(* Title: HOL/Finite_Set.thy ID: $Id: Finite_Set.thy,v 1.160 2007/11/06 07:47:25 haftmann Exp $ Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel with contributions by Jeremy Avigad *) header {* Finite sets *} theory Finite_Set imports Divides 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 prems 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{* 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" 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<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: 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_Collect_subset[simp]: "finite A ==> finite{x ∈ A. P x}" using finite_subset[of "{x ∈ A. P x}" "A"] by blast 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: 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 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 lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A" by simp 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) 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_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: finite) 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. *} inductive foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool" for f :: "'a => 'a => 'a" and g :: "'b => 'a" and z :: 'a where emptyI [intro]: "foldSet f g z {} z" | insertI [intro]: "[| x ∉ A; foldSet f g z A y |] ==> foldSet f g z (insert x A) (f (g x) y)" inductive_cases empty_foldSetE [elim!]: "foldSet f g z {} x" constdefs fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a" "fold f g z A == THE x. foldSet f g z A x" text{*A tempting alternative for the definiens is @{term "if finite A then THE x. foldSet f g e A x 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: "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)" by (erule insert_Diff [THEN subst], rule foldSet.intros, auto) lemma foldSet_imp_finite: "foldSet f g z A x==> finite A" by (induct set: foldSet) auto lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x" by (induct set: finite) auto subsubsection {* Commutative monoids *} (*FIXME integrate with Orderings.thy/OrderedGroup.thy*) locale ACf = fixes f :: "'a => 'a => 'a" (infixl "·" 70) assumes commute: "x · y = y · x" and assoc: "(x · y) · z = x · (y · z)" begin lemma 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 AC = assoc commute left_commute end locale ACe = ACf + fixes e :: 'a assumes ident [simp]: "x · e = x" begin lemma left_ident [simp]: "e · x = x" proof - have "x · e = x" by (rule ident) thus ?thesis by (subst commute) qed end locale ACIf = ACf + assumes idem: "x · x = x" begin lemma 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 ACI = AC idem idem2 end 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" by fact 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}; foldSet f g z A x; foldSet f g z A x' |] ==> 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}; foldSet f g z A x; foldSet f g z A x'|] ==> x' = x" by fact have Afoldx: "foldSet f g z A x" and Afoldx': "foldSet f g z A x'" and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+ show ?case proof (rule foldSet.cases [OF Afoldx]) assume "A = {}" and "x = z" with Afoldx' show "x' = x" by blast next fix B b u assume AbB: "A = insert b B" and x: "x = g b · u" and notinB: "b ∉ B" and Bu: "foldSet f g z B u" show "x'=x" proof (rule foldSet.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' = g c · v" and notinC: "c ∉ C" and Cv: "foldSet f g z C v" 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: "foldSet f g z ?D d" using finite_imp_foldSet by iprover moreover have cinB: "c ∈ B" using B by auto ultimately have "foldSet f g z B (g c · d)" 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 "foldSet f g z C (g b · d)" 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: "foldSet f g z A x ==> foldSet f g z A y ==> 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: "foldSet f g z A y ==> 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 ==> (foldSet f g z (insert x A) v) = (EX y. foldSet f g z A y & 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: finite) apply 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: finite) apply 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: finite, 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: 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 