(* Title: HOL/Finite_Set.thy Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel with contributions by Jeremy Avigad *) header {* Finite sets *} theory Finite_Set imports Nat Product_Type Power begin subsection {* Definition and basic properties *} inductive finite :: "'a set => bool" where emptyI [simp, intro!]: "finite {}" | insertI [simp, intro!]: "finite A ==> finite (insert a A)" 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 assms have "A \ UNIV" by blast thus ?thesis by blast qed lemma finite_induct [case_names empty insert, induct set: finite]: "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 {}" by fact 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 `P {x}` by simp next assume "F \ {}" thus ?thesis using insert by blast qed qed lemma finite_subset_induct [consumes 2, case_names empty insert]: assumes "finite F" and "F \ A" and empty: "P {}" and insert: "!!a F. finite F ==> a \ A ==> a \ F ==> P F ==> P (insert a F)" shows "P F" proof - from `finite F` and `F \ A` show ?thesis proof induct show "P {}" by fact next 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" . show "finite F" by fact show "x \ F" by fact qed qed qed text{* A finite choice principle. Does not need the SOME choice operator. *} lemma finite_set_choice: "finite A \ ALL x:A. (EX y. P x y) \ EX f. ALL x:A. P x (f x)" proof (induct set: finite) case empty thus ?case by simp next case (insert a A) then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto show ?case (is "EX f. ?P f") proof show "?P(%x. if x = a then b else f x)" using f ab by auto 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" by fact 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)" by (induct set: finite) 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})" by fact+ 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" using x by (rule insert_Diff) finally show ?thesis . next show "A \ F ==> ?thesis" by fact 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_Collect_disjI[simp]: "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})" by(simp add:Collect_disj_eq) lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" -- {* The converse obviously fails. *} by (blast intro: finite_subset) lemma finite_Collect_conjI [simp, intro]: "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}" -- {* The converse obviously fails. *} by(simp add:Collect_conj_eq) lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}" by(simp add: le_eq_less_or_eq) 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: assumes "finite A" and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" shows "P {}" proof - have "P (A - A)" proof - { fix c b :: "'a set" assume c: "finite c" and b: "finite b" and P1: "P b" and P2: "!!x y. finite y ==> x \ y ==> P y ==> P (y - {x})" have "c \ b ==> P (b - c)" using 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 } then show ?thesis by this (simp_all add: assms) qed then show ?thesis by simp qed lemma finite_Diff [simp]: "finite A ==> finite (A - B)" by (rule Diff_subset [THEN finite_subset]) lemma finite_Diff2 [simp]: assumes "finite B" shows "finite (A - B) = finite A" proof - have "finite A \ finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int) also have "\ \ finite(A-B)" using `finite B` by(simp) finally show ?thesis .. qed lemma finite_compl[simp]: "finite(A::'a set) \ finite(-A) = finite(UNIV::'a set)" by(simp add:Compl_eq_Diff_UNIV) lemma finite_Collect_not[simp]: "finite{x::'a. P x} \ finite{x. ~P x} = finite(UNIV::'a set)" by(simp add:Collect_neg_eq) 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: finite) 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 add: range_composition) 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: finite) apply 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: finite) 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) lemma finite_Collect_bex[simp]: "finite A \ finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})" apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})") apply auto done lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \ finite{x. EX y. P y & Q x y} = (ALL y. P y \ finite{x. Q x y})" apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})") apply auto done lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)" by (simp add: Plus_def) 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 finite{B. B \ A}" by(simp add: Pow_def[symmetric]) lemma finite_UnionD: "finite(\A) \ finite A" by(blast intro: finite_subset[OF subset_Pow_Union]) subsection {* Class @{text finite} *} setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*} class finite = assumes finite_UNIV: "finite (UNIV \ 'a set)" setup {* Sign.parent_path *} hide const finite context finite begin lemma finite [simp]: "finite (A \ 'a set)" by (rule subset_UNIV finite_UNIV finite_subset)+ end lemma UNIV_unit [noatp]: "UNIV = {()}" by auto instance unit :: finite by default (simp add: UNIV_unit) lemma UNIV_bool [noatp]: "UNIV = {False, True}" by auto instance bool :: finite by default (simp add: UNIV_bool) instance * :: (finite, finite) finite by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite) lemma inj_graph: "inj (%f. {(x, y). y = f x})" by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq) instance "fun" :: (finite, finite) finite proof show "finite (UNIV :: ('a => 'b) set)" proof (rule finite_imageD) let ?graph = "%f::'a => 'b. {(x, y). y = f x}" have "range ?graph \ Pow UNIV" by simp moreover have "finite (Pow (UNIV :: ('a * 'b) set))" by (simp only: finite_Pow_iff finite) ultimately show "finite (range ?graph)" by (rule finite_subset) show "inj ?graph" by (rule inj_graph) qed qed instance "+" :: (finite, finite) finite by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite) subsection {* A fold functional for finite sets *} text {* The intended behaviour is @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\ (f x\<^isub>n z)\)"} if @{text f} is ``left-commutative'': *} locale fun_left_comm = fixes f :: "'a \ 'b \ 'b" assumes fun_left_comm: "f x (f y z) = f y (f x z)" begin text{* On a functional level it looks much nicer: *} lemma fun_comp_comm: "f x \ f y = f y \ f x" by (simp add: fun_left_comm expand_fun_eq) end inductive fold_graph :: "('a \ 'b \ 'b) \ 'b \ 'a set \ 'b \ bool" for f :: "'a \ 'b \ 'b" and z :: 'b where emptyI [intro]: "fold_graph f z {} z" | insertI [intro]: "x \ A \ fold_graph f z A y \ fold_graph f z (insert x A) (f x y)" inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x" definition fold :: "('a \ 'b \ 'b) \ 'b \ 'a set \ 'b" where [code del]: "fold f z A = (THE y. fold_graph f z A y)" text{*A tempting alternative for the definiens is @{term "if finite A then THE y. fold_graph f z A y else e"}. It allows the removal of finiteness assumptions from the theorems @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}. The proofs become ugly. It is not worth the effort. (???) *} lemma Diff1_fold_graph: "fold_graph f z (A - {x}) y \ x \ A \ fold_graph f z A (f x y)" by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto) lemma fold_graph_imp_finite: "fold_graph f z A x \ finite A" by (induct set: fold_graph) auto lemma finite_imp_fold_graph: "finite A \ \x. fold_graph f z A x" by (induct set: finite) auto subsubsection{*From @{const fold_graph} 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 context fun_left_comm begin lemma fold_graph_determ_aux: "A = h`{i::nat. i inj_on h {i. i fold_graph f z A x \ fold_graph f z A x' \ x' = x" proof (induct n arbitrary: A x x' h rule: less_induct) case (less n) have IH: "\m h A x x'. m < n \ A = h ` {i. i inj_on h {i. i fold_graph f z A x \ fold_graph f z A x' \ x' = x" by fact have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'" and A: "A = h`{i. i B" and Bu: "fold_graph f z B u" show "x'=x" proof (rule fold_graph.cases [OF Afoldx']) assume "A = {}" and "x' = z" with AbB show "x' = x" by blast next fix C c v assume AcC: "A = insert c C" and x': "x' = f c v" and notinC: "c \ C" and Cv: "fold_graph f z C v" 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 fold_graph_imp_finite [OF Afoldx]) with AbB have "finite ?D" by simp then obtain d where Dfoldd: "fold_graph f z ?D d" using finite_imp_fold_graph by iprover moreover have cinB: "c \ B" using B by auto ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph) hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) moreover have "f b d = v" proof (rule IH[OF lessC Ceq inj_onC Cv]) show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp qed ultimately show ?thesis using fun_left_comm [of c b] x x' by (auto simp add: o_def) qed qed qed qed lemma fold_graph_determ: "fold_graph f z A x \ fold_graph f z A y \ y = x" apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) apply (blast intro: fold_graph_determ_aux [rule_format]) done lemma fold_equality: "fold_graph f z A y \ fold f z A = y" by (unfold fold_def) (blast intro: fold_graph_determ) text{* The base case for @{text fold}: *} lemma (in -) fold_empty [simp]: "fold f z {} = z" by (unfold fold_def) blast text{* The various recursion equations for @{const fold}: *} lemma fold_insert_aux: "x \ A \ fold_graph f z (insert x A) v \ (\y. fold_graph f z A y \ v = f x y)" apply auto apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE]) apply (fastsimp dest: fold_graph_imp_finite) apply (blast intro: fold_graph_determ) done lemma fold_insert [simp]: "finite A ==> x \ A ==> fold f z (insert x A) = f x (fold f z A)" apply (simp add: fold_def fold_insert_aux) apply (rule the_equality) apply (auto intro: finite_imp_fold_graph cong add: conj_cong simp add: fold_def[symmetric] fold_equality) done lemma fold_fun_comm: "finite A \ f x (fold f z A) = fold f (f x z) A" proof (induct rule: finite_induct) case empty then show ?case by simp next case (insert y A) then show ?case by (simp add: fun_left_comm[of x]) qed lemma fold_insert2: "finite A \ x \ A \ fold f z (insert x A) = fold f (f x z) A" by (simp add: fold_insert fold_fun_comm) lemma fold_rec: assumes "finite A" and "x \ A" shows "fold f z A = f x (fold f z (A - {x}))" proof - have A: "A = insert x (A - {x})" using `x \ A` by blast then have "fold f z A = fold f z (insert x (A - {x}))" by simp also have "\ = f x (fold f z (A - {x}))" by (rule fold_insert) (simp add: `finite A`)+ finally show ?thesis . qed lemma fold_insert_remove: assumes "finite A" shows "fold f z (insert x A) = f x (fold f z (A - {x}))" proof - from `finite A` have "finite (insert x A)" by auto moreover have "x \ insert x A" by auto ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))" by (rule fold_rec) then show ?thesis by simp qed end text{* A simplified version for idempotent functions: *} locale fun_left_comm_idem = fun_left_comm + assumes fun_left_idem: "f x (f x z) = f x z" begin text{* The nice version: *} lemma fun_comp_idem : "f x o f x = f x" by (simp add: fun_left_idem expand_fun_eq) lemma fold_insert_idem: assumes fin: "finite A" shows "fold f z (insert x A) = f x (fold f z A)" proof cases assume "x \ A" then obtain B where "A = insert x B" and "x \ B" by (rule set_insert) then show ?thesis using assms by (simp add:fun_left_idem) next assume "x \ A" then show ?thesis using assms by simp qed declare fold_insert[simp del] fold_insert_idem[simp] lemma fold_insert_idem2: "finite A \ fold f z (insert x A) = fold f (f x z) A" by(simp add:fold_fun_comm) end subsubsection{* The derived combinator @{text fold_image} *} definition fold_image :: "('b \ 'b \ 'b) \ ('a \ 'b) \ 'b \ 'a set \ 'b" where "fold_image f g = fold (%x y. f (g x) y)" lemma fold_image_empty[simp]: "fold_image f g z {} = z" by(simp add:fold_image_def) context ab_semigroup_mult begin lemma fold_image_insert[simp]: assumes "finite A" and "a \ A" shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)" proof - interpret I: fun_left_comm "%x y. (g x) * y" by unfold_locales (simp add: mult_ac) show ?thesis using assms by(simp add:fold_image_def I.fold_insert) qed (* lemma fold_commute: "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)" apply (induct set: finite) apply simp apply (simp add: mult_left_commute [of x]) done lemma fold_nest_Un_Int: "finite A ==> finite B ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)" apply (induct set: finite) apply simp apply (simp add: fold_commute Int_insert_left insert_absorb) done lemma fold_nest_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> fold times g z (A Un B) = fold times g (fold times g z B) A" by (simp add: fold_nest_Un_Int) *) lemma fold_image_reindex: assumes fin: "finite A" shows "inj_on h A \ fold_image times g z (h`A) = fold_image times (g\h) z A" using fin apply induct apply simp apply simp 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 fold_fusion: assumes "ab_semigroup_mult g" assumes fin: "finite A" and hyp: "\x y. h (g x y) = times x (h y)" shows "h (fold g j w A) = fold times j (h w) A" proof - class_interpret ab_semigroup_mult [g] by fact show ?thesis using fin hyp by (induct set: finite) simp_all qed *) lemma fold_image_cong: "finite A \ (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A" apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times 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 end context comm_monoid_mult begin lemma fold_image_Un_Int: "finite A ==> finite B ==> fold_image times g 1 A * fold_image times g 1 B = fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)" by (induct set: finite) (auto simp add: mult_ac insert_absorb Int_insert_left) corollary fold_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> fold_image times g 1 (A Un B) = fold_image times g 1 A * fold_image times g 1 B" by (simp add: fold_image_Un_Int) lemma fold_image_UN_disjoint: "\ finite I; ALL i:I. finite (A i); ALL i:I. ALL j:I. i \ j --> A i Int A j = {} \ \ fold_image times g 1 (UNION I A) = fold_image times (%i. fold_image times g 1 (A i)) 1 I" apply (induct set: finite, 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 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==> fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A = fold_image times (split g) 1 (SIGMA x:A. B x)" apply (subst Sigma_def) apply (subst fold_image_UN_disjoint, assumption, simp) apply blast apply (erule fold_image_cong) apply (subst fold_image_UN_disjoint, simp, simp) apply blast apply simp done lemma fold_image_distrib: "finite A \ fold_image times (%x. g x * h x) 1 A = fold_image times g 1 A * fold_image times h 1 A" by (erule finite_induct) (simp_all add: mult_ac) end subsection {* Generalized summation over a set *} interpretation comm_monoid_add!: comm_monoid_mult "0::'a::comm_monoid_add" "op +" proof qed (auto intro: add_assoc add_commute) definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" where "setsum f A == if finite A then fold_image (op +) f 0 A else 0" abbreviation Setsum ("\_" [1000] 999) where "\A == setsum (%x. x) A" text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is written @{text"\x\A. e"}. *} syntax "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) syntax (xsymbols) "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) syntax (HTML output) "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) translations -- {* Beware of argument permutation! *} "SUM i:A. b" == "CONST setsum (%i. b) A" "\i\A. b" == "CONST setsum (%i. b) A" text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter @{text"\x|P. e"}. *} syntax "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) syntax (xsymbols) "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) syntax (HTML output) "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) translations "SUM x|P. t" => "CONST setsum (%x. t) {x. P}" "\x|P. t" => "CONST setsum (%x. t) {x. P}" print_translation {* let fun 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) 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 comm_monoid_add.fold_image_reindex 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_reindex_nonzero: assumes fS: "finite S" and nz: "\ x y. x \ S \ y \ S \ x \ y \ f x = f y \ h (f x) = 0" shows "setsum h (f ` S) = setsum (h o f) S" using nz proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 x F) {assume fxF: "f x \ f ` F" hence "\y \ F . f y = f x" by auto then obtain y where y: "y \ F" "f x = f y" by auto from "2.hyps" y have xy: "x \ y" by auto from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto also have "\ = setsum (h o f) (insert x F)" unfolding setsum_insert[OF `finite F` `x\F`] using h0 apply simp apply (rule "2.hyps"(3)) apply (rule_tac y="y" in "2.prems") apply simp_all done finally have ?case .