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theory Finite_Set(* Title: HOL/Finite_Set.thy ID: $Id: Finite_Set.thy,v 1.96 2005/09/22 21:56:15 nipkow Exp $ Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel with contributions by Jeremy Avigad *) header {* Finite sets *} theory Finite_Set imports 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. i<n} & inj_on f {i. i<n}" using fin proof induct case empty show ?case proof show "∃f. {} = 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<n} ==> 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<n. f n = f k" hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq) thus ?thesis using finB by simp next assume "¬(∃ k<n. f n = f k)" hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq) thus ?thesis using finB by simp qed qed lemma finite_conv_nat_seg_image: "finite A = (∃ (n::nat) f. A = f ` {i::nat. i<n})" by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on) subsubsection{* Finiteness and set theoretic constructions *} lemma finite_UnI: "finite F ==> 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(\<Union>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) 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<n & f k = (x,y)") prefer 2 apply force apply clarify apply (rule_tac x=k in image_eqI, auto) done lemma finite_cartesian_productD2: "[| finite (A <*> 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<n & f k = (x,y)") prefer 2 apply force apply clarify apply (rule_tac x=k in image_eqI, auto) done 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(\<Union>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 {x1, ..., xn} = f (g x1) (… (f (g xn) 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{* Interpretation of locales: *} interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"] by(auto intro: ACf.intro ACe_axioms.intro add_assoc add_commute) interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"] apply - apply (fast intro: ACf.intro mult_assoc mult_commute) apply (fastsimp intro: ACe_axioms.intro mult_assoc mult_commute) done 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<n}" and anot: "a ∉ A" and inj_on: "inj_on h {i::nat. i<n}" shows "∃hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n" proof (cases n) case 0 thus ?thesis using aA by auto next case (Suc m) have nSuc: "n = Suc m" . have mlessn: "m<n" by (simp add: nSuc) from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE) let ?hm = "swap k m h" have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn by (simp add: inj_on_swap_iff inj_on) show ?thesis proof (intro exI conjI) show "inj_on ?hm {i. i < m}" using inj_hm by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on) show "m<n" by (rule mlessn) show "A = ?hm ` {i. i < m}" proof (rule insert_image_inj_on_eq) show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp show "?hm m ∉ 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<n}; inj_on h {i. i<n}; (A,x) : foldSet f g z; (A,x') : foldSet f g z |] ==> x' = x" proof (induct n rule: less_induct) case (less n) have IH: "!!m h A x x'. [|m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; (A,x) ∈ 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<n}" and injh: "inj_on h {i. i<n}" . show ?case proof (rule foldSet.cases [OF Afoldx]) assume "(A, x) = ({}, z)" with Afoldx' show "x' = x" by blast next fix B b u assume "(A,x) = (insert b B, g b · 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<n}" by simp from insert_inj_onE [OF Beq notinB injh] obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp from insert_inj_onE [OF Ceq notinC injh] obtain hC mC where inj_onC: "inj_on hC {i. i < mC}" and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto show "x'=x" proof cases assume "b=c" then moreover have "B = C" using AbB AcC notinB notinC by auto ultimately show ?thesis using Bu Cv x x' IH[OF lessC Ceq inj_onC] by auto next assume diff: "b ≠ 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 iprover 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 o 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" :: "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" == "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" :: "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" => "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) 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 o f) B" by(auto simp add: setsum_def AC_add.fold_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_cong: "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" by(fastsimp