hence "insert x F = F" by (rule insert_absorb) with P show ?thesis by (simp only:) next assume "x \ F" from F this P show ?thesis by (rule insert) qed qed qed lemma finite_ne_induct[case_names singleton insert, consumes 2]: assumes fin: "finite F" shows "F \ {} \ \ \x. P{x}; \x F. \ finite F; F \ {}; x \ F; P F \ \ P (insert x F) \ \ P F" using fin proof induct case empty thus ?case by simp next case (insert x F) show ?case proof cases assume "F = {}" thus ?thesis using insert(4) by simp next assume "F \ {}" thus ?thesis using insert by blast qed qed lemma finite_subset_induct [consumes 2, case_names empty insert]: "finite F ==> F \ A ==> P {} ==> (!!a F. finite F ==> a \ A ==> a \ F ==> P F ==> P (insert a F)) ==> P F" proof - assume "P {}" and insert: "!!a F. finite F ==> a \ A ==> a \ F ==> P F ==> P (insert a F)" assume "finite F" thus "F \ A ==> P F" proof induct show "P {}" . fix x F assume "finite F" and "x \ F" and P: "F \ A ==> P F" and i: "insert x F \ A" show "P (insert x F)" proof (rule insert) from i show "x \ A" by blast from i have "F \ A" by blast with P show "P F" . qed qed qed text{* Finite sets are the images of initial segments of natural numbers: *} lemma finite_imp_nat_seg_image_inj_on: assumes fin: "finite A" shows "\ (n::nat) f. A = f ` {i. if. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp qed next case (insert a A) have notinA: "a \ A" . from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast hence "insert a A = f(n:=a) ` {i. i < Suc n}" "inj_on (f(n:=a)) {i. i < Suc n}" using notinA by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq) thus ?case by blast qed lemma nat_seg_image_imp_finite: "!!f A. A = f ` {i::nat. i finite A" proof (induct n) case 0 thus ?case by simp next case (Suc n) let ?B = "f ` {i. i < n}" have finB: "finite ?B" by(rule Suc.hyps[OF refl]) show ?case proof cases assume "\k(\ k (n::nat) f. A = f ` {i::nat. i finite G ==> finite (F Un G)" -- {* The union of two finite sets is finite. *} by (induct set: Finites) simp_all lemma finite_subset: "A \ B ==> finite B ==> finite A" -- {* Every subset of a finite set is finite. *} proof - assume "finite B" thus "!!A. A \ B ==> finite A" proof induct case empty thus ?case by simp next case (insert x F A) have A: "A \ insert x F" and r: "A - {x} \ F ==> finite (A - {x})" . show "finite A" proof cases assume x: "x \ A" with A have "A - {x} \ F" by (simp add: subset_insert_iff) with r have "finite (A - {x})" . hence "finite (insert x (A - {x}))" .. also have "insert x (A - {x}) = A" by (rule insert_Diff) finally show ?thesis . next show "A \ F ==> ?thesis" . assume "x \ A" with A show "A \ F" by (simp add: subset_insert_iff) qed qed qed lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" -- {* The converse obviously fails. *} by (blast intro: finite_subset) lemma finite_insert [simp]: "finite (insert a A) = finite A" apply (subst insert_is_Un) apply (simp only: finite_Un, blast) done lemma finite_Union[simp, intro]: "\ finite A; !!M. M \ A \ finite M \ \ finite(\A)" by (induct rule:finite_induct) simp_all lemma finite_empty_induct: "finite A ==> P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}" proof - assume "finite A" and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" have "P (A - A)" proof - fix c b :: "'a set" presume c: "finite c" and b: "finite b" and P1: "P b" and P2: "!!x y. finite y ==> x \ y ==> P y ==> P (y - {x})" from c show "c \ b ==> P (b - c)" proof induct case empty from P1 show ?case by simp next case (insert x F) have "P (b - F - {x})" proof (rule P2) from _ b show "finite (b - F)" by (rule finite_subset) blast from insert show "x \ b - F" by simp from insert show "P (b - F)" by simp qed also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) finally show ?case . qed next show "A \ A" .. qed thus "P {}" by simp qed lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)" by (rule Diff_subset [THEN finite_subset]) lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" apply (subst Diff_insert) apply (case_tac "a : A - B") apply (rule finite_insert [symmetric, THEN trans]) apply (subst insert_Diff, simp_all) done text {* Image and Inverse Image over Finite Sets *} lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" -- {* The image of a finite set is finite. *} by (induct set: Finites) simp_all lemma finite_surj: "finite A ==> B <= f ` A ==> finite B" apply (frule finite_imageI) apply (erule finite_subset, assumption) done lemma finite_range_imageI: "finite (range g) ==> finite (range (%x. f (g x)))" apply (drule finite_imageI, simp) done lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" proof - have aux: "!!A. finite (A - {}) = finite A" by simp fix B :: "'a set" assume "finite B" thus "!!A. f`A = B ==> inj_on f A ==> finite A" apply induct apply simp apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") apply clarify apply (simp (no_asm_use) add: inj_on_def) apply (blast dest!: aux [THEN iffD1], atomize) apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) apply (frule subsetD [OF equalityD2 insertI1], clarify) apply (rule_tac x = xa in bexI) apply (simp_all add: inj_on_image_set_diff) done qed (rule refl) lemma inj_vimage_singleton: "inj f ==> f-`{a} \ {THE x. f x = a}" -- {* The inverse image of a singleton under an injective function is included in a singleton. *} apply (auto simp add: inj_on_def) apply (blast intro: the_equality [symmetric]) done lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)" -- {* The inverse image of a finite set under an injective function is finite. *} apply (induct set: Finites, simp_all) apply (subst vimage_insert) apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton]) done text {* The finite UNION of finite sets *} lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" by (induct set: Finites) simp_all text {* Strengthen RHS to @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \ {}})"}? We'd need to prove @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \ {}}"} by induction. *} lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" by (blast intro: finite_UN_I finite_subset) text {* Sigma of finite sets *} lemma finite_SigmaI [simp]: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" by (unfold Sigma_def) (blast intro!: finite_UN_I) lemma finite_cartesian_product: "[| finite A; finite B |] ==> finite (A <*> B)" by (rule finite_SigmaI) lemma finite_Prod_UNIV: "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") apply (erule ssubst) apply (erule finite_SigmaI, auto) done lemma finite_cartesian_productD1: "[| finite (A <*> B); B \ {} |] ==> finite A" apply (auto simp add: finite_conv_nat_seg_image) apply (drule_tac x=n in spec) apply (drule_tac x="fst o f" in spec) apply (auto simp add: o_def) prefer 2 apply (force dest!: equalityD2) apply (drule equalityD1) apply (rename_tac y x) apply (subgoal_tac "\k. k B); A \ {} |] ==> finite B" apply (auto simp add: finite_conv_nat_seg_image) apply (drule_tac x=n in spec) apply (drule_tac x="snd o f" in spec) apply (auto simp add: o_def) prefer 2 apply (force dest!: equalityD2) apply (drule equalityD1) apply (rename_tac x y) apply (subgoal_tac "\k. k 'b) set)" proof (rule finite_Prod_UNIV) show "finite (UNIV :: 'a set)" by (rule finite) show "finite (UNIV :: 'b set)" by (rule finite) qed qed text {* The powerset of a finite set *} lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" proof assume "finite (Pow A)" with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp next assume "finite A" thus "finite (Pow A)" by induct (simp_all add: finite_UnI finite_imageI Pow_insert) qed lemma finite_UnionD: "finite(\A) \ finite A" by(blast intro: finite_subset[OF subset_Pow_Union]) lemma finite_converse [iff]: "finite (r^-1) = finite r" apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") apply simp apply (rule iffI) apply (erule finite_imageD [unfolded inj_on_def]) apply (simp split add: split_split) apply (erule finite_imageI) apply (simp add: converse_def image_def, auto) apply (rule bexI) prefer 2 apply assumption apply simp done text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi Ehmety) *} lemma finite_Field: "finite r ==> finite (Field r)" -- {* A finite relation has a finite field (@{text "= domain \ range"}. *} apply (induct set: Finites) apply (auto simp add: Field_def Domain_insert Range_insert) done lemma trancl_subset_Field2: "r^+ <= Field r \ Field r" apply clarify apply (erule trancl_induct) apply (auto simp add: Field_def) done lemma finite_trancl: "finite (r^+) = finite r" apply auto prefer 2 apply (rule trancl_subset_Field2 [THEN finite_subset]) apply (rule finite_SigmaI) prefer 3 apply (blast intro: r_into_trancl' finite_subset) apply (auto simp add: finite_Field) done subsection {* A fold functional for finite sets *} text {* The intended behaviour is @{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\ (f (g x\<^isub>n) z)\)"} if @{text f} is associative-commutative. For an application of @{text fold} se the definitions of sums and products over finite sets. *} consts foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \ 'a) set" inductive "foldSet f g z" intros emptyI [intro]: "({}, z) : foldSet f g z" insertI [intro]: "\ x \ A; (A, y) : foldSet f g z \ \ (insert x A, f (g x) y) : foldSet f g z" inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z" constdefs fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a" "fold f g z A == THE x. (A, x) : foldSet f g z" text{*A tempting alternative for the definiens is @{term "if finite A then THE x. (A, x) : foldSet f g e else e"}. It allows the removal of finiteness assumptions from the theorems @{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}. The proofs become ugly, with @{text rule_format}. It is not worth the effort.*} lemma Diff1_foldSet: "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z" by (erule insert_Diff [THEN subst], rule foldSet.intros, auto) lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A" by (induct set: foldSet) auto lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z" by (induct set: Finites) auto subsubsection {* Commutative monoids *} locale ACf = fixes f :: "'a => 'a => 'a" (infixl "\" 70) assumes commute: "x \ y = y \ x" and assoc: "(x \ y) \ z = x \ (y \ z)" locale ACe = ACf + fixes e :: 'a assumes ident [simp]: "x \ e = x" locale ACIf = ACf + assumes idem: "x \ x = x" lemma (in ACf) left_commute: "x \ (y \ z) = y \ (x \ z)" proof - have "x \ (y \ z) = (y \ z) \ x" by (simp only: commute) also have "... = y \ (z \ x)" by (simp only: assoc) also have "z \ x = x \ z" by (simp only: commute) finally show ?thesis . qed lemmas (in ACf) AC = assoc commute left_commute lemma (in ACe) left_ident [simp]: "e \ x = x" proof - have "x \ e = x" by (rule ident) thus ?thesis by (subst commute) qed lemma (in ACIf) idem2: "x \ (x \ y) = x \ y" proof - have "x \ (x \ y) = (x \ x) \ y" by(simp add:assoc) also have "\ = x \ y" by(simp add:idem) finally show ?thesis . qed lemmas (in ACIf) ACI = AC idem idem2 text{* Instantiation of locales: *} lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \ 'a \ 'a)" by(fastsimp intro: ACf.intro add_assoc add_commute) lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)" by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add) lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \ 'a \ 'a)" by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute) lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)" by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult) subsubsection{*From @{term foldSet} to @{term fold}*} lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})" by (auto simp add: less_Suc_eq) lemma insert_image_inj_on_eq: "[|insert (h m) A = h ` {i. i < Suc m}; h m \ A; inj_on h {i. i < Suc m}|] ==> A = h ` {i. i < m}" apply (auto simp add: image_less_Suc inj_on_def) apply (blast intro: less_trans) done lemma insert_inj_onE: assumes aA: "insert a A = h`{i::nat. i A" and inj_on: "inj_on h {i::nat. ihm m. inj_on hm {i::nat. i A" by (simp add: swap_def hkeq anot) show "insert (?hm m) A = ?hm ` {i. i < Suc m}" using aA hkeq nSuc klessn by (auto simp add: swap_def image_less_Suc fun_upd_image less_Suc_eq inj_on_image_set_diff [OF inj_on]) qed qed qed lemma (in ACf) foldSet_determ_aux: "!!A x x' h. \ A = h`{i::nat. i \ x' = x" proof (induct n rule: less_induct) case (less n) have IH: "!!m h A x x'. \m foldSet f g z; (A, x') \ foldSet f g z\ \ x' = x" . have Afoldx: "(A,x) \ foldSet f g z" and Afoldx': "(A,x') \ foldSet f g z" and A: "A = h`{i. i u)" and notinB: "b \ B" and Bu: "(B,u) \ foldSet f g z" hence AbB: "A = insert b B" and x: "x = g b \ u" by auto show "x'=x" proof (rule foldSet.cases [OF Afoldx']) assume "(A, x') = ({}, z)" with AbB show "x' = x" by blast next fix C c v assume "(A,x') = (insert c C, g c \ v)" and notinC: "c \ C" and Cv: "(C,v) \ foldSet f g z" hence AcC: "A = insert c C" and x': "x' = g c \ v" by auto from A AbB have Beq: "insert b B = h`{i. i c" let ?D = "B - {c}" have B: "B = insert c ?D" and C: "C = insert b ?D" using AbB AcC notinB notinC diff by(blast elim!:equalityE)+ have "finite A" by(rule foldSet_imp_finite[OF Afoldx]) with AbB have "finite ?D" by simp then obtain d where Dfoldd: "(?D,d) \ foldSet f g z" using finite_imp_foldSet by rules moreover have cinB: "c \ B" using B by auto ultimately have "(B,g c \ d) \ foldSet f g z" by(rule Diff1_foldSet) hence "g c \ d = u" by (rule IH [OF lessB Beq inj_onB Bu]) moreover have "g b \ d = v" proof (rule IH[OF lessC Ceq inj_onC Cv]) show "(C, g b \ d) \ foldSet f g z" using C notinB Dfoldd by fastsimp qed ultimately show ?thesis using x x' by (auto simp: AC) qed qed qed qed lemma (in ACf) foldSet_determ: "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x" apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) apply (blast intro: foldSet_determ_aux [rule_format]) done lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y" by (unfold fold_def) (blast intro: foldSet_determ) text{* The base case for @{text fold}: *} lemma fold_empty [simp]: "fold f g z {} = z" by (unfold fold_def) blast lemma (in ACf) fold_insert_aux: "x \ A ==> ((insert x A, v) : foldSet f g z) = (EX y. (A, y) : foldSet f g z & v = f (g x) y)" apply auto apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE]) apply (fastsimp dest: foldSet_imp_finite) apply (blast intro: foldSet_determ) done text{* The recursion equation for @{text fold}: *} lemma (in ACf) fold_insert[simp]: "finite A ==> x \ A ==> fold f g z (insert x A) = f (g x) (fold f g z A)" apply (unfold fold_def) apply (simp add: fold_insert_aux) apply (rule the_equality) apply (auto intro: finite_imp_foldSet cong add: conj_cong simp add: fold_def [symmetric] fold_equality) done lemma (in ACf) fold_rec: assumes fin: "finite A" and a: "a:A" shows "fold f g z A = f (g a) (fold f g z (A - {a}))" proof- have A: "A = insert a (A - {a})" using a by blast hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp also have "\ = f (g a) (fold f g z (A - {a}))" by(rule fold_insert) (simp add:fin)+ finally show ?thesis . qed text{* A simplified version for idempotent functions: *} lemma (in ACIf) fold_insert_idem: assumes finA: "finite A" shows "fold f g z (insert a A) = g a \ fold f g z A" proof cases assume "a \ A" then obtain B where A: "A = insert a B" and disj: "a \ B" by(blast dest: mk_disjoint_insert) show ?thesis proof - from finA A have finB: "finite B" by(blast intro: finite_subset) have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp also have "\ = (g a) \ (fold f g z