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: finite) 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 text{* Interpretation of locales -- see OrderedGroup.thy *} interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"] by unfold_locales (auto intro: add_assoc add_commute) interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"] by unfold_locales (auto intro: mult_assoc mult_commute) 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" 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" == "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}" 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 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: 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: ring_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_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 prems 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 *} constdefs setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult" "setprod f A == if finite A then fold (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" == "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}" 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: 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" apply (erule finite_induct, auto) done 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)" 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::idom)) ==> setprod f A ≠ 0" apply (rule setprod_nonzero, auto) done lemma setprod_zero_eq_field: "finite A ==> (setprod f A = (0::'a::idom)) = (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) 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 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,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_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 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 ∩ \<Union> F = {}" by auto text{* 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 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: ring_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 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}.*} 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: 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 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 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]: "[| foldSet f id 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: 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. fold1Set f A x" 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: "foldSet f id b A y" shows "b ∉ A ==> foldSet f id z (insert b A) (z · y)" using fold proof (induct rule: foldSet.induct) case emptyI thus ?case by (force simp add: fold_insert_aux commute) next case (insertI x A y) have "foldSet f (λu. u) 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 AC) qed lemma (in ACf) foldSet_permute_diff: assumes fold: "foldSet f id b A x" shows "!!a. [|a ∈ A; b ∉ A|] ==> foldSet f id a (insert b (A-{a})) x" using fold proof (induct rule: foldSet.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: foldSet_insert_swap) next assume ainA: "a ∈ A" hence "foldSet f id a (insert x (insert b (A - {a}))) (x · y)" 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 "foldSet f id a A x") 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 lemma (in ACIf) hom_fold1_commute: assumes hom: "!!x y. h(f x y) = f (h x) (h y)" and N: "finite N" "N ≠ {}" shows "h(fold1 f N) = fold1 f (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 f (insert n N)) = h(f n (fold1 f N))" by simp also have "… = f (h n) (h(fold1 f N))" by(rule hom) also have "h(fold1 f N) = fold1 f (h ` N)" by(rule insert) also have "f (h n) … = fold1 f (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 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: "[|foldSet f id b (insert a A) x; a ∉ A; b ∉ A|] ==> foldSet f id a (insert b A) x" apply (case_tac "a=b") apply (auto dest: foldSet_permute_diff) done lemma (in ACf) fold1Set_determ: "fold1Set f A x ==> fold1Set f A y ==> y = x" proof (clarify elim!: fold1Set.cases) fix A x B y a b assume Ax: "foldSet f id a A x" assume By: "foldSet f 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: 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 "foldSet f id a (insert b ?D) y" by (auto intro: foldSet_permute simp add: insert_absorb) moreover have "foldSet f id a (insert b ?D) x" by (simp add: A [symmetric] Ax) ultimately show ?thesis by (blast intro: foldSet_determ) qed qed lemma (in ACf) fold1Set_equality: "fold1Set f A y ==> 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 proofs involve these relations. *} subsubsection{* Semi-Lattices *} locale ACIfSL = ord + ACIf + assumes below_def: "less_eq x y <-> x · y = x" and strict_below_def: "less x y <-> less_eq x y ∧ x ≠ y" begin notation less ("(_/ \<prec> _)" [51, 51] 50) notation (xsymbols) less_eq ("(_/ \<preceq> _)" [51, 51] 50) notation (HTML output) less_eq ("(_/ \<preceq> _)" [51, 51] 50) lemma below_refl [simp]: "x \<preceq> x" by (simp add: below_def idem) lemma below_antisym: assumes xy: "x \<preceq> y" and yx: "y \<preceq> x" shows "x = y" using xy [unfolded below_def, symmetric] yx [unfolded below_def commute] by (rule trans) lemma below_trans: assumes xy: "x \<preceq> y" and yz: "y \<preceq> z" shows "x \<preceq> z" proof - from xy have x_xy: "x · y = x" by (simp add: below_def) from yz have y_yz: "y · z = y" by (simp add: below_def) from y_yz have "x · y · z = x · y" by (simp add: assoc) with x_xy have "x · y · z = x" by simp moreover from x_xy have "x · z = x · y · z" by simp ultimately have "x · z = x" by simp then show ?thesis by (simp add: below_def) qed lemma below_f_conv [simp,noatp]: "x \<preceq> y · z = (x \<preceq> y ∧ x \<preceq> z)" proof assume "x \<preceq> 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 \<preceq> y ∧ x \<preceq> z" by(simp add: below_def) next assume a: "x \<preceq> y ∧ x \<preceq> 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 \<preceq> y · z" by(simp_all add: below_def) qed end interpretation ACIfSL < order by unfold_locales (simp add: strict_below_def, auto intro: below_refl below_trans below_antisym) locale ACIfSLlin = ACIfSL + assumes lin: "x·y ∈ {x,y}" begin lemma above_f_conv: "x · y \<preceq> z = (x \<preceq> z ∨ y \<preceq> z)" proof assume a: "x · y \<preceq> z" have "x · y = x ∨ x · y = y" using lin[of x y] by simp thus "x \<preceq> z ∨ y \<preceq> z" proof assume "x · y = x" hence "x \<preceq> z" by(rule subst)(rule a) thus ?thesis .. next assume "x · y = y" hence "y \<preceq> z" by(rule subst)(rule a) thus ?thesis .. qed next assume "x \<preceq> z ∨ y \<preceq> z" thus "x · y \<preceq> z" proof assume a: "x \<preceq> 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 \<preceq> z" by(simp add:below_def) next assume a: "y \<preceq> 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 \<preceq> z" by(simp add:below_def) qed qed lemma strict_below_f_conv[simp,noatp]: "x \<prec> y · z = (x \<prec> y ∧ x \<prec> z)" apply(simp add: strict_below_def) using lin[of y z] by (auto simp:below_def ACI) lemma strict_above_f_conv: "x · y \<prec> z = (x \<prec> z ∨ y \<prec> z)" apply(simp add: strict_below_def above_f_conv) using lin[of y z] lin[of x z] by (auto simp:below_def ACI) end interpretation ACIfSLlin < linorder by unfold_locales (insert lin [simplified insert_iff], simp add: below_def commute) 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 \<preceq> fold1 f A = (∀a∈A. x \<preceq> a)" using A by(induct rule:finite_ne_induct) simp_all lemma (in ACIfSLlin) strict_below_fold1_iff: "finite A ==> A ≠ {} ==> x \<prec> fold1 f A = (∀a∈A. x \<prec> 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 \<preceq> 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 \<preceq> 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 \<preceq> x = (∃a∈A. a \<preceq> x)" using A by(induct rule:finite_ne_induct)(simp_all add:above_f_conv) lemma (in ACIfSLlin) fold1_strict_below_iff: assumes A: "finite A" "A ≠ {}" shows "fold1 f A \<prec> x = (∃a∈A. a \<prec> x)" using A by(induct rule:finite_ne_induct)(simp_all add:strict_above_f_conv) lemma (in ACIfSLlin) fold1_antimono: assumes "A ≠ {}" and "A ⊆ B" and "finite B" shows "fold1 f B \<preceq> fold1 f A" proof(cases) assume "A = B" thus ?thesis by simp next assume "A ≠ B" have B: "B = A ∪ (B-A)" using `A ⊆ B` by blast have "fold1 f B = fold1 f (A ∪ (B-A))" by(subst B)(rule refl) also have "… = f (fold1 f A) (fold1 f (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 "… \<preceq> fold1 f A" by(simp add: above_f_conv) finally show ?thesis . 