simp: setsum_def intro: AC_add.fold_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: AC_add.fold_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 AC_add.fold_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) (*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 AC_add.fold_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 AC_add.fold_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 AC_add.fold_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: 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) 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 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_strict_mono: fixes f :: "'a => 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}" assumes fin_ne: "finite A" "A ≠ {}" shows "(!!x. x:A ==> f x < g x) ==> setsum f A < setsum g A" using fin_ne 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: 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_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 (* FIXME: this is distributitivty, name as such! *) (* suggested name: setsum_right_distrib (CB) *) 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_left_distrib: "setsum f A * (r::'a::semiring_0_cancel) = (∑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::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 lemma abs_setsum_abs[simp]: fixes f :: "'a => ('b::lordered_ab_group_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 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" :: "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" == "setprod (%i. b) A" "∏i∈A. b" == "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" => "setprod (%x. t) {x. P}" "∏x|P. t" => "setprod (%x. t) {x. P}" text{* Finally we abbreviate @{term"∏x∈A. x"} by @{text"∏A"}. *} 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) 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 o f) B" by(auto simp: setprod_def AC_mult.fold_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: AC_mult.fold_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: AC_mult.fold_cong) lemma setprod_reindex_cong: "inj_on f A ==> B = f ` A ==> g = h o 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 AC_mult.fold_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_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 AC_mult.fold_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 AC_mult.fold_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 AC_mult.fold_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: 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 o 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 o 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) 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) 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 ∩ \<Union> F = {}" by auto (* main cardinality theorem *) lemma card_partition [rule_format]: "finite C ==> finite (\<Union> C) --> (∀c∈C. card c = k) --> (∀c1 ∈ C. ∀c2 ∈ C. c1 ≠ c2 --> c1 ∩ c2 = {}) --> k * card(C) = card (\<Union> C)" apply (erule finite_induct, simp) apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition finite_subset [of _ "\<Union> (insert x F)"]) 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: ring_distrib add_ac) 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 lemma setsum_bounded: assumes le: "!!i. i∈A ==> f i ≤ (K::'a::{comm_semiring_1_cancel, 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 of_nat_id[simp]: "(of_nat n :: nat) = n" by(induct n, auto) 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}.*} 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 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, 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 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 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 (λu. 