B)" using finB disj by simp also have "\ = g a \ fold f g z A" using A finB disj by(simp add:idem assoc[symmetric]) finally show ?thesis . qed next assume "a \ A" with finA show ?thesis by simp qed lemma (in ACIf) foldI_conv_id: "finite A \ fold f g z A = fold f id z (g ` A)" by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert) subsubsection{*Lemmas about @{text fold}*} lemma (in ACf) fold_commute: "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)" apply (induct set: Finites, simp) apply (simp add: left_commute [of x]) done lemma (in ACf) fold_nest_Un_Int: "finite A ==> finite B ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)" apply (induct set: Finites, simp) apply (simp add: fold_commute Int_insert_left insert_absorb) done lemma (in ACf) fold_nest_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> fold f g z (A Un B) = fold f g (fold f g z B) A" by (simp add: fold_nest_Un_Int) lemma (in ACf) fold_reindex: assumes fin: "finite A" shows "inj_on h A \ fold f g z (h ` A) = fold f (g \ h) z A" using fin apply induct apply simp apply simp done lemma (in ACe) fold_Un_Int: "finite A ==> finite B ==> fold f g e A \ fold f g e B = fold f g e (A Un B) \ fold f g e (A Int B)" apply (induct set: Finites, simp) apply (simp add: AC insert_absorb Int_insert_left) done corollary (in ACe) fold_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> fold f g e (A Un B) = fold f g e A \ fold f g e B" by (simp add: fold_Un_Int) lemma (in ACe) fold_UN_disjoint: "\ finite I; ALL i:I. finite (A i); ALL i:I. ALL j:I. i \ j --> A i Int A j = {} \ \ fold f g e (UNION I A) = fold f (%i. fold f g e (A i)) e I" apply (induct set: Finites, simp, atomize) apply (subgoal_tac "ALL i:F. x \ i") prefer 2 apply blast apply (subgoal_tac "A x Int UNION F A = {}") prefer 2 apply blast apply (simp add: fold_Un_disjoint) done text{*Fusion theorem, as described in Graham Hutton's paper, A Tutorial on the Universality and Expressiveness of Fold, JFP 9:4 (355-372), 1999.*} lemma (in ACf) fold_fusion: includes ACf g shows "finite A ==> (!!x y. h (g x y) = f x (h y)) ==> h (fold g j w A) = fold f j (h w) A" by (induct set: Finites, simp_all) lemma (in ACf) fold_cong: "finite A \ (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A" apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C") apply simp apply (erule finite_induct, simp) apply (simp add: subset_insert_iff, clarify) apply (subgoal_tac "finite C") prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) apply (subgoal_tac "C = insert x (C - {x})") prefer 2 apply blast apply (erule ssubst) apply (drule spec) apply (erule (1) notE impE) apply (simp add: Ball_def del: insert_Diff_single) done lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==> fold f (%x. fold f (g x) e (B x)) e A = fold f (split g) e (SIGMA x:A. B x)" apply (subst Sigma_def) apply (subst fold_UN_disjoint, assumption, simp) apply blast apply (erule fold_cong) apply (subst fold_UN_disjoint, simp, simp) apply blast apply simp done lemma (in ACe) fold_distrib: "finite A \ fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)" apply (erule finite_induct, simp) apply (simp add:AC) done subsection {* Generalized summation over a set *} constdefs setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" "setsum f A == if finite A then fold (op +) f 0 A else 0" text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is written @{text"\x\A. e"}. *} syntax "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) syntax (xsymbols) "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) syntax (HTML output) "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) translations -- {* Beware of argument permutation! *} "SUM i:A. b" == "setsum (%i. b) A" "\i\A. b" == "setsum (%i. b) A" text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter @{text"\x|P. e"}. *} syntax "_qsetsum" :: "idt \ bool \ 'a \ 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) syntax (xsymbols) "_qsetsum" :: "idt \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) syntax (HTML output) "_qsetsum" :: "idt \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) translations "SUM x|P. t" => "setsum (%x. t) {x. P}" "\x|P. t" => "setsum (%x. t) {x. P}" text{* Finally we abbreviate @{term"\x\A. x"} by @{text"\A"}. *} syntax "_Setsum" :: "'a set => 'a::comm_monoid_mult" ("\_" [1000] 999) parse_translation {* let fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A in [("_Setsum", Setsum_tr)] end; *} print_translation {* let fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = if x<>y then raise Match else let val x' = Syntax.mark_bound x val t' = subst_bound(x',t) val P' = subst_bound(x',P) in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end in [("setsum", setsum_tr')] end *} lemma setsum_empty [simp]: "setsum f {} = 0" by (simp add: setsum_def) lemma setsum_insert [simp]: "finite F ==> a \ F ==> setsum f (insert a F) = f a + setsum f F" by (simp add: setsum_def ACf.fold_insert [OF ACf_add]) lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0" by (simp add: setsum_def) lemma setsum_reindex: "inj_on f B ==> setsum h (f ` B) = setsum (h \ f) B" by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD) lemma setsum_reindex_id: "inj_on f B ==> setsum f B = setsum id (f ` B)" by (auto simp add: setsum_reindex) lemma setsum_cong: "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add]) lemma setsum_reindex_cong: "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|] ==> setsum h B = setsum g A" by (simp add: setsum_reindex cong: setsum_cong) lemma setsum_0: "setsum (%i. 0) A = 0" apply (clarsimp simp: setsum_def) apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add]) done lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0" apply (subgoal_tac "setsum f F = setsum (%x. 0) F") apply (erule ssubst, rule setsum_0) apply (rule setsum_cong, auto) done lemma setsum_Un_Int: "finite A ==> finite B ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric]) lemma setsum_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" by (subst setsum_Un_Int [symmetric], auto) (*But we can't get rid of finite I. If infinite, although the rhs is 0, the lhs need not be, since UNION I A could still be finite.*) lemma setsum_UN_disjoint: "finite I ==> (ALL i:I. finite (A i)) ==> (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> setsum f (UNION I A) = (\i\I. setsum f (A i))" by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong) text{*No need to assume that @{term C} is finite. If infinite, the rhs is directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} lemma setsum_Union_disjoint: "[| (ALL A:C. finite A); (ALL A:C. ALL B:C. A \ B --> A Int B = {}) |] ==> setsum f (Union C) = setsum (setsum f) C" apply (cases "finite C") prefer 2 apply (force dest: finite_UnionD simp add: setsum_def) apply (frule setsum_UN_disjoint [of C id f]) apply (unfold Union_def id_def, assumption+) done (*But we can't get rid of finite A. If infinite, although the lhs is 0, the rhs need not be, since SIGMA A B could still be finite.*) lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> (\x\A. (\y\B x. f x y)) = (\z\(SIGMA x:A. B x). f (fst z) (snd z))" by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong) text{*Here we can eliminate the finiteness assumptions, by cases.