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}. *} lemma (in lower_semilattice) ACf_inf: "ACf inf" by (blast intro: ACf.intro inf_commute inf_assoc) lemma (in upper_semilattice) ACf_sup: "ACf sup" by (blast intro: ACf.intro sup_commute sup_assoc) lemma (in lower_semilattice) ACIf_inf: "ACIf inf" apply(rule ACIf.intro) apply(rule ACf_inf) apply(rule ACIf_axioms.intro) apply(rule inf_idem) done lemma (in upper_semilattice) ACIf_sup: "ACIf sup" apply(rule ACIf.intro) apply(rule ACf_sup) apply(rule ACIf_axioms.intro) apply(rule sup_idem) done lemma (in lower_semilattice) ACIfSL_inf: "ACIfSL (op ≤) (op <) inf" apply(rule ACIfSL.intro) apply(rule ACIf.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_greatest refl) apply(erule subst) apply(rule inf_le2) apply(rule less_le) done lemma (in upper_semilattice) ACIfSL_sup: "ACIfSL (%x y. y ≤ x) (%x y. y < x) sup" apply(rule ACIfSL.intro) apply(rule ACIf.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_least refl) apply(erule subst) apply(rule sup_ge2) apply(simp add: neq_commute less_le) done context lattice begin definition Inf_fin :: "'a set => 'a" ("\<Sqinter>fin_" [900] 900) where "Inf_fin = fold1 inf" definition Sup_fin :: "'a set => 'a" ("\<Squnion>fin_" [900] 900) where "Sup_fin = fold1 sup" lemma Inf_le_Sup [simp]: "[| finite A; A ≠ {} |] ==> \<Sqinter>finA ≤ \<Squnion>finA" 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 (2) ACIfSL.fold1_belowI[OF ACIfSL_inf]) apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup]) done lemma sup_Inf_absorb [simp]: "[| finite A; A ≠ {}; a ∈ A |] ==> (sup a (\<Sqinter>finA)) = a" apply(subst sup_commute) apply(simp add: Inf_fin_def sup_absorb2 ACIfSL.fold1_belowI [OF ACIfSL_inf]) done lemma inf_Sup_absorb [simp]: "[| finite A; A ≠ {}; a ∈ A |] ==> (inf a (\<Squnion>finA)) = a" by(simp add: Sup_fin_def inf_absorb1 ACIfSL.fold1_belowI [OF ACIfSL_sup]) end context distrib_lattice begin lemma sup_Inf1_distrib: "finite A ==> A ≠ {} ==> sup x (\<Sqinter>finA) = \<Sqinter>fin{sup x a|a. a ∈ A}" apply(simp add: Inf_fin_def image_def ACIf.hom_fold1_commute[OF ACIf_inf, where h="sup x", OF sup_inf_distrib1]) apply(rule arg_cong, blast) done lemma sup_Inf2_distrib: assumes A: "finite A" "A ≠ {}" and B: "finite B" "B ≠ {}" shows "sup (\<Sqinter>finA) (\<Sqinter>finB) = \<Sqinter>fin{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 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 (\<Sqinter>fin(insert x A)) (\<Sqinter>finB) = sup (inf x (\<Sqinter>finA)) (\<Sqinter>finB)" using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_fin_def]) also have "… = inf (sup x (\<Sqinter>finB)) (sup (\<Sqinter>finA) (\<Sqinter>finB))" by(rule sup_inf_distrib2) also have "… = inf (\<Sqinter>fin{sup x b|b. b ∈ B}) (\<Sqinter>fin{sup a b|a b. a ∈ A ∧ b ∈ B})" using insert by(simp add:sup_Inf1_distrib[OF B]) also have "… = \<Sqinter>fin({sup x b |b. b ∈ B} ∪ {sup a b |a b. a ∈ A ∧ b ∈ B})" (is "_ = \<Sqinter>fin?M") using B insert by (simp add: Inf_fin_def ACIf.fold1_Un2[OF ACIf_inf 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: "finite A ==> A ≠ {} ==> inf x (\<Squnion>finA) = \<Squnion>fin{inf x a|a. a ∈ A}" apply (simp add: Sup_fin_def image_def ACIf.hom_fold1_commute[OF ACIf_sup, where h="inf x", OF inf_sup_distrib1]) apply (rule arg_cong, blast) done lemma inf_Sup2_distrib: assumes A: "finite A" "A ≠ {}" and B: "finite B" "B ≠ {}" shows "inf (\<Squnion>finA) (\<Squnion>finB) = \<Squnion>fin{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 have "inf (\<Squnion>fin(insert x A)) (\<Squnion>finB) = inf (sup x (\<Squnion>finA)) (\<Squnion>finB)" using insert by (simp add: ACIf.fold1_insert_idem_def [OF ACIf_sup Sup_fin_def]) also have "… = sup (inf x (\<Squnion>finB)) (inf (\<Squnion>finA) (\<Squnion>finB))" by(rule inf_sup_distrib2) also have "… = sup (\<Squnion>fin{inf x b|b. b ∈ B}) (\<Squnion>fin{inf a b|a b. a ∈ A ∧ b ∈ B})" using insert by(simp add:inf_Sup1_distrib[OF B]) also have "… = \<Squnion>fin({inf x b |b. b ∈ B} ∪ {inf a b |a b. a ∈ A ∧ b ∈ B})" (is "_ = \<Squnion>fin?M") using B insert by (simp add: Sup_fin_def ACIf.fold1_Un2[OF ACIf_sup 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: "finite A ==> A ≠ {} ==> \<Sqinter>finA = Inf A" unfolding Inf_fin_def by (induct A set: finite) (simp_all add: Inf_insert_simp ACIf.fold1_insert_idem [OF ACIf_inf]) lemma Sup_fin_Sup: "finite A ==> A ≠ {} ==> \<Squnion>finA = Sup A" unfolding Sup_fin_def by (induct A set: finite) (simp_all add: Sup_insert_simp ACIf.fold1_insert_idem [OF ACIf_sup]) 