} moreover {assume fxF: "f x \ f ` F" have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" using fxF "2.hyps" by simp also have "\ = setsum (h o f) (insert x F)" unfolding setsum_insert[OF `finite F` `x\F`] apply simp apply (rule cong[OF refl[of "op + (h (f x))"]]) apply (rule "2.hyps"(3)) apply (rule_tac y="y" in "2.prems") apply simp_all done finally have ?case .} ultimately show ?case by blast qed lemma setsum_cong: "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong) lemma strong_setsum_cong[cong]: "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setsum (%x. f x) A = setsum (%x. g x) B" by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong) lemma setsum_cong2: "\\x. x \ A \ f x = g x\ \ setsum f A = setsum g A"; by (rule setsum_cong[OF refl], auto); lemma setsum_reindex_cong: "[|inj_on f A; B = f ` A; !!a. a:A \ g a = h (f a)|] ==> setsum h B = setsum g A" by (simp add: setsum_reindex cong: setsum_cong) lemma setsum_0[simp]: "setsum (%i. 0) A = 0" apply (clarsimp simp: setsum_def) apply (erule finite_induct, auto) done lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" by(simp add:setsum_cong) 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 comm_monoid_add.fold_image_Un_Int [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) lemma setsum_mono_zero_left: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. f i = 0" shows "setsum f S = setsum f T" proof- have eq: "T = S \ (T - S)" using ST by blast have d: "S \ (T - S) = {}" using ST by blast from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) show ?thesis by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) qed lemma setsum_mono_zero_right: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. f i = 0" shows "setsum f T = setsum f S" using setsum_mono_zero_left[OF fT ST z] by simp lemma setsum_mono_zero_cong_left: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. g i = 0" and fg: "\x. x \ S \ f x = g x" shows "setsum f S = setsum g T" proof- have eq: "T = S \ (T - S)" using ST by blast have d: "S \ (T - S) = {}" using ST by blast from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) show ?thesis using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) qed lemma setsum_mono_zero_cong_right: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. f i = 0" and fg: "\x. x \ S \ f x = g x" shows "setsum f T = setsum g S" using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto lemma setsum_delta: assumes fS: "finite S" shows "setsum (\k. if k=a then b k else 0) S = (if a \ S then b a else 0)" proof- let ?f = "(\k. if k=a then b k else 0)" {assume a: "a \ S" hence "\ k\ S. ?f k = 0" by simp hence ?thesis using a by simp} moreover {assume a: "a \ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A \ ?B" using a by blast have dj: "?A \ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a by simp} ultimately show ?thesis by blast qed lemma setsum_delta': assumes fS: "finite S" shows "setsum (\k. if a = k then b k else 0) S = (if a\ S then b a else 0)" using setsum_delta[OF fS, of a b, symmetric] by (auto intro: setsum_cong) (*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 comm_monoid_add.fold_image_UN_disjoint 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)) = (\(x,y)\(SIGMA x:A. B x). f x y)" by(simp add:setsum_def comm_monoid_add.fold_image_Sigma 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)) = (\(x,y) \ A <*> B. f x y)" apply (cases "finite A") apply (cases "finite B") apply (simp add: setsum_Sigma) apply (cases "A={}", simp) apply (simp) 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 comm_monoid_add.fold_image_distrib) subsubsection {* Properties in more restricted classes of structures *} lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" apply (case_tac "finite A") prefer 2 apply (simp add: setsum_def) apply (erule rev_mp) apply (erule finite_induct, auto) done lemma setsum_eq_0_iff [simp]: "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" by (induct set: finite) auto lemma setsum_Un_nat: "finite A ==> finite B ==> (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" -- {* For the natural numbers, we have subtraction. *} by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) lemma setsum_Un: "finite A ==> finite B ==> (setsum f (A Un B) :: 'a :: ab_group_add) = setsum f A + setsum f B - setsum f (A Int B)" by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) lemma 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) lemma setsum_diff1'[rule_format]: "finite A \ a \ A \ (\ x \ A. f x) = f a + (\ x \ (A - {a}). f x)" apply (erule finite_induct[where F=A and P="% A. (a \ A \ (\ x \ A. f x) = f a + (\ x \ (A - {a}). f x))"]) apply (auto simp add: insert_Diff_if add_ac) done (* By Jeremy Siek: *) lemma setsum_diff_nat: assumes "finite B" and "B \ A" shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" using assms proof induct show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp next fix F x assume finF: "finite F" and xnotinF: "x \ 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 fastsimp qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_strict_mono: fixes f :: "'a \ 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}" assumes "finite A" "A \ {}" and "!!x. x:A \ f x < g x" shows "setsum f A < setsum g A" using prems proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next case insert thus ?case by (auto simp: add_strict_mono) qed lemma setsum_negf: "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" proof (cases "finite A") case True thus ?thesis by (induct set: finite) auto next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_subtractf: "setsum (%x. ((f x)::'a::ab_group_add) - g x) A = setsum f A - setsum g A" proof (cases "finite A") 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 proof induct case empty then show ?case by simp next case (insert x F) then have "0 + 0 \ f x + setsum f F" by (blast intro: add_mono) with insert show ?case by simp qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_nonpos: 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 proof induct case empty then show ?case by simp next case (insert x F) then have "f x + setsum f F \ 0 + 0" by (blast intro: add_mono) with insert show ?case by simp qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_mono2: fixes f :: "'a \ 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}" assumes fin: "finite B" and sub: "A \ B" and nn: "\b. b \ B-A \ 0 \ f b" shows "setsum f A \ setsum f B" proof - have "setsum f A \ setsum f A + setsum f (B-A)" by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def) 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_mono3: "finite B ==> A <= B ==> ALL x: B - A. 0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==> setsum f