*} lemma setsum_cartesian_product: "(\x\A. (\y\B. f x y)) = (\z\A <*> B. f (fst z) (snd z))" apply (cases "finite A") apply (cases "finite B") apply (simp add: setsum_Sigma) apply (cases "A={}", simp) apply (simp add: setsum_0) apply (auto simp add: setsum_def dest: finite_cartesian_productD1 finite_cartesian_productD2) done lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" by(simp add:setsum_def ACe.fold_distrib[OF ACe_add]) subsubsection {* Properties in more restricted classes of structures *} lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" apply (case_tac "finite A") prefer 2 apply (simp add: setsum_def) apply (erule rev_mp) apply (erule finite_induct, auto) done lemma setsum_eq_0_iff [simp]: "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" by (induct set: Finites) auto lemma setsum_Un_nat: "finite A ==> finite B ==> (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" -- {* For the natural numbers, we have subtraction. *} by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps) lemma setsum_Un: "finite A ==> finite B ==> (setsum f (A Un B) :: 'a :: ab_group_add) = setsum f A + setsum f B - setsum f (A Int B)" by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps) lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = (if a:A then setsum f A - f a else setsum f A)" apply (case_tac "finite A") prefer 2 apply (simp add: setsum_def) apply (erule finite_induct) apply (auto simp add: insert_Diff_if) apply (drule_tac a = a in mk_disjoint_insert, auto) done lemma setsum_diff1: "finite A \ (setsum f (A - {a}) :: ('a::ab_group_add)) = (if a:A then setsum f A - f a else setsum f A)" by (erule finite_induct) (auto simp add: insert_Diff_if) (* By Jeremy Siek: *) lemma setsum_diff_nat: assumes finB: "finite B" shows "B \ A \ (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" using finB proof (induct) show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp next fix F x assume finF: "finite F" and xnotinF: "x \ F" and xFinA: "insert x F \ A" and IH: "F \ A \ setsum f (A - F) = setsum f A - setsum f F" from xnotinF xFinA have xinAF: "x \ (A - F)" by simp from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" by (simp add: setsum_diff1_nat) from xFinA have "F \ A" by simp with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" by simp from xnotinF have "A - insert x F = (A - F) - {x}" by auto with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" by simp from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp qed lemma setsum_diff: assumes le: "finite A" "B \ A" shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" proof - from le have finiteB: "finite B" using finite_subset by auto show ?thesis using finiteB le proof (induct) case empty thus ?case by auto next case (insert x F) thus ?case using le finiteB by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) qed qed lemma setsum_mono: assumes le: "\i. i\K \ f (i::'a) \ ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))" shows "(\i\K. f i) \ (\i\K. g i)" proof (cases "finite K") case True thus ?thesis using le proof (induct) case empty thus ?case by simp next case insert thus ?case using add_mono by force qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_mono2_nat: assumes fin: "finite B" and sub: "A \ B" shows "setsum f A \ (setsum f B :: nat)" proof - have "setsum f A \ setsum f A + setsum f (B-A)" by arith also have "\ = setsum f (A \ (B-A))" using fin finite_subset[OF sub fin] by (simp add:setsum_Un_disjoint del:Un_Diff_cancel) also have "A \ (B-A) = B" using sub by blast finally show ?thesis . qed lemma setsum_negf: "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" proof (cases "finite A") case True thus ?thesis by (induct set: Finites, auto) next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_subtractf: "setsum (%x. ((f x)::'a::ab_group_add) - g x) A = setsum f A - setsum g A" proof (cases "finite A") case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf) next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_nonneg: assumes nn: "\x\A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \ f x" shows "0 \ setsum f A" proof (cases "finite A") case True thus ?thesis using nn apply (induct set: Finites, auto) apply (subgoal_tac "0 + 0 \ f x + setsum f F", simp) apply (blast intro: add_mono) done next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_nonpos: assumes np: "\x\A. f x \ (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})" shows "setsum f A \ 0" proof (cases "finite A") case True thus ?thesis using np apply (induct set: Finites, auto) apply (subgoal_tac "f x + setsum f F \ 0 + 0", simp) apply (blast intro: add_mono) done next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_mult: fixes f :: "'a => ('b::semiring_0_cancel)" shows "r * setsum f A = setsum (%n. r * f n) A" proof (cases "finite A") case True thus ?thesis proof (induct) case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: right_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_abs[iff]: fixes f :: "'a => ('b::lordered_ab_group_abs)" shows "abs (setsum f A) \ setsum (%i. abs(f i)) A" proof (cases "finite A") case True thus ?thesis proof (induct) case empty thus ?case by simp next case (insert x A) thus ?case by (auto intro: abs_triangle_ineq order_trans) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_abs_ge_zero[iff]: fixes f :: "'a => ('b::lordered_ab_group_abs)" shows "0 \ setsum (%i. abs(f i)) A" proof (cases "finite A") case True thus ?thesis proof (induct) case empty thus ?case by simp next case (insert x A) thus ?case by (auto intro: order_trans) qed next case False thus ?thesis by (simp add: setsum_def) qed subsection {* Generalized product over a set *} constdefs setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult" "setprod f A == if finite A then fold (op *) f 1 A else 1" syntax "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_:_. _)" [0, 51, 10] 10) syntax (xsymbols) "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) syntax (HTML output) "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) translations "\i:A. b" == "setprod (%i. b) A" -- {* Beware of argument permutation! *} syntax "_Setprod" :: "'a set => 'a::comm_monoid_mult" ("\_" [1000] 999) parse_translation {* let fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A in [("_Setprod", Setprod_tr)] end; *} print_translation {* let fun setprod_tr' [Abs(x,Tx,t), A] = if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match in [("setprod", setprod_tr')] end *} lemma setprod_empty [simp]: "setprod f {} = 1" by (auto simp add: setprod_def) lemma setprod_insert [simp]: "[| finite A; a \ A |] ==> setprod f (insert a A) = f a * setprod f A" by (simp add: setprod_def ACf.fold_insert [OF ACf_mult]) lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1" by (simp add: setprod_def) lemma setprod_reindex: "inj_on f B ==> setprod h (f ` B) = setprod (h \ f) B" by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD) lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)" by (auto simp add: setprod_reindex) lemma setprod_cong: "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B" by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult]) lemma setprod_reindex_cong: "inj_on f A ==> B = f ` A ==> g = h \ f ==> setprod h B = setprod g A" by (frule setprod_reindex, simp) lemma setprod_1: "setprod (%i. 1) A = 1" apply (case_tac "finite A") apply (erule finite_induct, auto simp add: mult_ac) done lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1" apply (subgoal_tac "setprod f F = setprod (%x. 1) F") apply (erule ssubst, rule setprod_1) apply (rule setprod_cong, auto) done lemma setprod_Un_Int: "finite A ==> finite B ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B" by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric]) lemma setprod_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B" by (subst setprod_Un_Int [symmetric], auto) lemma setprod_UN_disjoint: "finite I ==> (ALL i:I. finite (A i)) ==> (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> setprod f (UNION I A) = setprod (%i. setprod f (A i)) I" by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong) lemma setprod_Union_disjoint: "[| (ALL A:C. finite A); (ALL A:C. ALL B:C. A \ B --> A Int B = {}) |] ==> setprod f (Union C) = setprod (setprod f) C" apply (cases "finite C") prefer 2 apply (force dest: finite_UnionD simp add: setprod_def) apply (frule setprod_UN_disjoint [of C id f]) apply (unfold Union_def id_def, assumption+) done lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==> (\x:A. (\y: B x. f x y)) = (\z:(SIGMA x:A. B x). f (fst z) (snd z))" by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong) text{*Here we can eliminate the finiteness assumptions, by cases.