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 definition Min :: "'a set => 'a" where "Min = fold1 min" definition Max :: "'a set => 'a" where "Max = fold1 max" end context linorder begin text {* recall: @{term min} and @{term max} behave like @{const inf} and @{const sup} *} lemma ACIf_min: "ACIf min" by (rule lower_semilattice.ACIf_inf, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) lemma ACf_min: "ACf min" by (rule lower_semilattice.ACf_inf, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) lemma ACIfSL_min: "ACIfSL (op ≤) (op <) min" by (rule lower_semilattice.ACIfSL_inf, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) lemma ACIfSLlin_min: "ACIfSLlin (op ≤) (op <) min" by (rule ACIfSLlin.intro, rule lower_semilattice.ACIfSL_inf, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) (unfold_locales, simp add: min_def) lemma ACIf_max: "ACIf max" by (rule upper_semilattice.ACIf_sup, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) lemma ACf_max: "ACf max" by (rule upper_semilattice.ACf_sup, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) lemma ACIfSL_max: "ACIfSL (λx y. y ≤ x) (λx y. y < x) max" by (rule upper_semilattice.ACIfSL_sup, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) lemma ACIfSLlin_max: "ACIfSLlin (λx y. y ≤ x) (λx y. y < x) max" by (rule ACIfSLlin.intro, rule upper_semilattice.ACIfSL_sup, rule lattice.axioms, rule distrib_lattice.axioms, rule distrib_lattice_min_max) (unfold_locales, simp add: max_def) lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def] lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def] lemmas Min_insert [simp] = ACIf.fold1_insert_idem_def [OF ACIf_min Min_def] lemmas Max_insert [simp] = ACIf.fold1_insert_idem_def [OF ACIf_max Max_def] lemma Min_in [simp]: shows "finite A ==> A ≠ {} ==> Min A ∈ A" using ACf.fold1_in [OF ACf_min] by (fastsimp simp: Min_def min_def) lemma Max_in [simp]: shows "finite A ==> A ≠ {} ==> Max A ∈ A" using ACf.fold1_in [OF ACf_max] by (fastsimp simp: Max_def max_def) lemma Min_antimono: "[| M ⊆ N; M ≠ {}; finite N |] ==> Min N ≤ Min M" by (simp add: Min_def ACIfSLlin.fold1_antimono [OF ACIfSLlin_min]) lemma Max_mono: "[| M ⊆ N; M ≠ {}; finite N |] ==> Max M ≤ Max N" by (simp add: Max_def ACIfSLlin.fold1_antimono [OF ACIfSLlin_max]) lemma Min_le [simp]: "[| finite A; A ≠ {}; x ∈ A |] ==> Min A ≤ x" by (simp add: Min_def ACIfSL.fold1_belowI [OF ACIfSL_min]) lemma Max_ge [simp]: "[| finite A; A ≠ {}; x ∈ A |] ==> x ≤ Max A" by (simp add: Max_def ACIfSL.fold1_belowI [OF ACIfSL_max]) lemma Min_ge_iff [simp,noatp]: "[| finite A; A ≠ {} |] ==> x ≤ Min A <-> (∀a∈A. x ≤ a)" by (simp add: Min_def ACIfSL.below_fold1_iff [OF ACIfSL_min]) lemma Max_le_iff [simp,noatp]: "[| finite A; A ≠ {} |] ==> Max A ≤ x <-> (∀a∈A. a ≤ x)" by (simp add: Max_def ACIfSL.below_fold1_iff [OF ACIfSL_max]) lemma Min_gr_iff [simp,noatp]: "[| finite A; A ≠ {} |] ==> x < Min A <-> (∀a∈A. x < a)" by (simp add: Min_def ACIfSLlin.strict_below_fold1_iff [OF ACIfSLlin_min]) lemma Max_less_iff [simp,noatp]: "[| finite A; A ≠ {} |] ==> Max A < x <-> (∀a∈A. a < x)" by (simp add: Max_def ACIfSLlin.strict_below_fold1_iff [OF ACIfSLlin_max]) lemma Min_le_iff [noatp]: "[| finite A; A ≠ {} |] ==> Min A ≤ x <-> (∃a∈A. a ≤ x)" by (simp add: Min_def ACIfSLlin.fold1_below_iff [OF ACIfSLlin_min]) lemma Max_ge_iff [noatp]: "[| finite A; A ≠ {} |] ==> x ≤ Max A <-> (∃a∈A. x ≤ a)" by (simp add: Max_def ACIfSLlin.fold1_below_iff [OF ACIfSLlin_max]) lemma Min_less_iff [noatp]: "[| finite A; A ≠ {} |] ==> Min A < x <-> (∃a∈A. a < x)" by (simp add: Min_def ACIfSLlin.fold1_strict_below_iff [OF ACIfSLlin_min]) lemma Max_gr_iff [noatp]: "[| finite A; A ≠ {} |] ==> x < Max A <-> (∃a∈A. x < a)" by (simp add: Max_def ACIfSLlin.fold1_strict_below_iff [OF ACIfSLlin_max]) lemma Min_Un: "[|finite A; A ≠ {}; finite B; B ≠ {}|] ==> Min (A ∪ B) = min (Min A) (Min B)" by (simp add: Min_def ACIf.fold1_Un2 [OF ACIf_min]) lemma Max_Un: "[|finite A; A ≠ {}; finite B; B ≠ {}|] ==> Max (A ∪ B) = max (Max A) (Max B)" by (simp add: Max_def ACIf.fold1_Un2 [OF ACIf_max]) lemma hom_Min_commute: "(!!x y. h (min x y) = min (h x) (h y)) ==> finite N ==> N ≠ {} ==> h (Min N) = Min (h ` N)" by (simp add: Min_def ACIf.hom_fold1_commute [OF ACIf_min]) lemma hom_Max_commute: "(!!x y. h (max x y) = max (h x) (h y)) ==> finite N ==> N ≠ {} ==> h (Max N) = Max (h ` N)" by (simp add: Max_def ACIf.hom_fold1_commute [OF ACIf_max]) 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 subsection {* Class @{text finite} and code generation *} lemma finite_code [code func]: "finite {} <-> True" "finite (insert a A) <-> finite A" by auto lemma card_code [code func]: "card {} = 0" "card (insert a A) = (if finite A then Suc (card (A - {a})) else card (insert a A))" by (auto simp add: card_insert) setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*} class finite (attach UNIV) = type + fixes itself :: "'a itself" assumes finite_UNIV: "finite (UNIV :: 'a set)" setup {* Sign.parent_path *} hide const finite lemma finite [simp]: "finite (A :: 'a::finite set)" by (rule finite_subset [OF subset_UNIV finite_UNIV]) lemma univ_unit [noatp]: "UNIV = {()}" by auto instance unit :: finite "Finite_Set.itself ≡ TYPE(unit)" proof have "finite {()}" by simp also note univ_unit [symmetric] finally show "finite (UNIV :: unit set)" . qed lemmas [code func] = univ_unit lemma univ_bool [noatp]: "UNIV = {False, True}" by auto instance bool :: finite "itself ≡ TYPE(bool)" proof have "finite {False, True}" by simp also note univ_bool [symmetric] finally show "finite (UNIV :: bool set)" . qed lemmas [code func] = univ_bool instance * :: (finite, finite) finite "itself ≡ TYPE('a::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 lemma univ_prod [noatp, code func]: "UNIV = (UNIV :: 'a::finite set) × (UNIV :: 'b::finite set)" unfolding UNIV_Times_UNIV .. instance "+" :: (finite, finite) finite "itself ≡ TYPE('a::finite + 'b::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 lemma univ_sum [noatp, code func]: "UNIV = (UNIV :: 'a::finite set) <+> (UNIV :: 'b::finite set)" unfolding UNIV_Plus_UNIV .. instance set :: (finite) finite "itself ≡ TYPE('a::finite set)" 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 univ_set [noatp, code func]: "UNIV = Pow (UNIV :: 'a::finite set)" unfolding Pow_UNIV .. 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 "itself ≡ TYPE('a::finite => 'b::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) show "inj ?graph" by (rule inj_graph) qed qed hide (open) const itself subsection {* Equality and order on functions *} instance "fun" :: (finite, eq) eq .. lemma eq_fun [code func]: fixes f g :: "'a::finite => 'b::eq" shows "f = g <-> (∀x∈UNIV. f x = g x)" unfolding expand_fun_eq by auto lemma order_fun [code func]: fixes f g :: "'a::finite => 'b::order" shows "f ≤ g <-> (∀x∈UNIV. f x ≤ g x)" and "f < g <-> f ≤ g ∧ (∃x∈UNIV. f x ≠ g x)" by (auto simp add: expand_fun_eq le_fun_def less_fun_def order_less_le) end
lemma ex_new_if_finite:
[| ¬ finite UNIV; 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_Collect_subset:
finite A ==> finite {x : A. P x}
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_Diff_singleton:
finite (A - {a}) = finite A
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 (λx. 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 B)
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
lemma Diff1_foldSet:
[| foldSet f g z (A - {x}) y; x ∈ A |] ==> foldSet f g z A (f (g x) y)
lemma foldSet_imp_finite:
foldSet f g z A x ==> finite A
lemma finite_imp_foldSet:
finite A ==> ∃x. foldSet f g z A x
lemma left_commute:
x · (y · z) = y · (x · z)
lemma AC:
x · y · z = x · (y · z)
x · y = y · x
x · (y · z) = y · (x · z)
lemma left_ident:
e · x = x
lemma idem2:
x · (x · y) = x · y
lemma ACI:
x · y · z = x · (y · z)
x · y = y · x
x · (y · z) = y · (x · z)
x · x = x
x · (x · y) = 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:
[| A = h ` {i. i < n}; inj_on h {i. i < n}; foldSet op · g z A x;
foldSet op · g z A x' |]
==> x' = x
lemma foldSet_determ:
[| foldSet op · g z A x; foldSet op · g z A y |] ==> y = x
lemma fold_equality:
foldSet op · g z A y ==> fold op · g z A = y
lemma fold_empty:
fold f g z {} = z
lemma fold_insert_aux:
x ∉ A
==> foldSet op · g z (insert x A) v = (∃y. foldSet op · g z A y ∧ v = g x · y)
lemma fold_insert:
[| finite A; x ∉ A |] ==> fold op · g z (insert x A) = g x · fold op · g z A
lemma fold_rec:
[| finite A; a ∈ A |] ==> fold op · g z A = g a · fold op · g z (A - {a})
lemma fold_insert_idem:
finite A ==> fold op · g z (insert a A) = g a · fold op · g z A
lemma foldI_conv_id:
finite A ==> fold op · g z A = fold op · id z (g ` A)
lemma fold_commute:
finite A ==> x · fold op · g z A = fold op · g (x · z) A