A <= setsum f B" apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)") apply (erule ssubst) apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)") apply simp apply (rule add_left_mono) apply (erule setsum_nonneg) apply (subst setsum_Un_disjoint [THEN sym]) apply (erule finite_subset, assumption) apply (rule finite_subset) prefer 2 apply assumption apply auto apply (rule setsum_cong) apply auto done lemma setsum_right_distrib: fixes f :: "'a => ('b::semiring_0)" shows "r * setsum f A = setsum (%n. r * f n) A" proof (cases "finite A") case True thus ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: right_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_left_distrib: "setsum f A * (r::'a::semiring_0) = (\n\A. f n * r)" proof (cases "finite A") case True then show ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: left_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_divide_distrib: "setsum f A / (r::'a::field) = (\n\A. f n / r)" proof (cases "finite A") case True then show ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: add_divide_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_abs[iff]: fixes f :: "'a => ('b::pordered_ab_group_add_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::pordered_ab_group_add_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 simp: add_nonneg_nonneg) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma abs_setsum_abs[simp]: fixes f :: "'a => ('b::pordered_ab_group_add_abs)" shows "abs (\a\A. abs(f a)) = (\a\A. abs(f a))" proof (cases "finite A") case True thus ?thesis proof induct case empty thus ?case by simp next case (insert a A) hence "\\a\insert a A. \f a\\ = \\f a\ + (\a\A. \f a\)\" by simp also have "\ = \\f a\ + \\a\A. \f a\\\" using insert by simp also have "\ = \f a\ + \\a\A. \f a\\" by (simp del: abs_of_nonneg) also have "\ = (\a\insert a A. \f a\)" using insert by simp finally show ?case . qed next case False thus ?thesis by (simp add: setsum_def) qed text {* Commuting outer and inner summation *} lemma swap_inj_on: "inj_on (%(i, j). (j, i)) (A \ B)" by (unfold inj_on_def) fast lemma swap_product: "(%(i, j). (j, i)) ` (A \ B) = B \ A" by (simp add: split_def image_def) blast lemma setsum_commute: "(\i\A. \j\B. f i j) = (\j\B. \i\A. f i j)" proof (simp add: setsum_cartesian_product) have "(\(x,y) \ A <*> B. f x y) = (\(y,x) \ (%(i, j). (j, i)) ` (A \ B). f x y)" (is "?s = _") apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on) apply (simp add: split_def) done also have "... = (\(y,x)\B \ A. f x y)" (is "_ = ?t") apply (simp add: swap_product) done finally show "?s = ?t" . qed lemma setsum_product: fixes f :: "'a => ('b::semiring_0)" shows "setsum f A * setsum g B = (\i\A. \j\B. f i * g j)" by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute) subsection {* Generalized product over a set *} definition setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult" where "setprod f A == if finite A then fold_image (op *) f 1 A else 1" abbreviation Setprod ("\_" [1000] 999) where "\A == setprod (%x. x) A" syntax "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3PROD _:_. _)" [0, 51, 10] 10) syntax (xsymbols) "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) syntax (HTML output) "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) translations -- {* Beware of argument permutation! *} "PROD i:A. b" == "CONST setprod (%i. b) A" "\i\A. b" == "CONST setprod (%i. b) A" text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter @{text"\x|P. e"}. *} syntax "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10) syntax (xsymbols) "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) syntax (HTML output) "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) translations "PROD x|P. t" => "CONST setprod (%x. t) {x. P}" "\x|P. t" => "CONST setprod (%x. t) {x. P}" 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) 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 fold_image_reindex 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: fold_image_cong) lemma strong_setprod_cong: "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B" by(fastsimp simp: simp_implies_def setprod_def intro: fold_image_cong) lemma setprod_reindex_cong: "inj_on f A ==> B = f ` A ==> g = h \ f ==> setprod h B = setprod g A" by (frule setprod_reindex, simp) lemma strong_setprod_reindex_cong: assumes i: "inj_on f A" and B: "B = f ` A" and eq: "\x. x \ A \ g x = (h \ f) x" shows "setprod h B = setprod g A" proof- have "setprod h B = setprod (h o f) A" by (simp add: B setprod_reindex[OF i, of h]) then show ?thesis apply simp apply (rule setprod_cong) apply simp by (erule eq[symmetric]) qed 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 fold_image_Un_Int[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_delta: assumes fS: "finite S" shows "setprod (\k. if k=a then b k else 1) S = (if a \ S then b a else 1)" proof- let ?f = "(\k. if k=a then b k else 1)" {assume a: "a \ S" hence "\ k\ S. ?f k = 1" by simp hence ?thesis using a by (simp add: setprod_1 cong add: setprod_cong) } moreover {assume a: "a \ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A \ ?B" using a by blast have dj: "?A \ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto have "setprod ?f ?A * setprod ?f ?B = setprod ?f S" using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a by (simp add: fA1 cong add: setprod_cong cong del: if_weak_cong)} ultimately show ?thesis by blast qed lemma setprod_delta': assumes fS: "finite S" shows "setprod (\k. if a = k then b k else 1) S = (if a\ S then b a else 1)" using setprod_delta[OF fS, of a b, symmetric] by (auto intro: setprod_cong) lemma setprod_UN_disjoint: "finite I ==> (ALL i:I. finite (A i)) ==> (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> setprod f (UNION I A) = setprod (%i. setprod f (A i)) I" by(simp add: setprod_def fold_image_UN_disjoint cong: setprod_cong) lemma setprod_Union_disjoint: "[| (ALL A:C. finite A); (ALL A:C. ALL B:C. A \ 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)) = (\(x,y)\(SIGMA x:A. B x). f x y)" by(simp add:setprod_def fold_image_Sigma 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)) = (\(x,y)\(A <*> B). f x y)" 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 fold_image_distrib) subsubsection {* Properties in more restricted classes of structures *} lemma setprod_eq_1_iff [simp]: "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))" by (induct set: finite) auto lemma setprod_zero: "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0" apply (induct set: finite, force, clarsimp) apply (erule disjE, auto) done lemma setprod_nonneg [rule_format]: "(ALL x: A. (0::'a::ordered_idom) \ f x) --> 0 \ setprod f A" apply (case_tac "finite A") apply (induct set: finite, 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: finite, 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) * y = 0 --> x = 0 | y = 0) ==> finite A ==> (ALL x: A. f x \ (0::'a)) --> setprod f A \ 0" by (erule finite_induct, auto) lemma setprod_zero_eq: "(ALL x y. (x::'a::comm_semiring_1) * y = 0 --> x = 0 | y = 0) ==> finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)" by (insert setprod_zero [of A f] setprod_nonzero [of A f], blast) lemma setprod_nonzero_field: "finite A ==> (ALL x: A. f x \ (0::'a::idom)) ==> setprod f A \ 0" by (rule setprod_nonzero, auto) lemma setprod_zero_eq_field: "finite A ==> (setprod f A = (0::'a::idom)) = (EX x: A. f x = 0)" by (rule setprod_zero_eq, auto) 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) 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)" by (erule finite_induct) (auto simp add: insert_Diff_if) lemma setprod_inversef: "finite A ==> ALL x: A. f x \ (0::'a::{field,division_by_zero}) ==> setprod (inverse \ f) A = inverse (setprod f