*} lemma setprod_cartesian_product: "(\x:A. (\y: B. f x y)) = (\z:(A <*> B). f (fst z) (snd z))" apply (cases "finite A") apply (cases "finite B") apply (simp add: setprod_Sigma) apply (cases "A={}", simp) apply (simp add: setprod_1) apply (auto simp add: setprod_def dest: finite_cartesian_productD1 finite_cartesian_productD2) done lemma setprod_timesf: "setprod (%x. f x * g x) A = (setprod f A * setprod g A)" by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult]) subsubsection {* Properties in more restricted classes of structures *} lemma setprod_eq_1_iff [simp]: "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))" by (induct set: Finites) auto lemma setprod_zero: "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0" apply (induct set: Finites, force, clarsimp) apply (erule disjE, auto) done lemma setprod_nonneg [rule_format]: "(ALL x: A. (0::'a::ordered_idom) \ f x) --> 0 \ setprod f A" apply (case_tac "finite A") apply (induct set: Finites, force, clarsimp) apply (subgoal_tac "0 * 0 \ f x * setprod f F", force) apply (rule mult_mono, assumption+) apply (auto simp add: setprod_def) done lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x) --> 0 < setprod f A" apply (case_tac "finite A") apply (induct set: Finites, force, clarsimp) apply (subgoal_tac "0 * 0 < f x * setprod f F", force) apply (rule mult_strict_mono, assumption+) apply (auto simp add: setprod_def) done lemma setprod_nonzero [rule_format]: "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==> finite A ==> (ALL x: A. f x \ (0::'a)) --> setprod f A \ 0" apply (erule finite_induct, auto) done lemma setprod_zero_eq: "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==> finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)" apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast) done lemma setprod_nonzero_field: "finite A ==> (ALL x: A. f x \ (0::'a::field)) ==> setprod f A \ 0" apply (rule setprod_nonzero, auto) done lemma setprod_zero_eq_field: "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)" apply (rule setprod_zero_eq, auto) done lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \ 0) ==> (setprod f (A Un B) :: 'a ::{field}) = setprod f A * setprod f B / setprod f (A Int B)" apply (subst setprod_Un_Int [symmetric], auto) apply (subgoal_tac "finite (A Int B)") apply (frule setprod_nonzero_field [of "A Int B" f], assumption) apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self) done lemma setprod_diff1: "finite A ==> f a \ 0 ==> (setprod f (A - {a}) :: 'a :: {field}) = (if a:A then setprod f A / f a else setprod f A)" apply (erule finite_induct) apply (auto simp add: insert_Diff_if) apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a") apply (erule ssubst) apply (subst times_divide_eq_right [THEN sym]) apply (auto simp add: mult_ac times_divide_eq_right divide_self) done lemma setprod_inversef: "finite A ==> ALL x: A. f x \ (0::'a::{field,division_by_zero}) ==> setprod (inverse \ f) A = inverse (setprod f A)" apply (erule finite_induct) apply (simp, simp) done lemma setprod_dividef: "[|finite A; \x \ A. g x \ (0::'a::{field,division_by_zero})|] ==> setprod (%x. f x / g x) A = setprod f A / setprod g A" apply (subgoal_tac "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \ g) x) A") apply (erule ssubst) apply (subst divide_inverse) apply (subst setprod_timesf) apply (subst setprod_inversef, assumption+, rule refl) apply (rule setprod_cong, rule refl) apply (subst divide_inverse, auto) done subsection {* Finite cardinality *} text {* This definition, although traditional, is ugly to work with: @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}. But now that we have @{text setsum} things are easy: *} constdefs card :: "'a set => nat" "card A == setsum (%x. 1::nat) A" lemma card_empty [simp]: "card {} = 0" by (simp add: card_def) lemma card_infinite [simp]: "~ finite A ==> card A = 0" by (simp add: card_def) lemma card_eq_setsum: "card A = setsum (%x. 1) A" by (simp add: card_def) lemma card_insert_disjoint [simp]: "finite A ==> x \ A ==> card (insert x A) = Suc(card A)" by(simp add: card_def ACf.fold_insert[OF ACf_add]) lemma card_insert_if: "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" by (simp add: insert_absorb) lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})" apply auto apply (drule_tac a = x in mk_disjoint_insert, clarify, auto) done lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)" by auto lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A" apply(rule_tac t = A in insert_Diff [THEN subst], assumption) apply(simp del:insert_Diff_single) done lemma card_Diff_singleton: "finite A ==> x: A ==> card (A - {x}) = card A - 1" by (simp add: card_Suc_Diff1 [symmetric]) lemma card_Diff_singleton_if: "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)" by (simp add: card_Diff_singleton) lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" by (simp add: card_insert_if card_Suc_Diff1) lemma card_insert_le: "finite A ==> card A <= card (insert x A)" by (simp add: card_insert_if) lemma card_mono: "\ finite B; A \ B \ \ card A \ card B" by (simp add: card_def setsum_mono2_nat) lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" apply (induct set: Finites, simp, clarify) apply (subgoal_tac "finite A & A - {x} <= F") prefer 2 apply (blast intro: finite_subset, atomize) apply (drule_tac x = "A - {x}" in spec) apply (simp add: card_Diff_singleton_if split add: split_if_asm) apply (case_tac "card A", auto) done lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" apply (simp add: psubset_def linorder_not_le [symmetric]) apply (blast dest: card_seteq) done lemma card_Un_Int: "finite A ==> finite B ==> card A + card B = card (A Un B) + card (A Int B)" by(simp add:card_def setsum_Un_Int) lemma card_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> card (A Un B) = card A + card B" by (simp add: card_Un_Int) lemma card_Diff_subset: "finite B ==> B <= A ==> card (A - B) = card A - card B" by(simp add:card_def setsum_diff_nat) lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" apply (rule Suc_less_SucD) apply (simp add: card_Suc_Diff1) done lemma card_Diff2_less: "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" apply (case_tac "x = y") apply (simp add: card_Diff1_less) apply (rule less_trans) prefer 2 apply (auto intro!: card_Diff1_less) done lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" apply (case_tac "x : A") apply (simp_all add: card_Diff1_less less_imp_le) done lemma card_psubset: "finite B ==> A \ B ==> card A < card B ==> A < B" by (erule psubsetI, blast) lemma insert_partition: "\ x \ F; \c1 \ insert x F. \c2 \ insert x F. c1 \ c2 \ c1 \ c2 = {} \ \ x \ \ F = {}" by auto (* main cardinality theorem *) lemma card_partition [rule_format]: "finite C ==> finite (\ C) --> (\c\C. card c = k) --> (\c1 \ C. \c2 \ C. c1 \ c2 --> c1 \ c2 = {}) --> k * card(C) = card (\ C)" apply (erule finite_induct, simp) apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition finite_subset [of _ "\ (insert x F)"]) done lemma setsum_constant_nat: "(\x\A. y) = (card A) * y" -- {* Generalized to any @{text comm_semiring_1_cancel} in @{text IntDef} as @{text setsum_constant}. *} apply (cases "finite A") apply (erule finite_induct, auto) done lemma setprod_constant: "finite A ==> (\x: A. (y::'a::recpower)) = y^(card A)" apply (erule finite_induct) apply (auto simp add: power_Suc) done subsubsection {* Cardinality of unions *} lemma card_UN_disjoint: "finite I ==> (ALL i:I. finite (A i)) ==> (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> card (UNION I A) = (\i\I. card(A i))" apply (simp add: card_def) apply (subgoal_tac "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I") apply (simp add: setsum_UN_disjoint) apply (simp add: setsum_constant_nat cong: setsum_cong) done lemma card_Union_disjoint: "finite C ==> (ALL A:C. finite A) ==> (ALL A:C. ALL B:C. A \ B --> A Int B = {}) ==> card (Union C) = setsum card C" apply (frule card_UN_disjoint [of C id]) apply (unfold Union_def id_def, assumption+) done subsubsection {* Cardinality of image *} text{*The image of a finite set can be expressed using @{term fold}.*} lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A" apply (erule finite_induct, simp) apply (subst ACf.fold_insert) apply (auto simp add: ACf_def) done lemma card_image_le: "finite A ==> card (f ` A) <= card A" apply (induct set: Finites, simp) apply (simp add: le_SucI finite_imageI card_insert_if) done lemma card_image: "inj_on f A ==> card (f ` A) = card A" by(simp add:card_def setsum_reindex o_def) lemma endo_inj_surj: "finite A ==> f ` A \ A ==> inj_on f A ==> f ` A = A" by (simp add: card_seteq card_image) lemma eq_card_imp_inj_on: "[| finite A; card(f ` A) = card A |] ==> inj_on f A" apply (induct rule:finite_induct, simp) apply(frule card_image_le[where f = f]) apply(simp add:card_insert_if split:if_splits) done lemma inj_on_iff_eq_card: "finite A ==> inj_on f A = (card(f ` A) = card A)" by(blast intro: card_image eq_card_imp_inj_on) lemma card_inj_on_le: "[|inj_on f A; f ` A \ B; finite B |] ==> card A \ card B" apply (subgoal_tac "finite A") apply (force intro: card_mono simp add: card_image [symmetric]) apply (blast intro: finite_imageD dest: finite_subset) done lemma card_bij_eq: "[|inj_on f A; f ` A \ B; inj_on g B; g ` B \ A; finite A; finite B |] ==> card A = card B" by (auto intro: le_anti_sym card_inj_on_le) subsubsection {* Cardinality of products *} (* lemma SigmaI_insert: "y \ A ==> (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \ (SIGMA x: A. B x))" by auto *) lemma card_SigmaI [simp]: "\ finite A; ALL a:A. finite (B a) \ \ card (SIGMA x: A. B x) = (\a\A. card (B a))" by(simp add:card_def setsum_Sigma) lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)" apply (cases "finite A") apply (cases "finite B") apply (simp add: setsum_constant_nat) apply (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2) done lemma card_cartesian_product_singleton: "card({x} <*> A) = card(A)" by (simp add: card_cartesian_product) subsubsection {* Cardinality of the Powerset *} lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) apply (induct set: Finites) apply (simp_all add: Pow_insert) apply (subst card_Un_disjoint, blast) apply (blast intro: finite_imageI, blast) apply (subgoal_tac "inj_on (insert x) (Pow F)") apply (simp add: card_image Pow_insert) apply (unfold inj_on_def) apply (blast elim!: equalityE) done text {* Relates to equivalence classes. Based on a theorem of F. Kammüller's. *} lemma dvd_partition: "finite (Union C) ==> ALL c : C. k dvd card c ==> (ALL c1: C. ALL c2: C. c1 \ c2 --> c1 Int c2 = {}) ==> k dvd card (Union C)" apply(frule finite_UnionD) apply(rotate_tac -1) apply (induct set: Finites, simp_all, clarify) apply (subst card_Un_disjoint) apply (auto simp add: dvd_add disjoint_eq_subset_Compl) done subsubsection {* Theorems about @{text "choose"} *} text {* \medskip Basic theorem about @{text "choose"}. By Florian Kamm\"uller, tidied by LCP. *} lemma card_s_0_eq_empty: "finite A ==> card {B. B \ A & card B = 0} = 1" apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) apply (simp cong add: rev_conj_cong) done lemma choose_deconstruct: "finite M ==> x \ M ==> {s. s <= insert x M & card(s) = Suc k} = {s. s <= M & card(s) = Suc k} Un {s. EX t. t <= M & card(t) = k & s = insert x t}" apply safe apply (auto intro: finite_subset [THEN card_insert_disjoint]) apply (drule_tac x = "xa - {x}" in spec) apply (subgoal_tac "x \ xa", auto) apply (erule rev_mp, subst card_Diff_singleton) apply (auto intro: finite_subset) done text{*There are as many subsets of @{term A} having cardinality @{term k} as there are sets obtained from the former by inserting a fixed element @{term x} into each.*} lemma constr_bij: "[|finite A; x \ A|] ==> card {B. EX C. C <= A & card(C) = k & B = insert x C} = card {B. B <= A & card(B) = k}" apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq) apply (auto elim!: equalityE simp add: inj_on_def) apply (subst Diff_insert0, auto) txt {* finiteness of the two sets *} apply (rule_tac [2] B = "Pow (A)" in finite_subset) apply (rule_tac B = "Pow (insert x A)" in finite_subset) apply fast+ done text {* Main theorem: combinatorial statement about number of subsets of a set. *} lemma n_sub_lemma: "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)" apply (induct k) apply (simp add: card_s_0_eq_empty, atomize) apply (rotate_tac -1, erule finite_induct) apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct) apply (subst card_Un_disjoint) prefer 4 apply (force simp add: constr_bij) prefer 3 apply force prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] finite_subset [of _ "Pow (insert x F)", standard]) apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) done theorem n_subsets: "finite A ==> card {B. B <= A & card B = k} = (card A choose k)" by (simp add: n_sub_lemma) subsection{* A fold functional for non-empty sets *} text{* Does not require start value. *} consts fold1Set :: "('a => 'a => 'a) => ('a set \ 'a) set" inductive "fold1Set f" intros fold1Set_insertI [intro]: "\ (A,x) \ foldSet f id a; a \ A \ \ (insert a A, x) \ fold1Set f" constdefs fold1 :: "('a => 'a => 'a) => 'a set => 'a" "fold1 f A == THE x. (A, x) : fold1Set f" lemma fold1Set_nonempty: "(A, x) : fold1Set f \ A \ {}" by(erule fold1Set.cases, simp_all) inductive_cases empty_fold1SetE [elim!]: "({}, x) : fold1Set f" inductive_cases insert_fold1SetE [elim!]: "(insert a X, x) : fold1Set f" lemma fold1Set_sing [iff]: "(({a},b) : fold1Set f) = (a = b)" by (blast intro: foldSet.intros elim: foldSet.cases) lemma fold1_singleton[simp]: "fold1 f {a} = a" by (unfold fold1_def) blast lemma finite_nonempty_imp_fold1Set: "\ finite A; A \ {} \ \ EX x. (A, x) : fold1Set f" apply (induct A rule: finite_induct) apply (auto dest: finite_imp_foldSet [of _ f id]) done text{*First, some lemmas about @{term foldSet}.*} lemma (in ACf) foldSet_insert_swap: assumes fold: "(A,y) \ foldSet f id b" shows "b \ A \ (insert b A, z \ y) \ foldSet f id z" using fold proof (induct rule: foldSet.induct) case emptyI thus ?case by (force simp add: fold_insert_aux commute) next case (insertI A x y) have "(insert x (insert b A), x \ (z \ y)) \ foldSet f (\u. u) z" using insertI by force --{*how does @{term id} get unfolded?