(* Title: HOL/Filter.thy Author: Brian Huffman Author: Johannes Hölzl *) section ‹Filters on predicates› theory Filter imports Set_Interval Lifting_Set begin subsection ‹Filters› text ‹ This definition also allows non-proper filters. › locale is_filter = fixes F :: "('a ⇒ bool) ⇒ bool" assumes True: "F (λx. True)" assumes conj: "F (λx. P x) ⟹ F (λx. Q x) ⟹ F (λx. P x ∧ Q x)" assumes mono: "∀x. P x ⟶ Q x ⟹ F (λx. P x) ⟹ F (λx. Q x)" typedef 'a filter = "{F :: ('a ⇒ bool) ⇒ bool. is_filter F}" proof show "(λx. True) ∈ ?filter" by (auto intro: is_filter.intro) qed lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" using Rep_filter [of F] by simp lemma Abs_filter_inverse': assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" using assms by (simp add: Abs_filter_inverse) subsubsection ‹Eventually› definition eventually :: "('a ⇒ bool) ⇒ 'a filter ⇒ bool" where "eventually P F ⟷ Rep_filter F P" syntax "_eventually" :: "pttrn => 'a filter => bool => bool" ("(3∀⇩_{F}_ in _./ _)" [0, 0, 10] 10) translations "∀⇩_{F}x in F. P" == "CONST eventually (λx. P) F" lemma eventually_Abs_filter: assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" unfolding eventually_def using assms by (simp add: Abs_filter_inverse) lemma filter_eq_iff: shows "F = F' ⟷ (∀P. eventually P F = eventually P F')" unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. lemma eventually_True [simp]: "eventually (λx. True) F" unfolding eventually_def by (rule is_filter.True [OF is_filter_Rep_filter]) lemma always_eventually: "∀x. P x ⟹ eventually P F" proof - assume "∀x. P x" hence "P = (λx. True)" by (simp add: ext) thus "eventually P F" by simp qed lemma eventuallyI: "(⋀x. P x) ⟹ eventually P F" by (auto intro: always_eventually) lemma filter_eqI: "(⋀P. eventually P F ⟷ eventually P G) ⟹ F = G" by (auto simp: filter_eq_iff) lemma eventually_mono: "⟦eventually P F; ⋀x. P x ⟹ Q x⟧ ⟹ eventually Q F" unfolding eventually_def by (blast intro: is_filter.mono [OF is_filter_Rep_filter]) lemma eventually_conj: assumes P: "eventually (λx. P x) F" assumes Q: "eventually (λx. Q x) F" shows "eventually (λx. P x ∧ Q x) F" using assms unfolding eventually_def by (rule is_filter.conj [OF is_filter_Rep_filter]) lemma eventually_mp: assumes "eventually (λx. P x ⟶ Q x) F" assumes "eventually (λx. P x) F" shows "eventually (λx. Q x) F" proof - have "eventually (λx. (P x ⟶ Q x) ∧ P x) F" using assms by (rule eventually_conj) then show ?thesis by (blast intro: eventually_mono) qed lemma eventually_rev_mp: assumes "eventually (λx. P x) F" assumes "eventually (λx. P x ⟶ Q x) F" shows "eventually (λx. Q x) F" using assms(2) assms(1) by (rule eventually_mp) lemma eventually_conj_iff: "eventually (λx. P x ∧ Q x) F ⟷ eventually P F ∧ eventually Q F" by (auto intro: eventually_conj elim: eventually_rev_mp) lemma eventually_elim2: assumes "eventually (λi. P i) F" assumes "eventually (λi. Q i) F" assumes "⋀i. P i ⟹ Q i ⟹ R i" shows "eventually (λi. R i) F" using assms by (auto elim!: eventually_rev_mp) lemma eventually_cong: assumes "eventually P F" and "⋀x. P x ⟹ Q x ⟷ R x" shows "eventually Q F ⟷ eventually R F" using assms eventually_elim2 by blast lemma eventually_ball_finite_distrib: "finite A ⟹ (eventually (λx. ∀y∈A. P x y) net) ⟷ (∀y∈A. eventually (λx. P x y) net)" by (induction A rule: finite_induct) (auto simp: eventually_conj_iff) lemma eventually_ball_finite: "finite A ⟹ ∀y∈A. eventually (λx. P x y) net ⟹ eventually (λx. ∀y∈A. P x y) net" by (auto simp: eventually_ball_finite_distrib) lemma eventually_all_finite: fixes P :: "'a ⇒ 'b::finite ⇒ bool" assumes "⋀y. eventually (λx. P x y) net" shows "eventually (λx. ∀y. P x y) net" using eventually_ball_finite [of UNIV P] assms by simp lemma eventually_ex: "(∀⇩_{F}x in F. ∃y. P x y) ⟷ (∃Y. ∀⇩_{F}x in F. P x (Y x))" proof assume "∀⇩_{F}x in F. ∃y. P x y" then have "∀⇩_{F}x in F. P x (SOME y. P x y)" by (auto intro: someI_ex eventually_mono) then show "∃Y. ∀⇩_{F}x in F. P x (Y x)" by auto qed (auto intro: eventually_mono) lemma not_eventually_impI: "eventually P F ⟹ ¬ eventually Q F ⟹ ¬ eventually (λx. P x ⟶ Q x) F" by (auto intro: eventually_mp) lemma not_eventuallyD: "¬ eventually P F ⟹ ∃x. ¬ P x" by (metis always_eventually) lemma eventually_subst: assumes "eventually (λn. P n = Q n) F" shows "eventually P F = eventually Q F" (is "?L = ?R") proof - from assms have "eventually (λx. P x ⟶ Q x) F" and "eventually (λx. Q x ⟶ P x) F" by (auto elim: eventually_mono) then show ?thesis by (auto elim: eventually_elim2) qed subsection ‹ Frequently as dual to eventually › definition frequently :: "('a ⇒ bool) ⇒ 'a filter ⇒ bool" where "frequently P F ⟷ ¬ eventually (λx. ¬ P x) F" syntax "_frequently" :: "pttrn ⇒ 'a filter ⇒ bool ⇒ bool" ("(3∃⇩_{F}_ in _./ _)" [0, 0, 10] 10) translations "∃⇩_{F}x in F. P" == "CONST frequently (λx. P) F" lemma not_frequently_False [simp]: "¬ (∃⇩_{F}x in F. False)" by (simp add: frequently_def) lemma frequently_ex: "∃⇩_{F}x in F. P x ⟹ ∃x. P x" by (auto simp: frequently_def dest: not_eventuallyD) lemma frequentlyE: assumes "frequently P F" obtains x where "P x" using frequently_ex[OF assms] by auto lemma frequently_mp: assumes ev: "∀⇩_{F}x in F. P x ⟶ Q x" and P: "∃⇩_{F}x in F. P x" shows "∃⇩_{F}x in F. Q x" proof - from ev have "eventually (λx. ¬ Q x ⟶ ¬ P x) F" by (rule eventually_rev_mp) (auto intro!: always_eventually) from eventually_mp[OF this] P show ?thesis by (auto simp: frequently_def) qed lemma frequently_rev_mp: assumes "∃⇩_{F}x in F. P x" assumes "∀⇩_{F}x in F. P x ⟶ Q x" shows "∃⇩_{F}x in F. Q x" using assms(2) assms(1) by (rule frequently_mp) lemma frequently_mono: "(∀x. P x ⟶ Q x) ⟹ frequently P F ⟹ frequently Q F" using frequently_mp[of P Q] by (simp add: always_eventually) lemma frequently_elim1: "∃⇩_{F}x in F. P x ⟹ (⋀i. P i ⟹ Q i) ⟹ ∃⇩_{F}x in F. Q x" by (metis frequently_mono) lemma frequently_disj_iff: "(∃⇩_{F}x in F. P x ∨ Q x) ⟷ (∃⇩_{F}x in F. P x) ∨ (∃⇩_{F}x in F. Q x)" by (simp add: frequently_def eventually_conj_iff) lemma frequently_disj: "∃⇩_{F}x in F. P x ⟹ ∃⇩_{F}x in F. Q x ⟹ ∃⇩_{F}x in F. P x ∨ Q x" by (simp add: frequently_disj_iff) lemma frequently_bex_finite_distrib: assumes "finite A" shows "(∃⇩_{F}x in F. ∃y∈A. P x y) ⟷ (∃y∈A. ∃⇩_{F}x in F. P x y)" using assms by induction (auto simp: frequently_disj_iff) lemma frequently_bex_finite: "finite A ⟹ ∃⇩_{F}x in F. ∃y∈A. P x y ⟹ ∃y∈A. ∃⇩_{F}x in F. P x y" by (simp add: frequently_bex_finite_distrib) lemma frequently_all: "(∃⇩_{F}x in F. ∀y. P x y) ⟷ (∀Y. ∃⇩_{F}x in F. P x (Y x))" using eventually_ex[of "λx y. ¬ P x y" F] by (simp add: frequently_def) lemma shows not_eventually: "¬ eventually P F ⟷ (∃⇩_{F}x in F. ¬ P x)" and not_frequently: "¬ frequently P F ⟷ (∀⇩_{F}x in F. ¬ P x)" by (auto simp: frequently_def) lemma frequently_imp_iff: "(∃⇩_{F}x in F. P x ⟶ Q x) ⟷ (eventually P F ⟶ frequently Q F)" unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] .. lemma frequently_eventually_conj: assumes "∃⇩_{F}x in F. P x" assumes "∀⇩_{F}x in F. Q x" shows "∃⇩_{F}x in F. Q x ∧ P x" using assms eventually_elim2 by (force simp add: frequently_def) lemma frequently_cong: assumes ev: "eventually P F" and QR: "⋀x. P x ⟹ Q x ⟷ R x" shows "frequently Q F ⟷ frequently R F" unfolding frequently_def using QR by (auto intro!: eventually_cong [OF ev]) lemma frequently_eventually_frequently: "frequently P F ⟹ eventually Q F ⟹ frequently (λx. P x ∧ Q x) F" using frequently_cong [of Q F P "λx. P x ∧ Q x"] by meson lemma eventually_frequently_const_simps: "(∃⇩_{F}x in F. P x ∧ C) ⟷ (∃⇩_{F}x in F. P x) ∧ C" "(∃⇩_{F}x in F. C ∧ P x) ⟷ C ∧ (∃⇩_{F}x in F. P x)" "(∀⇩_{F}x in F. P x ∨ C) ⟷ (∀⇩_{F}x in F. P x) ∨ C" "(∀⇩_{F}x in F. C ∨ P x) ⟷ C ∨ (∀⇩_{F}x in F. P x)" "(∀⇩_{F}x in F. P x ⟶ C) ⟷ ((∃⇩_{F}x in F. P x) ⟶ C)" "(∀⇩_{F}x in F. C ⟶ P x) ⟷ (C ⟶ (∀⇩_{F}x in F. P x))" by (cases C; simp add: not_frequently)+ lemmas eventually_frequently_simps = eventually_frequently_const_simps not_eventually eventually_conj_iff eventually_ball_finite_distrib eventually_ex not_frequently frequently_disj_iff frequently_bex_finite_distrib frequently_all frequently_imp_iff ML ‹ fun eventually_elim_tac facts = CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) => let val mp_facts = facts RL @{thms eventually_rev_mp} val rule = @{thm eventuallyI} |> fold (fn mp_fact => fn th => th RS mp_fact) mp_facts |> funpow (length facts) (fn th => @{thm impI} RS th) val cases_prop = Thm.prop_of (Rule_Cases.internalize_params (rule RS Goal.init (Thm.cterm_of ctxt goal))) val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])] in CONTEXT_CASES cases (resolve_tac ctxt [rule] i) (ctxt, st) end) › method_setup eventually_elim = ‹ Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1)) › "elimination of eventually quantifiers" subsubsection ‹Finer-than relation› text ‹\<^term>‹F ≤ F'› means that filter \<^term>‹F› is finer than filter \<^term>‹F'›.› instantiation filter :: (type) complete_lattice begin definition le_filter_def: "F ≤ F' ⟷ (∀P. eventually P F' ⟶ eventually P F)" definition "(F :: 'a filter) < F' ⟷ F ≤ F' ∧ ¬ F' ≤ F" definition "top = Abs_filter (λP. ∀x. P x)" definition "bot = Abs_filter (λP. True)" definition "sup F F' = Abs_filter (λP. eventually P F ∧ eventually P F')" definition "inf F F' = Abs_filter (λP. ∃Q R. eventually Q F ∧ eventually R F' ∧ (∀x. Q x ∧ R x ⟶ P x))" definition "Sup S = Abs_filter (λP. ∀F∈S. eventually P F)" definition "Inf S = Sup {F::'a filter. ∀F'∈S. F ≤ F'}" lemma eventually_top [simp]: "eventually P top ⟷ (∀x. P x)" unfolding top_filter_def by (rule eventually_Abs_filter, rule is_filter.intro, auto) lemma eventually_bot [simp]: "eventually P bot" unfolding bot_filter_def by (subst eventually_Abs_filter, rule is_filter.intro, auto) lemma eventually_sup: "eventually P (sup F F') ⟷ eventually P F ∧ eventually P F'" unfolding sup_filter_def by (rule