A)" by (erule finite_induct) auto 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 lemma setprod_dvd_setprod [rule_format]: "(ALL x : A. f x dvd g x) \ setprod f A dvd setprod g A" apply (cases "finite A") apply (induct set: finite) apply (auto simp add: dvd_def) apply (rule_tac x = "k * ka" in exI) apply (simp add: algebra_simps) done lemma setprod_dvd_setprod_subset: "finite B \ A <= B \ setprod f A dvd setprod f B" apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)") apply (unfold dvd_def, blast) apply (subst setprod_Un_disjoint [symmetric]) apply (auto elim: finite_subset intro: setprod_cong) done lemma setprod_dvd_setprod_subset2: "finite B \ A <= B \ ALL x : A. (f x::'a::comm_semiring_1) dvd g x \ setprod f A dvd setprod g B" apply (rule dvd_trans) apply (rule setprod_dvd_setprod, erule (1) bspec) apply (erule (1) setprod_dvd_setprod_subset) done lemma dvd_setprod: "finite A \ i:A \ (f i ::'a::comm_semiring_1) dvd setprod f A" by (induct set: finite) (auto intro: dvd_mult) lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \ (d::'a::comm_semiring_1) dvd (SUM x : A. f x)" apply (cases "finite A") apply (induct set: finite) apply 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: *} definition card :: "'a set \ nat" where "card A = setsum (\x. 1) 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) 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,noatp]: "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_Diff_insert[simp]: assumes "finite A" and "a:A" and "a ~: B" shows "card(A - insert a B) = card(A - B) - 1" proof - have "A - insert a B = (A - B) - {a}" using assms by blast then show ?thesis using assms by(simp add:card_Diff_singleton) qed lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert) lemma card_insert_le: "finite A ==> card A <= card (insert x A)" by (simp add: card_insert_if) lemma card_mono: "\ finite B; A \ B \ \ card A \ card B" by (simp add: card_def setsum_mono2) lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" apply (induct set: finite, simp, clarify) apply (subgoal_tac "finite A & A - {x} <= F") prefer 2 apply (blast intro: finite_subset, atomize) apply (drule_tac x = "A - {x}" in spec) apply (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_eq linorder_not_le [symmetric]) apply (blast dest: card_seteq) done lemma card_Un_Int: "finite A ==> finite B ==> card A + card B = card (A Un B) + card (A Int B)" 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 del:card_Diff_insert) done lemma card_Diff2_less: "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" apply (case_tac "x = y") apply (simp add: card_Diff1_less del:card_Diff_insert) apply (rule less_trans) prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert) done lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" apply (case_tac "x : A") apply (simp_all add: card_Diff1_less less_imp_le) done lemma card_psubset: "finite B ==> A \ 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 text{* 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 text{*The form of a finite set of given cardinality*} lemma card_eq_SucD: assumes "card A = Suc k" shows "\b B. A = insert b B & b \ B & card B = k & (k=0 \ B={})" proof - have fin: "finite A" using assms by (auto intro: ccontr) moreover have "card A \ 0" using assms by auto ultimately obtain b where b: "b \ A" by auto show ?thesis proof (intro exI conjI) show "A = insert b (A-{b})" using b by blast show "b \ A - {b}" by blast show "card (A - {b}) = k" and "k = 0 \ A - {b} = {}" using assms b fin by(fastsimp dest:mk_disjoint_insert)+ qed qed lemma card_Suc_eq: "(card A = Suc k) = (\b B. A = insert b B & b \ B & card B = k & (k=0 \ B={}))" apply(rule iffI) apply(erule card_eq_SucD) apply(auto) apply(subst card_insert) apply(auto intro:ccontr) done lemma setsum_constant [simp]: "(\x \ A. y) = of_nat(card A) * y" apply (cases "finite A") apply (erule finite_induct) apply (auto simp add: algebra_simps) done lemma setprod_constant: "finite A ==> (\x\ A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)" apply (erule finite_induct) apply (auto simp add: power_Suc) done lemma setprod_gen_delta: assumes fS: "finite S" shows "setprod (\k. if k=a then b k else c) S = (if a \ S then (b a ::'a::{comm_monoid_mult, recpower}) * c^ (card S - 1) else c^ card S)" proof- let ?f = "(\k. if k=a then b k else c)" {assume a: "a \ S" hence "\ k\ S. ?f k = c" by simp hence ?thesis using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) } moreover {assume a: "a \ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A \ ?B" using a by blast have dj: "?A \ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have fA0:"setprod ?f ?A = setprod (\i. c) ?A" apply (rule setprod_cong) by auto have cA: "card ?A = card S - 1" using fS a by auto have fA1: "setprod ?f ?A = c ^ card ?A" unfolding fA0 apply (rule setprod_constant) using fS by auto have "setprod ?f ?A * setprod ?f ?B = setprod ?f S" using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a cA by (simp add: fA1 ring_simps cong add: setprod_cong cong del: if_weak_cong)} ultimately show ?thesis by blast qed lemma setsum_bounded: assumes le: "\i. i\A \ f i \ (K::'a::{semiring_1, pordered_ab_semigroup_add})" shows "setsum f A \ of_nat(card A) * K" proof (cases "finite A") case True thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp next case False thus ?thesis by (simp add: setsum_def) qed 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 del: setsum_constant) apply (subgoal_tac "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I") apply (simp add: setsum_UN_disjoint del: setsum_constant) apply (simp cong: setsum_cong) done lemma card_Union_disjoint: "finite C ==> (ALL A:C. finite A) ==> (ALL A:C. ALL B:C. A \ B --> A Int B = {}) ==> card (Union C) = setsum card C" apply (frule card_UN_disjoint [of C id]) apply (unfold Union_def id_def, assumption+) done subsubsection {* Cardinality of image *} text{*The image of a finite set can be expressed using @{term fold_image}.*} lemma image_eq_fold_image: "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A" proof (induct rule: finite_induct) case empty then show ?case by simp next interpret ab_semigroup_mult "op Un" proof qed auto case insert then show ?case by simp qed lemma card_image_le: "finite A ==> card (f ` A) <= card A" apply (induct set: finite) apply simp 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 del:setsum_constant) 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) apply 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 del:setsum_constant) lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)" apply (cases "finite A") apply (cases "finite B") 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 sums *} lemma card_Plus: assumes "finite A" and "finite B" shows "card (A <+> B) = card A + card B" proof - have "Inl`A \ Inr`B = {}" by fast with assms show ?thesis unfolding Plus_def by (simp add: card_Un_disjoint card_image) qed subsubsection {* Cardinality of the Powerset *} lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) apply (induct set: finite) apply (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. *} 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: finite, simp_all, clarify) apply (subst card_Un_disjoint) apply (auto simp add: dvd_add disjoint_eq_subset_Compl) done subsubsection {* Relating injectivity and surjectivity *} lemma finite_surj_inj: "finite(A) \ A <= f`A \ inj_on f A" apply(rule eq_card_imp_inj_on, assumption) apply(frule finite_imageI) apply(drule (1) card_seteq) apply(erule card_image_le) apply simp done lemma finite_UNIV_surj_inj: fixes f :: "'a \ 'a" shows "finite(UNIV:: 'a set) \ surj f \ inj f" by (blast intro: finite_surj_inj subset_UNIV dest:surj_range) lemma finite_UNIV_inj_surj: fixes f :: "'a \ 