lemma fold_nest_Un_Int:
[| finite A; finite B |]
==> fold op · g (fold op · g z B) A =
fold op · g (fold op · g z (A ∩ B)) (A ∪ B)
lemma fold_nest_Un_disjoint:
[| finite A; finite B; A ∩ B = {} |]
==> fold op · g z (A ∪ B) = fold op · g (fold op · g z B) A
lemma fold_reindex:
[| finite A; inj_on h A |] ==> fold op · g z (h ` A) = fold op · (g o h) z A
lemma fold_Un_Int:
[| finite A; finite B |]
==> fold op · g e A · fold op · g e B =
fold op · g e (A ∪ B) · fold op · g e (A ∩ B)
corollary fold_Un_disjoint:
[| finite A; finite B; A ∩ B = {} |]
==> fold op · g e (A ∪ B) = fold op · g e A · fold op · g e B
lemma fold_UN_disjoint:
[| finite I; ∀i∈I. finite (A i); ∀i∈I. ∀j∈I. i ≠ j --> A i ∩ A j = {} |]
==> fold op · g e (UNION I A) = fold op · (λi. fold op · g e (A i)) e I
lemma fold_fusion:
[| ACf g; finite A; !!x y. h (g x y) = x · h y |]
==> h (fold g j w A) = fold op · j (h w) A
lemma fold_cong:
[| finite A; !!x. x ∈ A ==> g x = h x |] ==> fold op · g z A = fold op · h z A
lemma fold_Sigma:
[| finite A; ∀x∈A. finite (B x) |]
==> fold op · (λx. fold op · (g x) e (B x)) e A =
fold op · (λ(x, y). g x y) e (Sigma A B)
lemma fold_distrib:
finite A ==> fold op · (λx. g x · h x) e A = fold op · g e A · fold op · 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:
¬ finite A ==> 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 A B. 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_right_distrib:
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 (λ(i, j). (j, i)) (A × B)
lemma swap_product:
(λ(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 setsum_product:
setsum f A * setsum g B = (∑i∈A. ∑j∈B. f i * g 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:
¬ finite A ==> 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 A B. 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:
¬ finite A ==> 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 = {} ∨ ¬ finite A)
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_Diff_insert:
[| finite A; a ∈ A; a ∉ B |] ==> card (A - insert a B) = card (A - B) - 1
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 card_eq_SucD:
card A = Suc k
==> ∃b B. A = insert b B ∧ b ∉ B ∧ card B = k ∧ (k = 0 --> B = {})
lemma card_Suc_eq:
(card A = Suc k) =
(∃b B. A = insert b B ∧ b ∉ B ∧ card B = k ∧ (k = 0 --> B = {}))
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 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 ∪ (λx. {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 A B) = (∑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 finite_surj_inj:
[| finite A; A ⊆ f ` A |] ==> inj_on f A
lemma finite_UNIV_surj_inj:
[| finite UNIV; surj f |] ==> inj f
lemma finite_UNIV_inj_surj:
[| finite UNIV; inj f |] ==> surj f
corollary infinite_UNIV_nat:
¬ finite UNIV
lemma fold1Set_nonempty:
fold1Set f A x ==> A ≠ {}
lemma fold1Set_sing:
fold1Set f {a} b = (a = b)
lemma fold1_singleton:
fold1 f {a} = a
lemma finite_nonempty_imp_fold1Set:
[| finite A; A ≠ {} |] ==> ∃x. fold1Set f A x
lemma foldSet_insert_swap:
[| foldSet op · id b A y; b ∉ A |] ==> foldSet op · id z (insert b A) (z · y)
lemma foldSet_permute_diff:
[| foldSet op · id b A x; a ∈ A; b ∉ A |]
==> foldSet op · id a (insert b (A - {a})) x
lemma fold1_eq_fold:
[| finite A; a ∉ A |] ==> fold1 op · (insert a A) = fold op · id a A
lemma nonempty_iff:
(A ≠ {}) = (∃x B. A = insert x B ∧ x ∉ B)
lemma fold1_insert:
[| A ≠ {}; finite A; x ∉ A |] ==> fold1 op · (insert x A) = x · fold1 op · A
lemma fold1_insert_idem:
[| A ≠ {}; finite A |] ==> fold1 op · (insert x A) = x · fold1 op · A
lemma hom_fold1_commute:
[| !!x y. h (x · y) = h x · h y; finite N; N ≠ {} |]
==> h (fold1 op · N) = fold1 op · (h ` N)
lemma fold1_singleton_def:
g = fold1 f ==> g {a} = a
lemma fold1_insert_def:
[| g = fold1 op ·; finite A; x ∉ A; A ≠ {} |] ==> g (insert x A) = x · g A
lemma fold1_insert_idem_def:
[| g = fold1 op ·; finite A; A ≠ {} |] ==> g (insert x A) = x · g A
lemma foldSet_permute:
[| foldSet op · id b (insert a A) x; a ∉ A; b ∉ A |]
==> foldSet op · id a (insert b A) x
lemma fold1Set_determ:
[| fold1Set op · A x; fold1Set op · A y |] ==> y = x
lemma fold1Set_equality:
fold1Set op · A y ==> fold1 op · A = y
lemma below_refl:
x \<preceq> x
lemma below_antisym:
[| x \<preceq> y; y \<preceq> x |] ==> x = y
lemma below_trans:
[| x \<preceq> y; y \<preceq> z |] ==> x \<preceq> z
lemma below_f_conv:
(x \<preceq> y · z) = (x \<preceq> y ∧ x \<preceq> z)
lemma above_f_conv:
(x · y \<preceq> z) = (x \<preceq> z ∨ y \<preceq> z)
lemma strict_below_f_conv:
(x \<prec> y · z) = (x \<prec> y ∧ x \<prec> z)
lemma strict_above_f_conv:
(x · y \<prec> z) = (x \<prec> z ∨ y \<prec> z)
lemma fold1_Un:
[| finite A; A ≠ {}; finite B; B ≠ {}; A ∩ B = {} |]
==> fold1 op · (A ∪ B) = fold1 op · A · fold1 op · B
lemma fold1_Un2:
[| finite A; A ≠ {}; finite B; B ≠ {} |]
==> fold1 op · (A ∪ B) = fold1 op · A · fold1 op · B
lemma fold1_in:
[| finite A; A ≠ {}; !!x y. x · y ∈ {x, y} |] ==> fold1 op · A ∈ A
lemma below_fold1_iff:
[| finite A; A ≠ {} |] ==> (x \<preceq> fold1 op · A) = (∀a∈A. x \<preceq> a)
lemma strict_below_fold1_iff:
[| finite A; A ≠ {} |] ==> (x \<prec> fold1 op · A) = (∀a∈A. x \<prec> a)
lemma fold1_belowI:
[| finite A; A ≠ {}; a ∈ A |] ==> fold1 op · A \<preceq> a
lemma fold1_below_iff:
[| finite A; A ≠ {} |] ==> (fold1 op · A \<preceq> x) = (∃a∈A. a \<preceq> x)
lemma fold1_strict_below_iff:
[| finite A; A ≠ {} |] ==> (fold1 op · A \<prec> x) = (∃a∈A. a \<prec> x)
lemma fold1_antimono:
[| A ≠ {}; A ⊆ B; finite B |] ==> fold1 op · B \<preceq> fold1 op · A
lemma ACf_inf:
ACf inf
lemma ACf_sup:
ACf sup
lemma ACIf_inf:
ACIf inf
lemma ACIf_sup:
ACIf sup
lemma ACIfSL_inf:
ACIfSL op ≤ op < inf
lemma ACIfSL_sup:
ACIfSL greater_eq greater sup
lemma Inf_le_Sup:
[| finite A; A ≠ {} |] ==> \<Sqinter>finA ≤ \<Squnion>finA
lemma sup_Inf_absorb:
[| finite A; A ≠ {}; a ∈ A |] ==> sup a (\<Sqinter>finA) = a
lemma inf_Sup_absorb:
[| finite A; A ≠ {}; a ∈ A |] ==> inf a (\<Squnion>finA) = a
lemma sup_Inf1_distrib:
[| finite A; A ≠ {} |]
==> sup x (\<Sqinter>finA) = \<Sqinter>fin{sup x a |a. a ∈ A}
lemma sup_Inf2_distrib:
[| finite A; A ≠ {}; finite B; B ≠ {} |]
==> sup (\<Sqinter>finA) (\<Sqinter>finB) =
\<Sqinter>fin{sup a b |a b. a ∈ A ∧ b ∈ B}
lemma inf_Sup1_distrib:
[| finite A; A ≠ {} |]
==> inf x (\<Squnion>finA) = \<Squnion>fin{inf x a |a. a ∈ A}
lemma inf_Sup2_distrib:
[| finite A; A ≠ {}; finite B; B ≠ {} |]
==> inf (\<Squnion>finA) (\<Squnion>finB) =
\<Squnion>fin{inf a b |a b. a ∈ A ∧ b ∈ B}
lemma Inf_fin_Inf:
[| finite A; A ≠ {} |] ==> \<Sqinter>finA = Inf A
lemma Sup_fin_Sup:
[| finite A; A ≠ {} |] ==> \<Squnion>finA = Sup A
lemma ACIf_min:
ACIf min
lemma ACf_min:
ACf min
lemma ACIfSL_min:
ACIfSL op ≤ op < min
lemma ACIfSLlin_min:
ACIfSLlin op ≤ op < min
lemma ACIf_max:
ACIf max
lemma ACf_max:
ACf max
lemma ACIfSL_max:
ACIfSL greater_eq greater max
lemma ACIfSLlin_max:
ACIfSLlin greater_eq greater max
lemma Min_singleton:
Min {a} = a
lemma Max_singleton:
Max {a} = a
lemma Min_insert:
[| finite A; A ≠ {} |] ==> Min (insert x A) = min x (Min A)
lemma 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_antimono:
[| M ⊆ N; M ≠ {}; finite N |] ==> Min N ≤ Min M
lemma Max_mono:
[| M ⊆ N; M ≠ {}; finite N |] ==> Max M ≤ Max N
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_gr_iff:
[| finite A; A ≠ {} |] ==> (x < Min A) = (∀a∈A. x < a)
lemma Max_less_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 Min_less_iff:
[| finite A; A ≠ {} |] ==> (Min A < x) = (∃a∈A. a < x)
lemma Max_gr_iff:
[| finite A; A ≠ {} |] ==> (x < Max A) = (∃a∈A. x < a)
lemma Min_Un:
[| finite A; A ≠ {}; finite B; B ≠ {} |] ==> Min (A ∪ B) = min (Min A) (Min B)
lemma Max_Un:
[| finite A; A ≠ {}; finite B; B ≠ {} |] ==> Max (A ∪ B) = max (Max A) (Max B)
lemma hom_Min_commute:
[| !!x y. h (min x y) = min (h x) (h y); finite N; N ≠ {} |]
==> h (Min N) = Min (h ` N)
lemma hom_Max_commute:
[| !!x y. h (max x y) = max (h x) (h y); finite N; N ≠ {} |]
==> h (Max N) = Max (h ` N)
lemma add_Min_commute:
[| finite N; N ≠ {} |] ==> k + Min N = Min {k + m |m. m ∈ N}
lemma add_Max_commute:
[| finite N; N ≠ {} |] ==> k + Max N = Max {k + m |m. m ∈ N}
lemma finite_code:
finite {} = True
finite (insert a A) = finite A
lemma card_code:
card {} = 0
card (insert a A) =
(if finite A then Suc (card (A - {a})) else card (insert a A))
lemma finite:
finite A
lemma univ_unit:
UNIV = {()}
lemma
UNIV = {()}
lemma univ_bool:
UNIV = {False, True}
lemma
UNIV = {False, True}
lemma univ_prod:
UNIV = UNIV × UNIV
lemma univ_sum:
UNIV = UNIV <+> UNIV
lemma univ_set:
UNIV = Pow UNIV
lemma inj_graph:
inj (λf. {(x, y). y = f x})
lemma eq_fun:
(f = g) = (∀x∈UNIV. f x = g x)
lemma order_fun(1):
(f ≤ g) = (∀x∈UNIV. f x ≤ g x)
and order_fun(2):
(f < g) = (f ≤ g ∧ (∃x∈UNIV. f x ≠ g x))