eventually_Abs_filter, rule is_filter.intro) (auto elim!: eventually_rev_mp) lemma eventually_inf: "eventually P (inf F F') ⟷ (∃Q R. eventually Q F ∧ eventually R F' ∧ (∀x. Q x ∧ R x ⟶ P x))" unfolding inf_filter_def apply (rule eventually_Abs_filter [OF is_filter.intro]) apply (blast intro: eventually_True) apply (force elim!: eventually_conj)+ done lemma eventually_Sup: "eventually P (Sup S) ⟷ (∀F∈S. eventually P F)" unfolding Sup_filter_def apply (rule eventually_Abs_filter [OF is_filter.intro]) apply (auto intro: eventually_conj elim!: eventually_rev_mp) done instance proof fix F F' F'' :: "'a filter" and S :: "'a filter set" { show "F < F' ⟷ F ≤ F' ∧ ¬ F' ≤ F" by (rule less_filter_def) } { show "F ≤ F" unfolding le_filter_def by simp } { assume "F ≤ F'" and "F' ≤ F''" thus "F ≤ F''" unfolding le_filter_def by simp } { assume "F ≤ F'" and "F' ≤ F" thus "F = F'" unfolding le_filter_def filter_eq_iff by fast } { show "inf F F' ≤ F" and "inf F F' ≤ F'" unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } { assume "F ≤ F'" and "F ≤ F''" thus "F ≤ inf F' F''" unfolding le_filter_def eventually_inf by (auto intro: eventually_mono [OF eventually_conj]) } { show "F ≤ sup F F'" and "F' ≤ sup F F'" unfolding le_filter_def eventually_sup by simp_all } { assume "F ≤ F''" and "F' ≤ F''" thus "sup F F' ≤ F''" unfolding le_filter_def eventually_sup by simp } { assume "F'' ∈ S" thus "Inf S ≤ F''" unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } { assume "⋀F'. F' ∈ S ⟹ F ≤ F'" thus "F ≤ Inf S" unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } { assume "F ∈ S" thus "F ≤ Sup S" unfolding le_filter_def eventually_Sup by simp } { assume "⋀F. F ∈ S ⟹ F ≤ F'" thus "Sup S ≤ F'" unfolding le_filter_def eventually_Sup by simp } { show "Inf {} = (top::'a filter)" by (auto simp: top_filter_def Inf_filter_def Sup_filter_def) (metis (full_types) top_filter_def always_eventually eventually_top) } { show "Sup {} = (bot::'a filter)" by (auto simp: bot_filter_def Sup_filter_def) } qed end instance filter :: (type) distrib_lattice proof fix F G H :: "'a filter" show "sup F (inf G H) = inf (sup F G) (sup F H)" proof (rule order.antisym) show "inf (sup F G) (sup F H) ≤ sup F (inf G H)" unfolding le_filter_def eventually_sup proof safe fix P assume 1: "eventually P F" and 2: "eventually P (inf G H)" from 2 obtain Q R where QR: "eventually Q G" "eventually R H" "⋀x. Q x ⟹ R x ⟹ P x" by (auto simp: eventually_inf) define Q' where "Q' = (λx. Q x ∨ P x)" define R' where "R' = (λx. R x ∨ P x)" from 1 have "eventually Q' F" by (elim eventually_mono) (auto simp: Q'_def) moreover from 1 have "eventually R' F" by (elim eventually_mono) (auto simp: R'_def) moreover from QR(1) have "eventually Q' G" by (elim eventually_mono) (auto simp: Q'_def) moreover from QR(2) have "eventually R' H" by (elim eventually_mono)(auto simp: R'_def) moreover from QR have "P x" if "Q' x" "R' x" for x using that by (auto simp: Q'_def R'_def) ultimately show "eventually P (inf (sup F G) (sup F H))" by (auto simp: eventually_inf eventually_sup) qed qed (auto intro: inf.coboundedI1 inf.coboundedI2) qed lemma filter_leD: "F ≤ F' ⟹ eventually P F' ⟹ eventually P F" unfolding le_filter_def by simp lemma filter_leI: "(⋀P. eventually P F' ⟹ eventually P F) ⟹ F ≤ F'" unfolding le_filter_def by simp lemma eventually_False: "eventually (λx. False) F ⟷ F = bot" unfolding filter_eq_iff by (auto elim: eventually_rev_mp) lemma eventually_frequently: "F ≠ bot ⟹ eventually P F ⟹ frequently P F" using eventually_conj[of P F "λx. ¬ P x"] by (auto simp add: frequently_def eventually_False) lemma eventually_frequentlyE: assumes "eventually P F" assumes "eventually (λx. ¬ P x ∨ Q x) F" "F≠bot" shows "frequently Q F" proof - have "eventually Q F" using eventually_conj[OF assms(1,2),simplified] by (auto elim:eventually_mono) then show ?thesis using eventually_frequently[OF ‹F≠bot›] by auto qed lemma eventually_const_iff: "eventually (λx. P) F ⟷ P ∨ F = bot" by (cases P) (auto simp: eventually_False) lemma eventually_const[simp]: "F ≠ bot ⟹ eventually (λx. P) F ⟷ P" by (simp add: eventually_const_iff) lemma frequently_const_iff: "frequently (λx. P) F ⟷ P ∧ F ≠ bot" by (simp add: frequently_def eventually_const_iff) lemma frequently_const[simp]: "F ≠ bot ⟹ frequently (λx. P) F ⟷ P" by (simp add: frequently_const_iff) lemma eventually_happens: "eventually P net ⟹ net = bot ∨ (∃x. P x)" by (metis frequentlyE eventually_frequently) lemma eventually_happens': assumes "F ≠ bot" "eventually P F" shows "∃x. P x" using assms eventually_frequently frequentlyE by blast abbreviation (input) trivial_limit :: "'a filter ⇒ bool" where "trivial_limit F ≡ F = bot" lemma trivial_limit_def: "trivial_limit F ⟷ eventually (λx. False) F" by (rule eventually_False [symmetric]) lemma False_imp_not_eventually: "(∀x. ¬ P x ) ⟹ ¬ trivial_limit net ⟹ ¬ eventually (λx. P x) net" by (simp add: eventually_False) lemma trivial_limit_eventually: "trivial_limit net ⟹ eventually P net" by simp lemma trivial_limit_eq: "trivial_limit net ⟷ (∀P. eventually P net)" by (simp add: filter_eq_iff) lemma eventually_Inf: "eventually P (Inf B) ⟷ (∃X⊆B. finite X ∧ eventually P (Inf X))" proof - let ?F = "λP. ∃X⊆B. finite X ∧ eventually P (Inf X)" have eventually_F: "eventually P (Abs_filter ?F) ⟷ ?F P" for P proof (rule eventually_Abs_filter is_filter.intro)+ show "?F (λx. True)" by (rule exI[of _ "{}"]) (simp add: le_fun_def) next fix P Q assume "?F P" "?F Q" then obtain X Y where "X ⊆ B" "finite X" "eventually P (⨅ X)" "Y ⊆ B" "finite Y" "eventually Q (⨅ Y)" by blast then show "?F (λx. P x ∧ Q x)" by (intro exI[of _ "X ∪ Y"]) (auto simp: Inf_union_distrib eventually_inf) next fix P Q assume "?F P" then obtain X where "X ⊆ B" "finite X" "eventually P (⨅ X)" by blast moreover assume "∀x. P x ⟶ Q x" ultimately show "?F Q" by (intro exI[of _ X]) (auto elim: eventually_mono) qed have "Inf B = Abs_filter ?F" proof (intro antisym Inf_greatest) show "Inf B ≤ Abs_filter ?F" by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono) next fix F assume "F ∈ B" then show "Abs_filter ?F ≤ F" by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"]) qed then show ?thesis by (simp add: eventually_F) qed lemma eventually_INF: "eventually P (⨅b∈B. F b) ⟷ (∃X⊆B. finite X ∧ eventually P (⨅b∈X. F b))" unfolding eventually_Inf [of P "F`B"] by (metis finite_imageI image_mono finite_subset_image) lemma Inf_filter_not_bot: fixes B :: "'a filter set" shows "(⋀X. X ⊆ B ⟹ finite X ⟹ Inf X ≠ bot) ⟹ Inf B ≠ bot" unfolding trivial_limit_def eventually_Inf[of _ B] bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp lemma INF_filter_not_bot: fixes F :: "'i ⇒ 'a filter" shows "(⋀X. X ⊆ B ⟹ finite X ⟹ (⨅b∈X. F b) ≠ bot) ⟹ (⨅b∈B. F b) ≠ bot" unfolding trivial_limit_def eventually_INF [of _ _ B] bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp lemma eventually_Inf_base: assumes "B ≠ {}" and base: "⋀F G. F ∈ B ⟹ G ∈ B ⟹ ∃x∈B. x ≤ inf F G" shows "eventually P (Inf B) ⟷ (∃b∈B. eventually P b)" proof (subst eventually_Inf, safe) fix X assume "finite X" "X ⊆ B" then have "∃b∈B. ∀x∈X. b ≤ x" proof induct case empty then show ?case using ‹B ≠ {}› by auto next case (insert x X) then obtain b where "b ∈ B" "⋀x. x ∈ X ⟹ b ≤ x" by auto with ‹insert x X ⊆ B› base[of b x] show ?case by (auto intro: order_trans) qed then obtain b where "b ∈ B" "b ≤ Inf X" by (auto simp: le_Inf_iff) then show "eventually P (Inf X) ⟹ Bex B (eventually P)" by (intro bexI[of _ b]) (auto simp: le_filter_def) qed (auto intro!: exI[of _ "{x}" for x]) lemma eventually_INF_base: "B ≠ {} ⟹ (⋀a b. a ∈ B ⟹ b ∈ B ⟹ ∃x∈B. F x ≤ inf (F a) (F b)) ⟹ eventually P (⨅b∈B. F b) ⟷ (∃b∈B. eventually P (F b))" by (subst eventually_Inf_base) auto lemma eventually_INF1: "i ∈ I ⟹ eventually P (F i) ⟹ eventually P (⨅i∈I. F i)" using filter_leD[OF INF_lower] . lemma eventually_INF_finite: assumes "finite A" shows "eventually P (⨅ x∈A. F x) ⟷ (∃Q. (∀x∈A. eventually (Q x) (F x)) ∧ (∀y. (∀x∈A. Q x y) ⟶ P y))" using assms proof (induction arbitrary: P rule: finite_induct) case (insert a A P) from insert.hyps have [simp]: "x ≠ a" if "x ∈ A" for x using that by auto have "eventually P (⨅ x∈insert a A. F x) ⟷ (∃Q R S. eventually Q (F a) ∧ (( (∀x∈A. eventually (S x) (F x)) ∧ (∀y. (∀x∈A. S x y) ⟶ R y)) ∧ (∀x. Q x ∧ R x ⟶ P x)))" unfolding ex_simps by (simp add: eventually_inf insert.IH) also have "… ⟷ (∃Q. (∀x∈insert a A. eventually (Q x) (F x)) ∧ (∀y. (∀x∈insert a A. Q x y) ⟶ P y))" proof (safe, goal_cases) case (1 Q R S) thus ?case using 1 by (intro exI[of _ "S(a := Q)"]) auto next case (2 Q) show ?case by (rule exI[of _ "Q a"], rule exI[of _ "λy. ∀x∈A. Q x y"], rule exI[of _ "Q(a := (λ_. True))"]) (use 2 in auto) qed finally show ?case . qed auto lemma eventually_le_le: fixes P :: "'a ⇒ ('b :: preorder)" assumes "eventually (λx. P x ≤ Q x) F" assumes "eventually (λx. Q x ≤ R x) F" shows "eventually (λx. P x ≤ R x) F" using assms by eventually_elim (rule order_trans) subsubsection ‹Map function for filters› definition filtermap :: "('a ⇒ 'b) ⇒ 'a filter ⇒ 'b filter" where "filtermap f F = Abs_filter (λP. eventually (λx. P (f x)) F)" lemma eventually_filtermap: "eventually P (filtermap f F) = eventually (λx. P (f x)) F" unfolding filtermap_def apply (rule eventually_Abs_filter [OF is_filter.intro]) apply (auto elim!: eventually_rev_mp) done lemma eventually_comp_filtermap: "eventually (P ∘ f) F ⟷ eventually P (filtermap f F)" unfolding comp_def using eventually_filtermap by auto lemma filtermap_compose: "filtermap (f ∘ g) F = filtermap f (filtermap g F)" unfolding filter_eq_iff by (simp add: eventually_filtermap) lemma filtermap_ident: "filtermap (λx. x) F = F" by (simp add: filter_eq_iff eventually_filtermap) lemma filtermap_filtermap: "filtermap f (filtermap g F) = filtermap (λx. f (g x)) F" by (simp add: filter_eq_iff eventually_filtermap) lemma