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 "\<sqsubseteq>" 50) assumes below_def: "(x \<sqsubseteq> y) = (x·y = x)" locale ACIfSLlin = ACIfSL + assumes lin: "x·y ∈ {x,y}" lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x" by(simp add: below_def idem) lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y · z = (x \<sqsubseteq> y ∧ x \<sqsubseteq> z)" proof assume "x \<sqsubseteq> 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 \<sqsubseteq> y ∧ x \<sqsubseteq> z" by(simp add: below_def) next assume a: "x \<sqsubseteq> y ∧ x \<sqsubseteq> 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 \<sqsubseteq> y · z" by(simp_all add: below_def) qed lemma (in ACIfSLlin) above_f_conv: "x · y \<sqsubseteq> z = (x \<sqsubseteq> z ∨ y \<sqsubseteq> z)" proof assume a: "x · y \<sqsubseteq> z" have "x · y = x ∨ x · y = y" using lin[of x y] by simp thus "x \<sqsubseteq> z ∨ y \<sqsubseteq> z" proof assume "x · y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis .. next assume "x · y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis .. qed next assume "x \<sqsubseteq> z ∨ y \<sqsubseteq> z" thus "x · y \<sqsubseteq> z" proof assume a: "x \<sqsubseteq> 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 \<sqsubseteq> z" by(simp add:below_def) next assume a: "y \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> fold1 f A = (∀a∈A. x \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> x = (∃a∈A. a \<sqsubseteq> 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" ("\<Sqinter>_" [900] 900) and Sup :: "'a set => 'a" ("\<Squnion>_" [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 \<sqsubseteq>)" 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 \<sqsubseteq> 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[simp]: "[| finite A; A ≠ {} |] ==> \<Sqinter>A \<sqsubseteq> \<Squnion>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[simp]: "[| finite A; A ≠ {}; a ∈ A |] ==> (a \<squnion> \<Sqinter>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[simp]: "[| finite A; A ≠ {}; a ∈ A |] ==> (a \<sqinter> \<Squnion>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 \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> 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 \<squnion> a |a. a ∈ A}" by(fast intro: finite_surj[where f = "%a. x \<squnion> a", OF insert(1)]) have "x \<squnion> \<Sqinter> (insert y A) = x \<squnion> (y \<sqinter> \<Sqinter> A)" using insert by(simp add:ACf.fold1_insert_def[OF ACf_inf Inf_def]) also have "… = (x \<squnion> y) \<sqinter> (x \<squnion> \<Sqinter> A)" by(rule sup_inf_distrib1) also have "x \<squnion> \<Sqinter> A = \<Sqinter>{x \<squnion> a|a. a ∈ A}" using insert by simp also have "(x \<squnion> y) \<sqinter> … = \<Sqinter> (insert (x \<squnion> y) {x \<squnion> a |a. a ∈ A})" using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def fin]) also have "insert (x\<squnion>y) {x\<squnion>a |a. a ∈ A} = {x\<squnion>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 "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> 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 \<squnion> b |b. b ∈ B}" by(fast intro: finite_surj[where f = "%b. x \<squnion> b", OF B(1)]) have finAB: "finite {a \<squnion> b |a b. a ∈ A ∧ b ∈ B}" proof - have "{a \<squnion> b |a b. a ∈ A ∧ b ∈ B} = (UN a:A. UN b:B. {a \<squnion> b})" by blast thus ?thesis by(simp add: insert(1) B(1)) qed have ne: "{a \<squnion> b |a b. a ∈ A ∧ b ∈ B} ≠ {}" using insert B by blast have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B" using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def]) also have "… = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2) also have "… = \<Sqinter>{x \<squnion> b|b. b ∈ B} \<sqinter> \<Sqinter>{a \<squnion> b|a b. a ∈ A ∧ b ∈ B}" using insert by(simp add:sup_Inf1_distrib[OF B]) also have "… = \<Sqinter>({x\<squnion>b |b. b ∈ B} ∪ {a\<squnion>b |a b. a ∈ A ∧ b ∈ B})" (is "_ = \<Sqinter>?M") using B insert by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne]) also have "?M = {a \<squnion> 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: *} interpretation min: ACf ["min:: 'a::linorder => 'a => 'a"] apply(rule ACf.intro) apply(auto simp:min_def) done interpretation min: ACIf ["min:: 'a::linorder => 'a => 'a"] apply(rule ACIf_axioms.intro) apply(auto simp:min_def) done interpretation max: ACf ["max :: 'a::linorder => 'a => 'a"] apply(rule ACf.intro) apply(auto simp:max_def) done interpretation max: ACIf ["max:: 'a::linorder => 'a => 'a"] apply(rule ACIf_axioms.intro) apply(auto simp:max_def) done interpretation min: ACIfSL ["min:: 'a::linorder => 'a => 'a" "op ≤"] apply(rule ACIfSL_axioms.intro) apply(auto simp:min_def) done interpretation min: ACIfSLlin ["min :: 'a::linorder => 'a => 'a" "op ≤"] apply(rule ACIfSLlin_axioms.intro) apply(auto simp:min_def) done interpretation max: ACIfSL ["max :: 'a::linorder => 'a => 'a" "%x y. y≤x"] apply(rule ACIfSL_axioms.intro) apply(auto simp:max_def) done interpretation max: ACIfSLlin ["max :: 'a::linorder => 'a => 'a" "%x y. y≤x"] apply(rule ACIfSLlin_axioms.intro) apply(auto simp:max_def) done interpretation min_max: Lattice ["op ≤" "min :: 'a::linorder => 'a => 'a" "max" "Min" "Max"] apply - apply(rule Min_def) apply(rule Max_def) done interpretation min_max: Distrib_Lattice ["op ≤" "min :: 'a::linorder => 'a => 'a" "max" "Min" "Max"] . text{* Now we instantiate the recursion equations and declare them simplification rules: *} (* Making Min or Max a defined parameter of a locale, suitably extending ACIf, could make the following interpretations more automatic. *) lemmas Min_singleton = fold1_singleton_def [OF Min_def] lemmas Max_singleton = fold1_singleton_def [OF Max_def] lemmas Min_insert = min.fold1_insert_idem_def [OF Min_def] lemmas Max_insert = max.fold1_insert_idem_def [OF Max_def] declare Min_singleton [simp] Max_singleton [simp] declare Min_insert [simp] Max_insert [simp] text{* Now we instantiate some @{text fold1} properties: *} lemma Min_in [simp]: shows "finite A ==> A ≠ {} ==> Min A ∈ A" using min.fold1_in by(fastsimp simp: Min_def min_def) lemma Max_in [simp]: shows "finite A ==> A ≠ {} ==> Max A ∈ A" using max.fold1_in by(fastsimp simp: Max_def max_def) lemma Min_le [simp]: "[| finite A; A ≠ {}; x ∈ A |] ==> Min A ≤ x" by(simp add: Min_def min.fold1_belowI) lemma Max_ge [simp]: "[| finite A; A ≠ {}; x ∈ A |] ==> x ≤ Max A" by(simp add: Max_def max.fold1_belowI) lemma Min_ge_iff[simp]: "[| finite A; A ≠ {} |] ==> (x ≤ Min A) = (∀a∈A. x ≤ a)" by(simp add: Min_def min.below_fold1_iff) lemma Max_le_iff[simp]: "[| finite A; A ≠ {} |] ==> (Max A ≤ x) = (∀a∈A. a ≤ x)" by(simp add: Max_def max.below_fold1_iff) lemma Min_le_iff: "[| finite A; A ≠ {} |] ==> (Min A ≤ x) = (∃a∈A. a ≤ x)" by(simp add: Min_def min.fold1_below_iff) lemma Max_ge_iff: "[| finite A; A ≠ {} |] ==> (x ≤ Max A) = (∃a∈A. x ≤ a)" by(simp add: Max_def max.fold1_below_iff) subsection {* Properties of axclass @{text finite} *} text{* Many of these are by Brian Huffman. *} lemma finite_set: "finite (A::'a::finite set)" by (rule finite_subset [OF subset_UNIV finite]) instance unit :: finite proof have "finite {()}" by simp also have "{()} = UNIV" by auto finally show "finite (UNIV :: unit set)" . qed instance bool :: finite proof have "finite {True, False}" by simp also have "{True, False} = UNIV" by auto finally show "finite (UNIV :: bool set)" . qed instance * :: (finite, finite) finite proof show "finite (UNIV :: ('a × '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 instance "+" :: (finite, finite) finite proof have a: "finite (UNIV :: 'a set)" by (rule finite) have b: "finite (UNIV :: 'b set)" by (rule finite) from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))" by (rule finite_Plus) thus "finite (UNIV :: ('a + 'b) set)" by simp qed instance set :: (finite) finite proof have "finite (UNIV :: 'a set)" by (rule finite) hence "finite (Pow (UNIV :: 'a set))" by (rule finite_Pow_iff [THEN iffD2]) thus "finite (UNIV :: 'a set set)" by simp qed 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}" show "finite (range ?graph)" by (rule finite_set) show "inj ?graph" by (rule inj_graph) qed qed end
lemma ex_new_if_finite:
[| UNIV ∉ Finites; finite A |] ==> ∃a. a ∉ A
lemma finite_induct:
[| finite F; P {}; !!x F. [| finite F; x ∉ F; P F |] ==> P (insert x F) |] ==> P F
lemma finite_ne_induct:
[| finite F; F ≠ {}; !!x. P {x}; !!x F. [| finite F; F ≠ {}; x ∉ F; P F |] ==> P (insert x F) |] ==> P F
lemma finite_subset_induct:
[| finite F; F ⊆ A; P {}; !!a F. [| finite F; a ∈ A; a ∉ F; P F |] ==> P (insert a F) |] ==> P F
lemma finite_imp_nat_seg_image_inj_on:
finite A ==> ∃n f. A = f ` {i. i < n} ∧ inj_on f {i. i < n}
lemma nat_seg_image_imp_finite:
A = f ` {i. i < n} ==> finite A
lemma finite_conv_nat_seg_image:
finite A = (∃n f. A = f ` {i. i < n})
lemma finite_UnI:
[| finite F; finite G |] ==> finite (F ∪ G)
lemma finite_subset:
[| A ⊆ B; finite B |] ==> finite A
lemma finite_Un:
finite (F ∪ G) = (finite F ∧ finite G)
lemma finite_Int:
finite F ∨ finite G ==> finite (F ∩ G)
lemma finite_insert:
finite (insert a A) = finite A
lemma finite_Union:
[| finite A; !!M. M ∈ A ==> finite M |] ==> finite (Union A)
lemma finite_empty_induct:
[| finite A; P A; !!a A. [| finite A; a ∈ A; P A |] ==> P (A - {a}) |] ==> P {}
lemma finite_Diff:
finite B ==> finite (B - Ba)
lemma finite_Diff_insert:
finite (A - insert a B) = finite (A - B)
lemma finite_imageI:
finite F ==> finite (h ` F)
lemma finite_surj:
[| finite A; B ⊆ f ` A |] ==> finite B
lemma finite_range_imageI:
finite (range g) ==> finite (range (%x. f (g x)))
lemma finite_imageD:
[| finite (f ` A); inj_on f A |] ==> finite A
lemma inj_vimage_singleton:
inj f ==> f -` {a} ⊆ {THE x. f x = a}
lemma finite_vimageI:
[| finite F; inj h |] ==> finite (h -` F)
lemma finite_UN_I:
[| finite A; !!a. a ∈ A ==> finite (B a) |] ==> finite (UN a:A. B a)
lemma finite_UN:
finite A ==> finite (UNION A B) = (∀x∈A. finite (B x))
lemma finite_Plus:
[| finite A; finite B |] ==> finite (A <+> B)
lemma finite_SigmaI:
[| finite A; !!a. a ∈ A ==> finite (B a) |] ==> finite (SIGMA a:A. B a)
lemma finite_cartesian_product:
[| finite A; finite B |] ==> finite (A × B)
lemma finite_Prod_UNIV:
[| finite UNIV; finite UNIV |] ==> finite UNIV
lemma finite_cartesian_productD1:
[| finite (A × B); B ≠ {} |] ==> finite A
lemma finite_cartesian_productD2:
[| finite (A × B); A ≠ {} |] ==> finite B
lemma finite_Pow_iff:
finite (Pow A) = finite A
lemma finite_UnionD:
finite (Union A) ==> finite A
lemma finite_converse:
finite (r^-1) = finite r
lemma finite_Field:
finite r ==> finite (Field r)
lemma trancl_subset_Field2:
r+ ⊆ Field r × Field r
lemma finite_trancl:
finite (r+) = finite r
lemmas empty_foldSetE:
[| ({}, x) ∈ foldSet f g z; x = z ==> P |] ==> P
lemma Diff1_foldSet:
[| (A - {x}, y) ∈ foldSet f g z; x ∈ A |] ==> (A, f (g x) y) ∈ foldSet f g z
lemma foldSet_imp_finite:
(A, x) ∈ foldSet f g z ==> finite A
lemma finite_imp_foldSet:
finite A ==> ∃x. (A, x) ∈ foldSet f g z
lemma left_commute:
ACf f ==> f x (f y z) = f y (f x z)
lemmas AC:
ACf f ==> f (f x y) z = f x (f y z)
ACf f ==> f x y = f y x
ACf f ==> f x (f y z) = f y (f x z)
lemma left_ident:
ACe f e ==> f e x = x
lemma idem2:
ACIf f ==> f x (f x y) = f x y
lemmas ACI:
ACIf f ==> f (f x y) z = f x (f y z)
ACIf f ==> f x y = f y x
ACIf f ==> f x (f y z) = f y (f x z)
ACIf f ==> f x x = x
ACIf f ==> f x (f x y) = f x y
lemma image_less_Suc:
h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})
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}
lemma insert_inj_onE:
[| insert a A = h ` {i. i < n}; a ∉ A; inj_on h {i. i < n} |] ==> ∃hm m. inj_on hm {i. i < m} ∧ A = hm ` {i. i < m} ∧ m < n
lemma foldSet_determ_aux:
[| ACf f; A = h ` {i. i < n}; inj_on h {i. i < n}; (A, x) ∈ foldSet f g z; (A, x') ∈ foldSet f g z |] ==> x' = x
lemma foldSet_determ:
[| ACf f; (A, x) ∈ foldSet f g z; (A, y) ∈ foldSet f g z |] ==> y = x
lemma fold_equality:
[| ACf f; (A, y) ∈ foldSet f g z |] ==> fold f g z A = y
lemma fold_empty:
fold f g z {} = z
lemma fold_insert_aux:
[| ACf f; x ∉ A |] ==> ((insert x A, v) ∈ foldSet f g z) = (∃y. (A, y) ∈ foldSet f g z ∧ v = f (g x) y)
lemma fold_insert:
[| ACf f; finite A; x ∉ A |] ==> fold f g z (insert x A) = f (g x) (fold f g z A)
lemma fold_rec:
[| ACf f; finite A; a ∈ A |] ==> fold f g z A = f (g a) (fold f g z (A - {a}))
lemma fold_insert_idem:
[| ACIf f; finite A |] ==> fold f g z (insert a A) = f (g a) (fold f g z A)
lemma foldI_conv_id:
[| ACIf f; finite A |] ==> fold f g z A = fold f id z (g ` A)
lemma fold_commute:
[| ACf f; finite A |] ==> f x (fold f g z A) = fold f g (f x z) A
lemma fold_nest_Un_Int:
[| ACf f; finite A; finite B |] ==> fold f g (fold f g z B) A = fold f g (fold f g z (A ∩ B)) (A ∪ B)
lemma fold_nest_Un_disjoint:
[| ACf f; finite A; finite B; A ∩ B = {} |] ==> fold f g z (A ∪ B) = fold f g (fold f g z B) A
lemma fold_reindex:
[| ACf f; finite A; inj_on h A |] ==> fold f g z (h ` A) = fold f (g o h) z A
lemma fold_Un_Int:
[| ACe f e; finite A; finite B |] ==> f (fold f g e A) (fold f g e B) = f (fold f g e (A ∪ B)) (fold f g e (A ∩ B))
corollary fold_Un_disjoint:
[| ACe f e; finite A; finite B; A ∩ B = {} |] ==> fold f g e (A ∪ B) = f (fold f g e A) (fold f g e B)
lemma fold_UN_disjoint:
[| ACe f e; finite I; ∀i∈I. finite (A i); ∀i∈I. ∀j∈I. i ≠ j --> A i ∩ A j = {} |] ==> fold f g e (UNION I A) = fold f (%i. fold f g e (A i)) e I
lemma fold_fusion:
[| ACf f; ACf g; finite A; !!x y. h (g x y) = f x (h y) |] ==> h (fold g j w A) = fold f j (h w) A
lemma fold_cong:
[| ACf f; finite A; !!x. x ∈ A ==> g x = h x |] ==> fold f g z A = fold f h z A
lemma fold_Sigma:
[| ACe f e; finite A; ∀x∈A. finite (B x) |] ==> fold f (%x. fold f (g x) e (B x)) e A = fold f (%(x, y). g x y) e (SIGMA x:A. B x)
lemma fold_distrib:
[| ACe f e; finite A |] ==> fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)
lemma setsum_empty:
setsum f {} = (0::'a)
lemma setsum_insert:
[| finite F; a ∉ F |] ==> setsum f (insert a F) = f a + setsum f F
lemma setsum_infinite:
A ∉ Finites ==> setsum f A = (0::'b)
lemma setsum_reindex:
inj_on f B ==> setsum h (f ` B) = setsum (h o f) B
lemma setsum_reindex_id:
inj_on f B ==> setsum f B = setsum id (f ` B)
lemma setsum_cong:
[| A = B; !!x. x ∈ B ==> f x = g x |] ==> setsum f A = setsum g B
lemma strong_setsum_cong:
[| A = B; !!x. x ∈ B =simp=> f x = g x |] ==> setsum f A = setsum g B
lemma setsum_cong2:
(!!x. x ∈ A ==> f x = g x) ==> setsum f A = setsum g A
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
lemma setsum_0:
(∑i∈A. (0::'a)) = (0::'a)
lemma setsum_0':
∀a∈A. f a = (0::'b) ==> setsum f A = (0::'b)
lemma setsum_Un_Int:
[| finite A; finite B |] ==> setsum g (A ∪ B) + setsum g (A ∩ B) = setsum g A + setsum g B
lemma setsum_Un_disjoint:
[| finite A; finite B; A ∩ B = {} |] ==> setsum g (A ∪ B) = setsum g A + setsum g B
lemma setsum_UN_disjoint:
[| finite I; ∀i∈I. finite (A i); ∀i∈I. ∀j∈I. i ≠ j --> A i ∩ A j = {} |] ==> setsum f (UNION I A) = (∑i∈I. setsum f (A i))
lemma setsum_Union_disjoint:
[| ∀A∈C. finite A; ∀A∈C. ∀B∈C. A ≠ B --> A ∩ B = {} |] ==> setsum f (Union C) = setsum (setsum f) C
lemma setsum_Sigma:
[| finite A; ∀x∈A. finite (B x) |] ==> (∑x∈A. setsum (f x) (B x)) = (∑(x, y)∈(SIGMA x:A. B x). f x y)
lemma setsum_cartesian_product:
(∑x∈A. setsum (f x) B) = (∑(x, y)∈A × B. f x y)
lemma setsum_addf:
(∑x∈A. f x + g x) = setsum f A + setsum g A
lemma setsum_SucD:
setsum f A = Suc n ==> ∃a∈A. 0 < f a
lemma setsum_eq_0_iff:
finite F ==> (setsum f F = 0) = (∀a∈F. f a = 0)
lemma setsum_Un_nat:
[| finite A; finite B |] ==> setsum f (A ∪ B) = setsum f A + setsum f B - setsum f (A ∩ B)
lemma setsum_Un:
[| finite A; finite B |] ==> setsum f (A ∪ B) = setsum f A + setsum f B - setsum f (A ∩ B)
lemma setsum_diff1_nat:
setsum f (A - {a}) = (if a ∈ A then setsum f A - f a else setsum f A)
lemma setsum_diff1:
finite A ==> setsum f (A - {a}) = (if a ∈ A then setsum f A - f a else setsum f A)
lemma setsum_diff1':
[| finite A; a ∈ A |] ==> setsum f A = f a + setsum f (A - {a})
lemma setsum_diff_nat:
[| finite B; B ⊆ A |] ==> setsum f (A - B) = setsum f A - setsum f B
lemma setsum_diff:
[| finite A; B ⊆ A |] ==> setsum f (A - B) = setsum f A - setsum f B
lemma setsum_mono:
(!!i. i ∈ K ==> f i ≤ g i) ==> setsum f K ≤ setsum g K
lemma setsum_strict_mono:
[| finite A; A ≠ {}; !!x. x ∈ A ==> f x < g x |] ==> setsum f A < setsum g A
lemma setsum_negf:
(∑x∈A. - f x) = - setsum f A
lemma setsum_subtractf:
(∑x∈A. f x - g x) = setsum f A - setsum g A
lemma setsum_nonneg:
∀x∈A. (0::'a) ≤ f x ==> (0::'a) ≤ setsum f A
lemma setsum_nonpos:
∀x∈A. f x ≤ (0::'a) ==> setsum f A ≤ (0::'a)
lemma setsum_mono2:
[| finite B; A ⊆ B; !!b. b ∈ B - A ==> (0::'b) ≤ f b |] ==> setsum f A ≤ setsum f B
lemma setsum_mono3:
[| finite B; A ⊆ B; ∀x∈B - A. (0::'a) ≤ f x |] ==> setsum f A ≤ setsum f B
lemma setsum_mult:
r * setsum f A = (∑n∈A. r * f n)
lemma setsum_left_distrib:
setsum f A * r = (∑n∈A. f n * r)
lemma setsum_divide_distrib:
setsum f A / r = (∑n∈A. f n / r)
lemma setsum_abs:
¦setsum f A¦ ≤ (∑i∈A. ¦f i¦)
lemma setsum_abs_ge_zero:
(0::'b) ≤ (∑i∈A. ¦f i¦)
lemma abs_setsum_abs:
¦∑a∈A. ¦f a¦¦ = (∑a∈A. ¦f a¦)
lemma swap_inj_on:
inj_on (%(i, j). (j, i)) (A × B)
lemma swap_product:
(%(i, j). (j, i)) ` (A × B) = B × A
lemma setsum_commute:
(∑i∈A. setsum (f i) B) = (∑j∈B. ∑i∈A. f i j)
lemma setprod_empty:
setprod f {} = (1::'a)
lemma setprod_insert:
[| finite A; a ∉ A |] ==> setprod f (insert a A) = f a * setprod f A
lemma setprod_infinite:
A ∉ Finites ==> setprod f A = (1::'b)
lemma setprod_reindex:
inj_on f B ==> setprod h (f ` B) = setprod (h o f) B
lemma setprod_reindex_id:
inj_on f B ==> setprod f B = setprod id (f ` B)
lemma setprod_cong:
[| A = B; !!x. x ∈ B ==> f x = g x |] ==> setprod f A = setprod g B
lemma strong_setprod_cong:
[| A = B; !!x. x ∈ B =simp=> f x = g x |] ==> setprod f A = setprod g B
lemma setprod_reindex_cong:
[| inj_on f A; B = f ` A; g = h o f |] ==> setprod h B = setprod g A
lemma setprod_1:
(∏i∈A. (1::'a)) = (1::'a)
lemma setprod_1':
∀a∈F. f a = (1::'b) ==> setprod f F = (1::'b)
lemma setprod_Un_Int:
[| finite A; finite B |] ==> setprod g (A ∪ B) * setprod g (A ∩ B) = setprod g A * setprod g B
lemma setprod_Un_disjoint:
[| finite A; finite B; A ∩ B = {} |] ==> setprod g (A ∪ B) = setprod g A * setprod g B
lemma setprod_UN_disjoint:
[| finite I; ∀i∈I. finite (A i); ∀i∈I. ∀j∈I. i ≠ j --> A i ∩ A j = {} |] ==> setprod f (UNION I A) = (∏i∈I. setprod f (A i))
lemma setprod_Union_disjoint:
[| ∀A∈C. finite A; ∀A∈C. ∀B∈C. A ≠ B --> A ∩ B = {} |] ==> setprod f (Union C) = setprod (setprod f) C
lemma setprod_Sigma:
[| finite A; ∀x∈A. finite (B x) |] ==> (∏x∈A. setprod (f x) (B x)) = (∏(x, y)∈(SIGMA x:A. B x). f x y)
lemma setprod_cartesian_product:
(∏x∈A. setprod (f x) B) = (∏(x, y)∈A × B. f x y)
lemma setprod_timesf:
(∏x∈A. f x * g x) = setprod f A * setprod g A
lemma setprod_eq_1_iff:
finite F ==> (setprod f F = 1) = (∀a∈F. f a = 1)
lemma setprod_zero:
[| finite A; ∃x∈A. f x = (0::'a) |] ==> setprod f A = (0::'a)
lemma setprod_nonneg:
(!!x. x ∈ A ==> (0::'a) ≤ f x) ==> (0::'a) ≤ setprod f A
lemma setprod_pos:
(!!x. x ∈ A ==> (0::'a) < f x) ==> (0::'a) < setprod f A
lemma setprod_nonzero:
[| !!x y. x * y = (0::'a) ==> x = (0::'a) ∨ y = (0::'a); finite A; !!x. x ∈ A ==> f x ≠ (0::'a) |] ==> setprod f A ≠ (0::'a)
lemma setprod_zero_eq:
[| ∀x y. x * y = (0::'a) --> x = (0::'a) ∨ y = (0::'a); finite A |] ==> (setprod f A = (0::'a)) = (∃x∈A. f x = (0::'a))
lemma setprod_nonzero_field:
[| finite A; ∀x∈A. f x ≠ (0::'a) |] ==> setprod f A ≠ (0::'a)
lemma setprod_zero_eq_field:
finite A ==> (setprod f A = (0::'a)) = (∃x∈A. f x = (0::'a))
lemma setprod_Un:
[| finite A; finite B; ∀x∈A ∩ B. f x ≠ (0::'a) |] ==> setprod f (A ∪ B) = setprod f A * setprod f B / setprod f (A ∩ B)
lemma setprod_diff1:
[| finite A; f a ≠ (0::'a) |] ==> setprod f (A - {a}) = (if a ∈ A then setprod f A / f a else setprod f A)
lemma setprod_inversef:
[| finite A; ∀x∈A. f x ≠ (0::'a) |] ==> setprod (inverse o f) A = inverse (setprod f A)
lemma setprod_dividef:
[| finite A; ∀x∈A. g x ≠ (0::'a) |] ==> (∏x∈A. f x / g x) = setprod f A / setprod g A
lemma card_empty:
card {} = 0
lemma card_infinite:
A ∉ Finites ==> card A = 0
lemma card_eq_setsum:
card A = (∑x∈A. 1)
lemma card_insert_disjoint:
[| finite A; x ∉ A |] ==> card (insert x A) = Suc (card A)
lemma card_insert_if:
finite A ==> card (insert x A) = (if x ∈ A then card A else Suc (card A))
lemma card_0_eq:
finite A ==> (card A = 0) = (A = {})
lemma card_eq_0_iff:
(card A = 0) = (A = {} ∨ A ∉ Finites)
lemma card_Suc_Diff1:
[| finite A; x ∈ A |] ==> Suc (card (A - {x})) = card A
lemma card_Diff_singleton:
[| finite A; x ∈ A |] ==> card (A - {x}) = card A - 1
lemma card_Diff_singleton_if:
finite A ==> card (A - {x}) = (if x ∈ A then card A - 1 else card A)
lemma card_insert:
finite A ==> card (insert x A) = Suc (card (A - {x}))
lemma card_insert_le:
finite A ==> card A ≤ card (insert x A)
lemma card_mono:
[| finite B; A ⊆ B |] ==> card A ≤ card B
lemma card_seteq:
[| finite B; A ⊆ B; card B ≤ card A |] ==> A = B
lemma psubset_card_mono:
[| finite B; A ⊂ B |] ==> card A < card B
lemma card_Un_Int:
[| finite A; finite B |] ==> card A + card B = card (A ∪ B) + card (A ∩ B)
lemma card_Un_disjoint:
[| finite A; finite B; A ∩ B = {} |] ==> card (A ∪ B) = card A + card B
lemma card_Diff_subset:
[| finite B; B ⊆ A |] ==> card (A - B) = card A - card B
lemma card_Diff1_less:
[| finite A; x ∈ A |] ==> card (A - {x}) < card A
lemma card_Diff2_less:
[| finite A; x ∈ A; y ∈ A |] ==> card (A - {x} - {y}) < card A
lemma card_Diff1_le:
finite A ==> card (A - {x}) ≤ card A
lemma card_psubset:
[| finite B; A ⊆ B; card A < card B |] ==> A ⊂ B
lemma insert_partition:
[| x ∉ F; ∀c1∈insert x F. ∀c2∈insert x F. c1 ≠ c2 --> c1 ∩ c2 = {} |] ==> x ∩ Union F = {}
lemma card_partition:
[| finite C; finite (Union C); !!c. c ∈ C ==> card c = k; !!c1 c2. [| c1 ∈ C; c2 ∈ C; c1 ≠ c2 |] ==> c1 ∩ c2 = {} |] ==> k * card C = card (Union C)
lemma setsum_constant:
(∑x∈A. y) = of_nat (card A) * y
lemma setprod_constant:
finite A ==> (∏x∈A. y) = y ^ card A
lemma setsum_bounded:
(!!i. i ∈ A ==> f i ≤ K) ==> setsum f A ≤ of_nat (card A) * K
lemma of_nat_id:
of_nat n = n
lemma card_UN_disjoint:
[| finite I; ∀i∈I. finite (A i); ∀i∈I. ∀j∈I. i ≠ j --> A i ∩ A j = {} |] ==> card (UNION I A) = (∑i∈I. card (A i))
lemma card_Union_disjoint:
[| finite C; ∀A∈C. finite A; ∀A∈C. ∀B∈C. A ≠ B --> A ∩ B = {} |] ==> card (Union C) = setsum card C
lemma image_eq_fold:
finite A ==> f ` A = fold op ∪ (%x. {f x}) {} A
lemma card_image_le:
finite A ==> card (f ` A) ≤ card A
lemma card_image:
inj_on f A ==> card (f ` A) = card A
lemma endo_inj_surj:
[| finite A; f ` A ⊆ A; inj_on f A |] ==> f ` A = A
lemma eq_card_imp_inj_on:
[| finite A; card (f ` A) = card A |] ==> inj_on f A
lemma inj_on_iff_eq_card:
finite A ==> inj_on f A = (card (f ` A) = card A)
lemma card_inj_on_le:
[| inj_on f A; f ` A ⊆ B; finite B |] ==> card A ≤ card B
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
lemma card_SigmaI:
[| finite A; ∀a∈A. finite (B a) |] ==> card (SIGMA x:A. B x) = (∑a∈A. card (B a))
lemma card_cartesian_product:
card (A × B) = card A * card B
lemma card_cartesian_product_singleton:
card ({x} × A) = card A
lemma card_Pow:
finite A ==> card (Pow A) = Suc (Suc 0) ^ card A