*} thus ?case by (simp add: insert_commute AC) qed lemma (in ACf) foldSet_permute_diff: assumes fold: "(A,x) \ foldSet f id b" shows "!!a. \a \ A; b \ A\ \ (insert b (A-{a}), x) \ foldSet f id a" using fold proof (induct rule: foldSet.induct) case emptyI thus ?case by simp next case (insertI A x y) have "a = x \ a \ A" using insertI by simp thus ?case proof assume "a = x" with insertI show ?thesis by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap) next assume ainA: "a \ A" hence "(insert x (insert b (A - {a})), x \ y) \ foldSet f id a" using insertI by (force simp: id_def) moreover have "insert x (insert b (A - {a})) = insert b (insert x A - {a})" using ainA insertI by blast ultimately show ?thesis by (simp add: id_def) qed qed lemma (in ACf) fold1_eq_fold: "[|finite A; a \ A|] ==> fold1 f (insert a A) = fold f id a A" apply (simp add: fold1_def fold_def) apply (rule the_equality) apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id]) apply (rule sym, clarify) apply (case_tac "Aa=A") apply (best intro: the_equality foldSet_determ) apply (subgoal_tac "(A,x) \ foldSet f id a") apply (best intro: the_equality foldSet_determ) apply (subgoal_tac "insert aa (Aa - {a}) = A") prefer 2 apply (blast elim: equalityE) apply (auto dest: foldSet_permute_diff [where a=a]) done lemma nonempty_iff: "(A \ {}) = (\x B. A = insert x B & x \ B)" apply safe apply simp apply (drule_tac x=x in spec) apply (drule_tac x="A-{x}" in spec, auto) done lemma (in ACf) fold1_insert: assumes nonempty: "A \ {}" and A: "finite A" "x \ A" shows "fold1 f (insert x A) = f x (fold1 f A)" proof - from nonempty obtain a A' where "A = insert a A' & a ~: A'" by (auto simp add: nonempty_iff) with A show ?thesis by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) qed lemma (in ACIf) fold1_insert_idem [simp]: assumes nonempty: "A \ {}" and A: "finite A" shows "fold1 f (insert x A) = f x (fold1 f A)" proof - from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" by (auto simp add: nonempty_iff) show ?thesis proof cases assume "a = x" thus ?thesis proof cases assume "A' = {}" with prems show ?thesis by (simp add: idem) next assume "A' \ {}" with prems show ?thesis by (simp add: fold1_insert assoc [symmetric] idem) qed next assume "a \ x" with prems show ?thesis by (simp add: insert_commute fold1_eq_fold fold_insert_idem) qed qed text{* Now the recursion rules for definitions: *} lemma fold1_singleton_def: "g \ fold1 f \ g {a} = a" by(simp add:fold1_singleton) lemma (in ACf) fold1_insert_def: "\ g \ fold1 f; finite A; x \ A; A \ {} \ \ g(insert x A) = x \ (g A)" by(simp add:fold1_insert) lemma (in ACIf) fold1_insert_idem_def: "\ g \ fold1 f; finite A; A \ {} \ \ g(insert x A) = x \ (g A)" by(simp add:fold1_insert_idem) subsubsection{* Determinacy for @{term fold1Set} *} text{*Not actually used!!*} lemma (in ACf) foldSet_permute: "[|(insert a A, x) \ foldSet f id b; a \ A; b \ A|] ==> (insert b A, x) \ foldSet f id a" apply (case_tac "a=b") apply (auto dest: foldSet_permute_diff) done lemma (in ACf) fold1Set_determ: "(A, x) \ fold1Set f ==> (A, y) \ fold1Set f ==> y = x" proof (clarify elim!: fold1Set.cases) fix A x B y a b assume Ax: "(A, x) \ foldSet f id a" assume By: "(B, y) \ foldSet f id b" assume anotA: "a \ A" assume bnotB: "b \ B" assume eq: "insert a A = insert b B" show "y=x" proof cases assume same: "a=b" hence "A=B" using anotA bnotB eq by (blast elim!: equalityE) thus ?thesis using Ax By same by (blast intro: foldSet_determ) next assume diff: "a\b" let ?D = "B - {a}" have B: "B = insert a ?D" and A: "A = insert b ?D" and aB: "a \ B" and bA: "b \ A" using eq anotA bnotB diff by (blast elim!:equalityE)+ with aB bnotB By have "(insert b ?D, y) \ foldSet f id a" by (auto intro: foldSet_permute simp add: insert_absorb) moreover have "(insert b ?D, x) \ foldSet f id a" by (simp add: A [symmetric] Ax) ultimately show ?thesis by (blast intro: foldSet_determ) qed qed lemma (in ACf) fold1Set_equality: "(A, y) : fold1Set f ==> fold1 f A = y" by (unfold fold1_def) (blast intro: fold1Set_determ) declare empty_foldSetE [rule del] foldSet.intros [rule del] empty_fold1SetE [rule del] insert_fold1SetE [rule del] -- {* No more proves involve these relations. *} subsubsection{* Semi-Lattices *} locale ACIfSL = ACIf + fixes below :: "'a \ 'a \ bool" (infixl "\" 50) assumes below_def: "(x \ y) = (x\y = x)" locale ACIfSLlin = ACIfSL + assumes lin: "x\y \ {x,y}" lemma (in ACIfSL) below_refl[simp]: "x \ x" by(simp add: below_def idem) lemma (in ACIfSL) below_f_conv[simp]: "x \ y \ z = (x \ y \ x \ z)" proof assume "x \ y \ z" hence xyzx: "x \ (y \ z) = x" by(simp add: below_def) have "x \ y = x" proof - have "x \ y = (x \ (y \ z)) \ y" by(rule subst[OF xyzx], rule refl) also have "\ = x \ (y \ z)" by(simp add:ACI) also have "\ = x" by(rule xyzx) finally show ?thesis . qed moreover have "x \ z = x" proof - have "x \ z = (x \ (y \ z)) \ z" by(rule subst[OF xyzx], rule refl) also have "\ = x \ (y \ z)" by(simp add:ACI) also have "\ = x" by(rule xyzx) finally show ?thesis . qed ultimately show "x \ y \ x \ z" by(simp add: below_def) next assume a: "x \ y \ x \ z" hence y: "x \ y = x" and z: "x \ z = x" by(simp_all add: below_def) have "x \ (y \ z) = (x \ y) \ z" by(simp add:assoc) also have "x \ y = x" using a by(simp_all add: below_def) also have "x \ z = x" using a by(simp_all add: below_def) finally show "x \ y \ z" by(simp_all add: below_def) qed lemma (in ACIfSLlin) above_f_conv: "x \ y \ z = (x \ z \ y \ z)" proof assume a: "x \ y \ z" have "x \ y = x \ x \ y = y" using lin[of x y] by simp thus "x \ z \ y \ z" proof assume "x \ y = x" hence "x \ z" by(rule subst)(rule a) thus ?thesis .. next assume "x \ y = y" hence "y \ z" by(rule subst)(rule a) thus ?thesis .. qed next assume "x \ z \ y \ z" thus "x \ y \ z" proof assume a: "x \ z" have "(x \ y) \ z = (x \ z) \ y" by(simp add:ACI) also have "x \ z = x" using a by(simp add:below_def) finally show "x \ y \ z" by(simp add:below_def) next assume a: "y \ z" have "(x \ y) \ z = x \ (y \ z)" by(simp add:ACI) also have "y \ z = y" using a by(simp add:below_def) finally show "x \ y \ z" by(simp add:below_def) qed qed subsubsection{* Lemmas about @{text fold1} *} lemma (in ACf) fold1_Un: assumes A: "finite A" "A \ {}" shows "finite B \ B \ {} \ A Int B = {} \ fold1 f (A Un B) = f (fold1 f A) (fold1 f B)" using A proof(induct rule:finite_ne_induct) case singleton thus ?case by(simp add:fold1_insert) next case insert thus ?case by (simp add:fold1_insert assoc) qed lemma (in ACIf) fold1_Un2: assumes A: "finite A" "A \ {}" shows "finite B \ B \ {} \ fold1 f (A Un B) = f (fold1 f A) (fold1 f B)" using A proof(induct rule:finite_ne_induct) case singleton thus ?case by(simp add:fold1_insert_idem) next case insert thus ?case by (simp add:fold1_insert_idem assoc) qed lemma (in ACf) fold1_in: assumes A: "finite (A)" "A \ {}" and elem: "\x y. x\y \ {x,y}" shows "fold1 f A \ A" using A proof (induct rule:finite_ne_induct) case singleton thus ?case by simp next case insert thus ?case using elem by (force simp add:fold1_insert) qed lemma (in ACIfSL) below_fold1_iff: assumes A: "finite A" "A \ {}" shows "x \ fold1 f A = (\a\A. x \ a)" using A by(induct rule:finite_ne_induct) simp_all lemma (in ACIfSL) fold1_belowI: assumes A: "finite A" "A \ {}" shows "a \ A \ fold1 f A \ a" using A proof (induct rule:finite_ne_induct) case singleton thus ?case by simp next case (insert x F) from insert(5) have "a = x \ a \ F" by simp thus ?case proof assume "a = x" thus ?thesis using insert by(simp add:below_def ACI) next assume "a \ F" hence bel: "fold1 f F \ a" by(rule insert) have "fold1 f (insert x F) \ a = x \ (fold1 f F \ a)" using insert by(simp add:below_def ACI) also have "fold1 f F \ a = fold1 f F" using bel by(simp add:below_def ACI) also have "x \ \ = fold1 f (insert x F)" using insert by(simp add:below_def ACI) finally show ?thesis by(simp add:below_def) qed qed lemma (in ACIfSLlin) fold1_below_iff: assumes A: "finite A" "A \ {}" shows "fold1 f A \ x = (\a\A. a \ x)" using A by(induct rule:finite_ne_induct)(simp_all add:above_f_conv) subsubsection{* Lattices *} locale Lattice = lattice + fixes Inf :: "'a set \ 'a" ("\_" [900] 900) and Sup :: "'a set \ 'a" ("\_" [900] 900) defines "Inf == fold1 inf" and "Sup == fold1 sup" locale Distrib_Lattice = distrib_lattice + Lattice text{* Lattices are semilattices *} lemma (in Lattice) ACf_inf: "ACf inf" by(blast intro: ACf.intro inf_commute inf_assoc) lemma (in Lattice) ACf_sup: "ACf sup" by(blast intro: ACf.intro sup_commute sup_assoc) lemma (in Lattice) ACIf_inf: "ACIf inf" apply(rule ACIf.intro) apply(rule ACf_inf) apply(rule ACIf_axioms.intro) apply(rule inf_idem) done lemma (in Lattice) ACIf_sup: "ACIf sup" apply(rule ACIf.intro) apply(rule ACf_sup) apply(rule ACIf_axioms.intro) apply(rule sup_idem) done lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \)" apply(rule ACIfSL.intro) apply(rule ACf_inf) apply(rule ACIf.axioms[OF ACIf_inf]) apply(rule ACIfSL_axioms.intro) apply(rule iffI) apply(blast intro: antisym inf_le1 inf_le2 inf_least refl) apply(erule subst) apply(rule inf_le2) done lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \ x)" apply(rule ACIfSL.intro) apply(rule ACf_sup) apply(rule ACIf.axioms[OF ACIf_sup]) apply(rule ACIfSL_axioms.intro) apply(rule iffI) apply(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl) apply(erule subst) apply(rule sup_ge2) done subsubsection{* Fold laws in lattices *} lemma (in Lattice) Inf_le_Sup: "\ finite A; A \ {} \ \ \A \ \A" apply(unfold Sup_def Inf_def) apply(subgoal_tac "EX a. a:A") prefer 2 apply blast apply(erule exE) apply(rule trans) apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf]) apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup]) done lemma (in Lattice) sup_Inf_absorb: "\ finite A; A \ {}; a \ A \ \ (a \ \A) = a" apply(subst sup_commute) apply(simp add:Inf_def sup_absorb ACIfSL.fold1_belowI[OF ACIfSL_inf]) done lemma (in Lattice) inf_Sup_absorb: "\ finite A; A \ {}; a \ A \ \ (a \ \A) = a" by(simp add:Sup_def inf_absorb ACIfSL.fold1_belowI[OF ACIfSL_sup]) lemma (in Distrib_Lattice) sup_Inf1_distrib: assumes A: "finite A" "A \ {}" shows "(x \ \A) = \{x \ a|a. a \ A}" using A proof (induct rule: finite_ne_induct) case singleton thus ?case by(simp add:Inf_def) next case (insert y A) have fin: "finite {x \ a |a. a \ A}" by(fast intro: finite_surj[where f = "%a. x \ a", OF insert(1)]) have "x \ \ (insert y A) = x \ (y \ \ A)" using insert by(simp add:ACf.fold1_insert_def[OF ACf_inf Inf_def]) also have "\ = (x \ y) \ (x \ \ A)" by(rule sup_inf_distrib1) also have "x \ \ A = \{x \ a|a. a \ A}" using insert by simp also have "(x \ y) \ \ = \ (insert (x \ y) {x \ a |a. a \ A})" using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def fin]) also have "insert (x\y) {x\a |a. a \ A} = {x\a |a. a \ insert y A}" by blast finally show ?case . qed lemma (in Distrib_Lattice) sup_Inf2_distrib: assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" shows "(\A \ \B) = \{a \ b|a b. a \ A \ b \ B}" using A proof (induct rule: finite_ne_induct) case singleton thus ?case by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def]) next case (insert x A) have finB: "finite {x \ b |b. b \ B}" by(fast intro: finite_surj[where f = "%b. x \ b", OF B(1)]) have finAB: "finite {a \ b |a b. a \ A \ b \ B}" proof - have "{a \ b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {a \ b})" by blast thus ?thesis by(simp add: insert(1) B(1)) qed have ne: "{a \ b |a b. a \ A \ b \ B} \ {}" using insert B by blast have "\(insert x A) \ \B = (x \ \A) \ \B" using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def]) also have "\ = (x \ \B) \ (\A \ \B)" by(rule sup_inf_distrib2) also have "\ = \{x \ b|b. b \ B} \ \{a \ b|a b. a \ A \ b \ B}" using insert by(simp add:sup_Inf1_distrib[OF B]) also have "\ = \({x\b |b. b \ B} \ {a\b |a b. a \ A \ b \ B})" (is "_ = \?M") using B insert by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne]) also have "?M = {a \ b |a b. a \ insert x A \ b \ B}" by blast finally show ?case . qed subsection{*Min and Max*} text{* As an application of @{text fold1} we define the minimal and maximal element of a (non-empty) set over a linear order. *} constdefs Min :: "('a::linorder)set => 'a" "Min == fold1 min" Max :: "('a::linorder)set => 'a" "Max == fold1 max" text{* Before we can do anything, we need to show that @{text min} and @{text max} are ACI and the ordering is linear: *} lemma ACf_min: "ACf(min :: 'a::linorder \ 'a \ 'a)" apply(rule ACf.intro) apply(auto simp:min_def) done lemma ACIf_min: "ACIf(min:: 'a::linorder \ 'a \ 'a)" apply(rule ACIf.intro[OF ACf_min]) apply(rule ACIf_axioms.intro) apply(auto simp:min_def) done lemma ACf_max: "ACf(max :: 'a::linorder \ 'a \ 'a)" apply(rule ACf.intro) apply(auto simp:max_def) done lemma ACIf_max: "ACIf(max:: 'a::linorder \ 'a \ 'a)" apply(rule ACIf.intro[OF ACf_max]) apply(rule ACIf_axioms.intro) apply(auto simp:max_def) done lemma ACIfSL_min: "ACIfSL(min :: 'a::linorder \ 'a \ 'a) (op \)" apply(rule ACIfSL.intro) apply(rule ACf_min) apply(rule ACIf.axioms[OF ACIf_min]) apply(rule ACIfSL_axioms.intro) apply(auto simp:min_def) done lemma ACIfSLlin_min: "ACIfSLlin(min :: 'a::linorder \ 'a \ 'a) (op \)" apply(rule ACIfSLlin.intro) apply(rule ACf_min) apply(rule ACIf.axioms[OF ACIf_min]) apply(rule ACIfSL.axioms[OF ACIfSL_min]) apply(rule ACIfSLlin_axioms.intro) apply(auto simp:min_def) done lemma ACIfSL_max: "ACIfSL(max :: 'a::linorder \ 'a \ 'a) (%x y. y\x)" apply(rule ACIfSL.intro) apply(rule ACf_max) apply(rule ACIf.axioms[OF ACIf_max]) apply(rule ACIfSL_axioms.intro) apply(auto simp:max_def) done lemma ACIfSLlin_max: "ACIfSLlin(max :: 'a::linorder \ 'a \ 'a) (%x y. y\x)" apply(rule ACIfSLlin.intro) apply(rule ACf_max) apply(rule ACIf.axioms[OF ACIf_max]) apply(rule ACIfSL.axioms[OF ACIfSL_max]) apply(rule ACIfSLlin_axioms.intro) apply(auto simp:max_def) done lemma Lattice_min_max: "Lattice (op \) (min :: 'a::linorder \ 'a \ 'a) max" apply(rule Lattice.intro) apply(rule partial_order_order) apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min]) apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max]) done lemma Distrib_Lattice_min_max: "Distrib_Lattice (op \) (min :: 'a::linorder \ 'a \ 'a) max" apply(rule Distrib_Lattice.intro) apply(rule partial_order_order) apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min]) apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max]) apply(rule distrib_lattice.axioms[OF distrib_lattice_min_max]) done text{* Now we instantiate the recursion equations and declare them simplification rules: *} declare fold1_singleton_def[OF Min_def, simp] ACIf.fold1_insert_idem_def[OF ACIf_min Min_def, simp] fold1_singleton_def[OF Max_def, simp] ACIf.fold1_insert_idem_def[OF ACIf_max Max_def, simp] text{* Now we instantiate some @{text fold1} properties: *} lemma Min_in [simp]: shows "finite A \ A \ {} \ Min A \ A" using ACf.fold1_in[OF ACf_min] by(fastsimp simp: Min_def min_def) lemma Max_in [simp]: shows "finite A \ A \ {} \ Max A \ A" using ACf.fold1_in[OF ACf_max] by(fastsimp simp: Max_def max_def) lemma Min_le [simp]: "\ finite A; A \ {}; x \ A \ \ Min A \ x" by(simp add: Min_def ACIfSL.fold1_belowI[OF ACIfSL_min]) lemma Max_ge [simp]: "\ finite A; A \ {}; x \ A \ \ x \ Max A" by(simp add: Max_def ACIfSL.fold1_belowI[OF ACIfSL_max]) lemma Min_ge_iff[simp]: "\ finite A; A \ {} \ \ (x \ Min A) = (\a\A. x \ a)" by(simp add: Min_def ACIfSL.below_fold1_iff[OF ACIfSL_min]) lemma Max_le_iff[simp]: "\ finite A; A \ {} \ \ (Max A \ x) = (\a\A. a \ x)" by(simp add: Max_def ACIfSL.below_fold1_iff[OF ACIfSL_max]) lemma Min_le_iff: "\ finite A; A \ {} \ \ (Min A \ x) = (\a\A. a \ x)" by(simp add: Min_def ACIfSLlin.fold1_below_iff[OF ACIfSLlin_min]) lemma Max_ge_iff: "\ finite A; A \ {} \ \ (x \ Max A) = (\a\A. x \ a)" by(simp add: Max_def ACIfSLlin.fold1_below_iff[OF ACIfSLlin_max]) lemma Min_le_Max: "\ finite A; A \ {} \ \ Min A \ Max A" by(simp add: Min_def Max_def Lattice.Inf_le_Sup[OF Lattice_min_max]) lemma max_Min2_distrib: "\ finite A; A \ {}; finite B; B \ {} \ \ max (Min A) (Min B) = Min{ max a b |a b. a \ A \ b \ B}" by(simp add: Min_def Distrib_Lattice.sup_Inf2_distrib[OF Distrib_Lattice_min_max]) end