'a" shows "finite(UNIV:: 'a set) \ inj f \ surj f" by(fastsimp simp:surj_def dest!: endo_inj_surj) corollary infinite_UNIV_nat: "~finite(UNIV::nat set)" proof assume "finite(UNIV::nat set)" with finite_UNIV_inj_surj[of Suc] show False by simp (blast dest: Suc_neq_Zero surjD) qed lemma infinite_UNIV_char_0: "\ finite (UNIV::'a::semiring_char_0 set)" proof assume "finite (UNIV::'a set)" with subset_UNIV have "finite (range of_nat::'a set)" by (rule finite_subset) moreover have "inj (of_nat::nat \ 'a)" by (simp add: inj_on_def) ultimately have "finite (UNIV::nat set)" by (rule finite_imageD) then show "False" by (simp add: infinite_UNIV_nat) qed subsection{* A fold functional for non-empty sets *} text{* Does not require start value. *} inductive fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool" for f :: "'a => 'a => 'a" where fold1Set_insertI [intro]: "\ fold_graph f a A x; a \ A \ \ fold1Set f (insert a A) x" constdefs fold1 :: "('a => 'a => 'a) => 'a set => 'a" "fold1 f A == THE x. fold1Set f A x" lemma fold1Set_nonempty: "fold1Set f A x \ A \ {}" by(erule fold1Set.cases, simp_all) inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x" inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x" lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)" by (blast intro: fold_graph.intros elim: fold_graph.cases) lemma fold1_singleton [simp]: "fold1 f {a} = a" by (unfold fold1_def) blast lemma finite_nonempty_imp_fold1Set: "\ finite A; A \ {} \ \ EX x. fold1Set f A x" apply (induct A rule: finite_induct) apply (auto dest: finite_imp_fold_graph [of _ f]) done text{*First, some lemmas about @{const fold_graph}.*} context ab_semigroup_mult begin lemma fun_left_comm: "fun_left_comm(op *)" by unfold_locales (simp add: mult_ac) lemma fold_graph_insert_swap: assumes fold: "fold_graph times (b::'a) A y" and "b \ A" shows "fold_graph times z (insert b A) (z * y)" proof - interpret fun_left_comm "op *::'a \ 'a \ 'a" by (rule fun_left_comm) from assms show ?thesis proof (induct rule: fold_graph.induct) case emptyI thus ?case by (force simp add: fold_insert_aux mult_commute) next case (insertI x A y) have "fold_graph times z (insert x (insert b A)) (x * (z * y))" using insertI by force --{*how does @{term id} get unfolded?*} thus ?case by (simp add: insert_commute mult_ac) qed qed lemma fold_graph_permute_diff: assumes fold: "fold_graph times b A x" shows "!!a. \a \ A; b \ A\ \ fold_graph times a (insert b (A-{a})) x" using fold proof (induct rule: fold_graph.induct) case emptyI thus ?case by simp next case (insertI x A 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: fold_graph_insert_swap) next assume ainA: "a \ A" hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)" using insertI by force moreover have "insert x (insert b (A - {a})) = insert b (insert x A - {a})" using ainA insertI by blast ultimately show ?thesis by simp qed qed lemma fold1_eq_fold: assumes "finite A" "a \ A" shows "fold1 times (insert a A) = fold times a A" proof - interpret fun_left_comm "op *::'a \ 'a \ 'a" by (rule fun_left_comm) from assms show ?thesis apply (simp add: fold1_def fold_def) apply (rule the_equality) apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times]) apply (rule sym, clarify) apply (case_tac "Aa=A") apply (best intro: the_equality fold_graph_determ) apply (subgoal_tac "fold_graph times a A x") apply (best intro: the_equality fold_graph_determ) apply (subgoal_tac "insert aa (Aa - {a}) = A") prefer 2 apply (blast elim: equalityE) apply (auto dest: fold_graph_permute_diff [where a=a]) done qed 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 fold1_insert: assumes nonempty: "A \ {}" and A: "finite A" "x \ A" shows "fold1 times (insert x A) = x * fold1 times A" proof - interpret fun_left_comm "op *::'a \ 'a \ 'a" by (rule fun_left_comm) 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 end context ab_semigroup_idem_mult begin lemma fun_left_comm_idem: "fun_left_comm_idem(op *)" apply unfold_locales apply (simp add: mult_ac) apply (simp add: mult_idem mult_assoc[symmetric]) done lemma fold1_insert_idem [simp]: assumes nonempty: "A \ {}" and A: "finite A" shows "fold1 times (insert x A) = x * fold1 times A" proof - interpret fun_left_comm_idem "op *::'a \ 'a \ 'a" by (rule fun_left_comm_idem) 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: mult_idem) next assume "A' \ {}" with prems show ?thesis by (simp add: fold1_insert mult_assoc [symmetric] mult_idem) qed next assume "a \ x" with prems show ?thesis by (simp add: insert_commute fold1_eq_fold fold_insert_idem) qed qed lemma hom_fold1_commute: assumes hom: "!!x y. h (x * y) = h x * h y" and N: "finite N" "N \ {}" shows "h (fold1 times N) = fold1 times (h ` N)" using N proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next case (insert n N) then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp also have "\ = h n * h (fold1 times N)" by(rule hom) also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert) also have "times (h n) \ = fold1 times (insert (h n) (h ` N))" using insert by(simp) also have "insert (h n) (h ` N) = h ` insert n N" by simp finally show ?case . qed end text{* Now the recursion rules for definitions: *} lemma fold1_singleton_def: "g = fold1 f \ g {a} = a" by(simp add:fold1_singleton) lemma (in ab_semigroup_mult) fold1_insert_def: "\ g = fold1 times; finite A; x \ A; A \ {} \ \ g (insert x A) = x * g A" by (simp add:fold1_insert) lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def: "\ g = fold1 times; finite A; A \ {} \ \ g (insert x A) = x * g A" by simp subsubsection{* Determinacy for @{term fold1Set} *} (*Not actually used!!*) (* context ab_semigroup_mult begin lemma fold_graph_permute: "[|fold_graph times id b (insert a A) x; a \ A; b \ A|] ==> fold_graph times id a (insert b A) x" apply (cases "a=b") apply (auto dest: fold_graph_permute_diff) done lemma fold1Set_determ: "fold1Set times A x ==> fold1Set times A y ==> y = x" proof (clarify elim!: fold1Set.cases) fix A x B y a b assume Ax: "fold_graph times id a A x" assume By: "fold_graph times id b B y" 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: fold_graph_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 "fold_graph times id a (insert b ?D) y" by (auto intro: fold_graph_permute simp add: insert_absorb) moreover have "fold_graph times id a (insert b ?D) x" by (simp add: A [symmetric] Ax) ultimately show ?thesis by (blast intro: fold_graph_determ) qed qed lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y" by (unfold fold1_def) (blast intro: fold1Set_determ) end *) declare empty_fold_graphE [rule del] fold_graph.intros [rule del] empty_fold1SetE [rule del] insert_fold1SetE [rule del] -- {* No more proofs involve these relations. *} subsubsection {* Lemmas about @{text fold1} *} context ab_semigroup_mult begin lemma fold1_Un: assumes A: "finite A" "A \ {}" shows "finite B \ B \ {} \ A Int B = {} \ fold1 times (A Un B) = fold1 times A * fold1 times B" using A by (induct rule: finite_ne_induct) (simp_all add: fold1_insert mult_assoc) lemma fold1_in: assumes A: "finite (A)" "A \ {}" and elem: "\x y. x * y \ {x,y}" shows "fold1 times 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 end lemma (in ab_semigroup_idem_mult) fold1_Un2: assumes A: "finite A" "A \ {}" shows "finite B \ B \ {} \ fold1 times (A Un B) = fold1 times A * fold1 times B" using A proof(induct rule:finite_ne_induct) case singleton thus ?case by simp next case insert thus ?case by (simp add: mult_assoc) qed subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *} text{* As an application of @{text fold1} we define infimum and supremum in (not necessarily complete!) lattices over (non-empty) sets by means of @{text fold1}. *} context lower_semilattice begin lemma ab_semigroup_idem_mult_inf: "ab_semigroup_idem_mult inf" proof qed (rule inf_assoc inf_commute inf_idem)+ lemma below_fold1_iff: assumes "finite A" "A \ {}" shows "x \ fold1 inf A \ (\a\A. x \ a)" proof - interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) show ?thesis using assms by (induct rule: finite_ne_induct) simp_all qed lemma fold1_belowI: assumes "finite A" and "a \ A" shows "fold1 inf A \ a" proof - from assms have "A \ {}" by auto from `finite A` `A \ {}` `a \ A` show ?thesis proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) case (insert x F) from insert(5) have "a = x \ a \ F" by simp thus ?case proof assume "a = x" thus ?thesis using insert by (simp add: mult_ac) next assume "a \ F" hence bel: "fold1 inf F \ a" by (rule insert) have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)" using insert by (simp add: mult_ac) also have "inf (fold1 inf F) a = fold1 inf F" using bel by (auto intro: antisym) also have "inf x \ = fold1 inf (insert x F)" using insert by (simp add: mult_ac) finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" . moreover have "inf (fold1 inf (insert x F)) a \ a" by simp ultimately show ?thesis by simp qed qed qed end lemma (in upper_semilattice) ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup" by (rule lower_semilattice.ab_semigroup_idem_mult_inf) (rule dual_lattice) context lattice begin definition Inf_fin :: "'a set \ 'a" ("\\<^bsub>fin\<^esub>_" [900] 900) where "Inf_fin = fold1 inf" definition Sup_fin :: "'a set \ 'a" ("\\<^bsub>fin\<^esub>_" [900] 900) where "Sup_fin = fold1 sup" lemma Inf_le_Sup [simp]: "\ finite A; A \ {} \ \ \\<^bsub>fin\<^esub>A \ \\<^bsub>fin\<^esub>A" apply(unfold Sup_fin_def Inf_fin_def) apply(subgoal_tac "EX a. a:A") prefer 2 apply blast apply(erule exE) apply(rule order_trans) apply(erule (1) fold1_belowI) apply(erule (1) lower_semilattice.fold1_belowI [OF dual_lattice]) done lemma sup_Inf_absorb [simp]: "finite A \ a \ A \ sup a (\\<^bsub>fin\<^esub>A) = a" apply(subst sup_commute) apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI) done lemma inf_Sup_absorb [simp]: "finite A \ a \ A \ inf a (\\<^bsub>fin\<^esub>A) = a" by (simp add: Sup_fin_def inf_absorb1 lower_semilattice.fold1_belowI [OF dual_lattice]) end context distrib_lattice begin lemma sup_Inf1_distrib: assumes "finite A" and "A \ {}" shows "sup x (\\<^bsub>fin\<^esub>A) = \\<^bsub>fin\<^esub>{sup x a|a. a \ A}" proof - interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) from assms show ?thesis by (simp add: Inf_fin_def image_def hom_fold1_commute [where h="sup x", OF sup_inf_distrib1]) (rule arg_cong [where f="fold1 inf"], blast) qed lemma sup_Inf2_distrib: assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" shows "sup (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B) = \\<^bsub>fin\<^esub>{sup 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_fin_def]) next interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) case (insert x A) have finB: "finite {sup x b |b. b \ B}" by(rule finite_surj[where f = "sup x", OF B(1)], auto) have finAB: "finite {sup a b |a b. a \ A \ b \ B}" proof - have "{sup a b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {sup a b})" by blast thus ?thesis by(simp add: insert(1) B(1)) qed have ne: "{sup a b |a b. a \ A \ b \ B} \ {}" using insert B by blast have "sup (\\<^bsub>fin\<^esub>(insert x A)) (\\<^bsub>fin\<^esub>B) = sup (inf x (\\<^bsub>fin\<^esub>A)) (\\<^bsub>fin\<^esub>B)" using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def]) also have "\ = inf (sup x (\\<^bsub>fin\<^esub>B)) (sup (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2) also have "\ = inf (\\<^bsub>fin\<^esub>{sup x b|b. b \ B}) (\\<^bsub>fin\<^esub>{sup a b|a b. a \ A \ b \ B})" using insert by(simp add:sup_Inf1_distrib[OF B]) also have "\ = \\<^bsub>fin\<^esub>({sup x b |b. b \ B} \ {sup a b |a b. a \ A \ b \ B})" (is "_ = \\<^bsub>fin\<^esub>?M") using B insert by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne]) also have "?M = {sup a b |a b. a \ insert x A \ b \ B}" by blast finally show ?case . qed lemma inf_Sup1_distrib: assumes "finite A" and "A \ {}" shows "inf x (\\<^bsub>fin\<^esub>A) = \\<^bsub>fin\<^esub>{inf x a|a. a \ A}" proof - interpret ab_semigroup_idem_mult sup by (rule ab_semigroup_idem_mult_sup) from assms show ?thesis by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1]) (rule arg_cong [where f="fold1 sup"], blast) qed lemma inf_Sup2_distrib: assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" shows "inf (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B) = \\<^bsub>fin\<^esub>{inf a b|a b. a \ A \ b \ B}" using A proof (induct rule: finite_ne_induct) case singleton thus ?case by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def]) next case (insert x A) have finB: "finite {inf x b |b. b \ B}" by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto) have finAB: "finite {inf a b |a b. a \ A \ b \ B}" proof - have "{inf a b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {inf a b})" by blast thus ?thesis by(simp add: insert(1) B(1)) qed have ne: "{inf a b |a b. a \ A \ b \ B} \ {}" using insert B by blast interpret ab_semigroup_idem_mult sup by (rule ab_semigroup_idem_mult_sup) have "inf (\\<^bsub>fin\<^esub>(insert x A)) (\\<^bsub>fin\<^esub>B) = inf (sup x (\\<^bsub>fin\<^esub>A)) (\\<^bsub>fin\<^esub>B)" using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def]) also have "\ = sup (inf x (\\<^bsub>fin\<^esub>B)) (inf (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2) also have "\ = sup (\\<^bsub>fin\<^esub>{inf x b|b. b \ B}) (\\<^bsub>fin\<^esub>{inf a b|a b. a \ A \ b \ B})" using insert by(simp add:inf_Sup1_distrib[OF B]) also have "\ = \\<^bsub>fin\<^esub>({inf x b |b. b \ B} \ {inf a b |a b. a \ A \ b \ B})" (is "_ = \\<^bsub>fin\<^esub>?M") using B insert by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne]) also have "?M = {inf a b |a b. a \ insert x A \ b \ B}" by blast finally show ?case . qed end context complete_lattice begin text {* Coincidence on finite sets in complete lattices: *} lemma Inf_fin_Inf: assumes "finite A" and "A \ {}" shows "\\<^bsub>fin\<^esub>A = Inf A" proof - interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) from assms show ?thesis unfolding Inf_fin_def by (induct A set: finite) (simp_all add: Inf_insert_simp) qed lemma Sup_fin_Sup: assumes "finite A" and "A \ {}" shows "\\<^bsub>fin\<^esub>A = Sup A" proof - interpret ab_semigroup_idem_mult sup by (rule ab_semigroup_idem_mult_sup) from assms show ?thesis unfolding Sup_fin_def by (induct A set: finite) (simp_all add: Sup_insert_simp) qed end subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *} text{* As an application of @{text fold1} we define minimum and maximum in (not necessarily complete!) linear orders over (non-empty) sets by means of @{text fold1}. *} context linorder begin lemma ab_semigroup_idem_mult_min: "ab_semigroup_idem_mult min" proof qed (auto simp add: min_def) lemma ab_semigroup_idem_mult_max: "ab_semigroup_idem_mult max" proof qed (auto simp add: max_def) lemma min_lattice: "lower_semilattice (op \) (op <) min" proof qed (auto simp add: min_def) lemma max_lattice: "lower_semilattice (op \) (op >) max" proof qed (auto simp add: max_def) lemma dual_max: "ord.max (op \) = min" by (auto simp add: ord.max_def_raw min_def_raw expand_fun_eq) lemma dual_min: "ord.min (op \) = max" by (auto simp add: ord.min_def_raw max_def_raw expand_fun_eq) lemma strict_below_fold1_iff: assumes "finite A" and "A \ {}" shows "x < fold1 min A \ (\a\A. x < a)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (induct rule: finite_ne_induct) (simp_all add: fold1_insert) qed lemma fold1_below_iff: assumes "finite A" and "A \ {}" shows "fold1 min A \ x \ (\a\A. a \ x)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (induct rule: finite_ne_induct) (simp_all add: fold1_insert min_le_iff_disj) qed lemma fold1_strict_below_iff: assumes "finite A" and "A \ {}" shows "fold1 min A < x \ (\a\A. a < x)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (induct rule: finite_ne_induct) (simp_all add: fold1_insert min_less_iff_disj) qed lemma fold1_antimono: assumes "A \ {}" and "A \ B" and "finite B" shows "fold1 min B \ fold1 min A" proof cases assume "A = B" thus ?thesis by simp next interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) assume "A \ B" have B: "B = A \ (B-A)" using `A \ B` by blast have "fold1 min B = fold1 min (A \ (B-A))" by(subst B)(rule refl) also have "\ = min (fold1 min A) (fold1 min (B-A))" proof - have "finite A" by(rule finite_subset[OF `A \ B` `finite B`]) moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *) moreover have "(B-A) \ {}" using prems by blast moreover have "A Int (B-A) = {}" using prems by blast ultimately show ?thesis using `A \ {}` by (rule_tac fold1_Un) qed also have "\ \ fold1 min A" by (simp add: min_le_iff_disj) finally show ?thesis . qed definition Min :: "'a set \ 'a" where "Min = fold1 min" definition Max :: "'a set \ 'a" where "Max = fold1 max" lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def] lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def] lemma Min_insert [simp]: assumes "finite A" and "A \ {}" shows "Min (insert x A) = min x (Min A)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (rule fold1_insert_idem_def [OF Min_def]) qed lemma Max_insert [simp]: assumes "finite A" and "A \ {}" shows "Max (insert x A) = max x (Max A)" proof - interpret ab_semigroup_idem_mult max by (rule ab_semigroup_idem_mult_max) from assms show ?thesis by (rule fold1_insert_idem_def [OF Max_def]) qed lemma Min_in [simp]: assumes "finite A" and "A \ {}" shows "Min A \ A" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def) qed lemma Max_in [simp]: assumes "finite A" and "A \ {}" shows "Max A \ A" proof - interpret ab_semigroup_idem_mult max by (rule ab_semigroup_idem_mult_max) from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def) qed lemma Min_Un: assumes "finite A" and "A \ {}" and "finite B" and "B \ {}" shows "Min (A \ B) = min (Min A) (Min B)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (simp add: Min_def fold1_Un2) qed lemma Max_Un: assumes "finite A" and "A \ {}" and "finite B" and "B \ {}" shows "Max (A \ B) = max (Max A) (Max B)" proof - interpret ab_semigroup_idem_mult max by (rule ab_semigroup_idem_mult_max) from assms show ?thesis by (simp add: Max_def fold1_Un2) qed lemma hom_Min_commute: assumes "\x y. h (min x y) = min (h x) (h y)" and "finite N" and "N \ {}" shows "h (Min N) = Min (h ` N)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (simp add: Min_def hom_fold1_commute) qed lemma hom_Max_commute: assumes "\x y. h (max x y) = max (h x) (h y)" and "finite N" and "N \ {}" shows "h (Max N) = Max (h ` N)" proof - interpret ab_semigroup_idem_mult max by (rule ab_semigroup_idem_mult_max) from assms show ?thesis by (simp add: Max_def hom_fold1_commute [of h]) qed lemma Min_le [simp]: assumes "finite A" and "x \ A" shows "Min A \ x" proof - interpret lower_semilattice "op \" "op <" min by (rule min_lattice) from assms show ?thesis by (simp add: Min_def fold1_belowI) qed lemma Max_ge [simp]: assumes "finite A" and "x \ A" shows "x \ Max A" proof - interpret lower_semilattice "op \" "op >" max by (rule max_lattice) from assms show ?thesis by (simp add: Max_def fold1_belowI) qed lemma Min_ge_iff [simp, noatp]: assumes "finite A" and "A \ {}" shows "x \ Min A \ (\a\A. x \ a)" proof - interpret lower_semilattice "op \" "op <" min by (rule min_lattice) from assms show ?thesis by (simp add: Min_def below_fold1_iff) qed lemma Max_le_iff [simp, noatp]: assumes "finite A" and "A \ {}" shows "Max A \ x \ (\a\A. a \ x)" proof - interpret lower_semilattice "op \" "op >" max by (rule max_lattice) from assms show ?thesis by (simp add: Max_def below_fold1_iff) qed lemma Min_gr_iff [simp, noatp]: assumes "finite A" and "A \ {}" shows "x < Min A \ (\a\A. x < a)" proof - interpret lower_semilattice "op \" "op <" min by (rule min_lattice) from assms show ?thesis by (simp add: Min_def strict_below_fold1_iff) qed lemma Max_less_iff [simp, noatp]: assumes "finite A" and "A \ {}" shows "Max A < x \ (\a\A. a < x)" proof - note Max = Max_def interpret linorder "op \" "op >" by (rule dual_linorder) from assms show ?thesis by (simp add: Max strict_below_fold1_iff [folded dual_max]) qed lemma Min_le_iff [noatp]: assumes "finite A" and "A \ {}" shows "Min A \ x \ (\a\A. a \ x)" proof - interpret lower_semilattice "op \" "op <" min by (rule min_lattice) from assms show ?thesis by (simp add: Min_def fold1_below_iff) qed lemma Max_ge_iff [noatp]: assumes "finite A" and "A \ {}" shows "x \ Max A \ (\a\A. x \ a)" proof - note Max = Max_def interpret linorder "op \" "op >" by (rule dual_linorder) from assms show ?thesis by (simp add: Max fold1_below_iff [folded dual_max]) qed lemma Min_less_iff [noatp]: assumes "finite A" and "A \ {}" shows "Min A < x \ (\a\A. a < x)" proof - interpret lower_semilattice "op \" "op <" min by (rule min_lattice) from assms show ?thesis by (simp add: Min_def fold1_strict_below_iff) qed lemma Max_gr_iff [noatp]: assumes "finite A" and "A \ {}" shows "x < Max A \ (\a\A. x < a)" proof - note Max = Max_def interpret linorder "op \" "op >" by (rule dual_linorder) from assms show ?thesis by (simp add: Max fold1_strict_below_iff [folded dual_max]) qed lemma Min_antimono: assumes "M \ N" and "M \ {}" and "finite N" shows "Min N \ Min M" proof - interpret distrib_lattice "op \" "op <" min max by (rule distrib_lattice_min_max) from assms show ?thesis by (simp add: Min_def fold1_antimono) qed lemma Max_mono: assumes "M \ N" and "M \ {}" and "finite N" shows "Max M \ Max N" proof - note Max = Max_def interpret linorder "op \" "op >" by (rule dual_linorder) from assms show ?thesis by (simp add: Max fold1_antimono [folded dual_max]) qed lemma finite_linorder_induct[consumes 1, case_names empty insert]: "finite A \ P {} \ (!!A b. finite A \ ALL a:A. a < b \ P A \ P(insert b A)) \ P A" proof (induct A rule: measure_induct_rule[where f=card]) fix A :: "'a set" assume IH: "!! B. card B < card A \ finite B \ P {} \ (!!A b. finite A \ (\a\A. a P A \ P (insert b A)) \ P B" and "finite A" and "P {}" and step: "!!A b. \finite A; \a\A. a < b; P A\ \ P (insert b A)" show "P A" proof (cases "A = {}") assume "A = {}" thus "P A" using `P {}` by simp next let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B" assume "A \ {}" with `finite A` have "Max A : A" by auto hence A: "?A = A" using insert_Diff_single insert_absorb by auto note card_Diff1_less[OF `finite A` `Max A : A`] moreover have "finite ?B" using `finite A` by simp ultimately have "P ?B" using `P {}` step IH by blast moreover have "\a\?B. a < Max A" using Max_ge [OF `finite A`] by fastsimp ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastsimp qed qed end context ordered_ab_semigroup_add begin lemma add_Min_commute: fixes k assumes "finite N" and "N \ {}" shows "k + Min N = Min {k + m | m. m \ N}" proof - have "\x y. k + min x y = min (k + x) (k + y)" by (simp add: min_def not_le) (blast intro: antisym less_imp_le add_left_mono) with assms show ?thesis using hom_Min_commute [of "plus k" N] by simp (blast intro: arg_cong [where f = Min]) qed lemma add_Max_commute: fixes k assumes "finite N" and "N \ {}" shows "k + Max N = Max {k + m | m. m \ N}" proof - have "\x y. k + max x y = max (k + x) (k + y)" by (simp add: max_def not_le) (blast intro: antisym less_imp_le add_left_mono) with assms show ?thesis using hom_Max_commute [of "plus k" N] by simp (blast intro: arg_cong [where f = Max]) qed end end