lemma dvd_partition:
[| finite (Union C); ∀c∈C. k dvd card c; ∀c1∈C. ∀c2∈C. c1 ≠ c2 --> c1 ∩ c2 = {} |] ==> k dvd card (Union C)
lemma fold1Set_nonempty:
(A, x) ∈ fold1Set f ==> A ≠ {}
lemmas empty_fold1SetE:
({}, x) ∈ fold1Set f ==> P
lemmas insert_fold1SetE:
[| (insert a X, x) ∈ fold1Set f; !!A a. [| (A, x) ∈ foldSet f id a; a ∉ A; insert a X = insert a A |] ==> P |] ==> P
lemma fold1Set_sing:
(({a}, b) ∈ fold1Set f) = (a = b)
lemma fold1_singleton:
fold1 f {a} = a
lemma finite_nonempty_imp_fold1Set:
[| finite A; A ≠ {} |] ==> ∃x. (A, x) ∈ fold1Set f
lemma foldSet_insert_swap:
[| ACf f; (A, y) ∈ foldSet f id b; b ∉ A |] ==> (insert b A, f z y) ∈ foldSet f id z
lemma foldSet_permute_diff:
[| ACf f; (A, x) ∈ foldSet f id b; a ∈ A; b ∉ A |] ==> (insert b (A - {a}), x) ∈ foldSet f id a
lemma fold1_eq_fold:
[| ACf f; finite A; a ∉ A |] ==> fold1 f (insert a A) = fold f id a A
lemma nonempty_iff:
(A ≠ {}) = (∃x B. A = insert x B ∧ x ∉ B)
lemma fold1_insert:
[| ACf f; A ≠ {}; finite A; x ∉ A |] ==> fold1 f (insert x A) = f x (fold1 f A)
lemma fold1_insert_idem:
[| ACIf f; A ≠ {}; finite A |] ==> fold1 f (insert x A) = f x (fold1 f A)
lemma fold1_singleton_def:
g == fold1 f ==> g {a} = a
lemma fold1_insert_def:
[| ACf f; g == fold1 f; finite A; x ∉ A; A ≠ {} |] ==> g (insert x A) = f x (g A)
lemma fold1_insert_idem_def:
[| ACIf f; g == fold1 f; finite A; A ≠ {} |] ==> g (insert x A) = f x (g A)
lemma foldSet_permute:
[| ACf f; (insert a A, x) ∈ foldSet f id b; a ∉ A; b ∉ A |] ==> (insert b A, x) ∈ foldSet f id a
lemma fold1Set_determ:
[| ACf f; (A, x) ∈ fold1Set f; (A, y) ∈ fold1Set f |] ==> y = x
lemma fold1Set_equality:
[| ACf f; (A, y) ∈ fold1Set f |] ==> fold1 f A = y
lemma below_refl:
ACIfSL f below ==> below x x
lemma below_f_conv:
ACIfSL f below ==> below x (f y z) = (below x y ∧ below x z)
lemma above_f_conv:
ACIfSLlin f below ==> below (f x y) z = (below x z ∨ below y z)
lemma fold1_Un:
[| ACf f; finite A; A ≠ {}; finite B; B ≠ {}; A ∩ B = {} |] ==> fold1 f (A ∪ B) = f (fold1 f A) (fold1 f B)
lemma fold1_Un2:
[| ACIf f; finite A; A ≠ {}; finite B; B ≠ {} |] ==> fold1 f (A ∪ B) = f (fold1 f A) (fold1 f B)
lemma fold1_in:
[| ACf f; finite A; A ≠ {}; !!x y. f x y ∈ {x, y} |] ==> fold1 f A ∈ A
lemma below_fold1_iff:
[| ACIfSL f below; finite A; A ≠ {} |] ==> below x (fold1 f A) = (∀a∈A. below x a)
lemma fold1_belowI:
[| ACIfSL f below; finite A; A ≠ {}; a ∈ A |] ==> below (fold1 f A) a
lemma fold1_below_iff:
[| ACIfSLlin f below; finite A; A ≠ {} |] ==> below (fold1 f A) x = (∃a∈A. below a x)
lemma ACf_inf:
Lattice below inf sup ==> ACf inf
lemma ACf_sup:
Lattice below inf sup ==> ACf sup
lemma ACIf_inf:
Lattice below inf sup ==> ACIf inf
lemma ACIf_sup:
Lattice below inf sup ==> ACIf sup
lemma ACIfSL_inf:
Lattice below inf sup ==> ACIfSL inf below
lemma ACIfSL_sup:
Lattice below inf sup ==> ACIfSL sup (%x y. below y x)
lemma Inf_le_Sup:
[| Lattice below inf sup; finite A; A ≠ {} |] ==> below (fold1 inf A) (fold1 sup A)
lemma sup_Inf_absorb:
[| Lattice below inf sup; finite A; A ≠ {}; a ∈ A |] ==> sup a (fold1 inf A) = a
lemma inf_Sup_absorb:
[| Lattice below inf sup; finite A; A ≠ {}; a ∈ A |] ==> inf a (fold1 sup A) = a
lemma sup_Inf1_distrib:
[| Distrib_Lattice below inf sup; finite A; A ≠ {} |] ==> sup x (fold1 inf A) = fold1 inf {sup x a |a. a ∈ A}
lemma sup_Inf2_distrib:
[| Distrib_Lattice below inf sup; finite A; A ≠ {}; finite B; B ≠ {} |] ==> sup (fold1 inf A) (fold1 inf B) = fold1 inf {sup a b |a b. a ∈ A ∧ b ∈ B}
lemmas Min_singleton:
Min {a} = a
lemmas Min_singleton:
Min {a} = a
lemmas Max_singleton:
Max {a} = a
lemmas Max_singleton:
Max {a} = a
lemmas Min_insert:
[| finite A; A ≠ {} |] ==> Min (insert x A) = min x (Min A)
lemmas Min_insert:
[| finite A; A ≠ {} |] ==> Min (insert x A) = min x (Min A)
lemmas Max_insert:
[| finite A; A ≠ {} |] ==> Max (insert x A) = max x (Max A)
lemmas Max_insert:
[| finite A; A ≠ {} |] ==> Max (insert x A) = max x (Max A)
lemma Min_in:
[| finite A; A ≠ {} |] ==> Min A ∈ A
lemma Max_in:
[| finite A; A ≠ {} |] ==> Max A ∈ A
lemma Min_le:
[| finite A; A ≠ {}; x ∈ A |] ==> Min A ≤ x
lemma Max_ge:
[| finite A; A ≠ {}; x ∈ A |] ==> x ≤ Max A
lemma Min_ge_iff:
[| finite A; A ≠ {} |] ==> (x ≤ Min A) = (∀a∈A. x ≤ a)
lemma Max_le_iff:
[| finite A; A ≠ {} |] ==> (Max A ≤ x) = (∀a∈A. a ≤ x)
lemma Min_le_iff:
[| finite A; A ≠ {} |] ==> (Min A ≤ x) = (∃a∈A. a ≤ x)
lemma Max_ge_iff:
[| finite A; A ≠ {} |] ==> (x ≤ Max A) = (∃a∈A. x ≤ a)
lemma finite_set:
finite A
lemma inj_graph:
inj (%f. {(x, y). y = f x})