filtermap_mono: "F ≤ F' ⟹ filtermap f F ≤ filtermap f F'" unfolding le_filter_def eventually_filtermap by simp lemma filtermap_bot [simp]: "filtermap f bot = bot" by (simp add: filter_eq_iff eventually_filtermap) lemma filtermap_bot_iff: "filtermap f F = bot ⟷ F = bot" by (simp add: trivial_limit_def eventually_filtermap) lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" by (simp add: filter_eq_iff eventually_filtermap eventually_sup) lemma filtermap_SUP: "filtermap f (⨆b∈B. F b) = (⨆b∈B. filtermap f (F b))" by (simp add: filter_eq_iff eventually_Sup eventually_filtermap) lemma filtermap_inf: "filtermap f (inf F1 F2) ≤ inf (filtermap f F1) (filtermap f F2)" by (intro inf_greatest filtermap_mono inf_sup_ord) lemma filtermap_INF: "filtermap f (⨅b∈B. F b) ≤ (⨅b∈B. filtermap f (F b))" by (rule INF_greatest, rule filtermap_mono, erule INF_lower) lemma frequently_filtermap: "frequently P (filtermap f F) = frequently (λx. P (f x)) F" by (simp add: frequently_def eventually_filtermap) subsubsection ‹Contravariant map function for filters› definition filtercomap :: "('a ⇒ 'b) ⇒ 'b filter ⇒ 'a filter" where "filtercomap f F = Abs_filter (λP. ∃Q. eventually Q F ∧ (∀x. Q (f x) ⟶ P x))" lemma eventually_filtercomap: "eventually P (filtercomap f F) ⟷ (∃Q. eventually Q F ∧ (∀x. Q (f x) ⟶ P x))" unfolding filtercomap_def proof (intro eventually_Abs_filter, unfold_locales, goal_cases) case 1 show ?case by (auto intro!: exI[of _ "λ_. True"]) next case (2 P Q) then obtain P' Q' where P'Q': "eventually P' F" "∀x. P' (f x) ⟶ P x" "eventually Q' F" "∀x. Q' (f x) ⟶ Q x" by (elim exE conjE) show ?case by (rule exI[of _ "λx. P' x ∧ Q' x"]) (use P'Q' in ‹auto intro!: eventually_conj›) next case (3 P Q) thus ?case by blast qed lemma filtercomap_ident: "filtercomap (λx. x) F = F" by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono) lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (λx. g (f x)) F" unfolding filter_eq_iff by (auto simp: eventually_filtercomap) lemma filtercomap_mono: "F ≤ F' ⟹ filtercomap f F ≤ filtercomap f F'" by (auto simp: eventually_filtercomap le_filter_def) lemma filtercomap_bot [simp]: "filtercomap f bot = bot" by (auto simp: filter_eq_iff eventually_filtercomap) lemma filtercomap_top [simp]: "filtercomap f top = top" by (auto simp: filter_eq_iff eventually_filtercomap) lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)" unfolding filter_eq_iff proof safe fix P assume "eventually P (filtercomap f (F1 ⊓ F2))" then obtain Q R S where *: "eventually Q F1" "eventually R F2" "⋀x. Q x ⟹ R x ⟹ S x" "⋀x. S (f x) ⟹ P x" unfolding eventually_filtercomap eventually_inf by blast from * have "eventually (λx. Q (f x)) (filtercomap f F1)" "eventually (λx. R (f x)) (filtercomap f F2)" by (auto simp: eventually_filtercomap) with * show "eventually P (filtercomap f F1 ⊓ filtercomap f F2)" unfolding eventually_inf by blast next fix P assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))" then obtain Q Q' R R' where *: "eventually Q F1" "eventually R F2" "⋀x. Q (f x) ⟹ Q' x" "⋀x. R (f x) ⟹ R' x" "⋀x. Q' x ⟹ R' x ⟹ P x" unfolding eventually_filtercomap eventually_inf by blast from * have "eventually (λx. Q x ∧ R x) (F1 ⊓ F2)" by (auto simp: eventually_inf) with * show "eventually P (filtercomap f (F1 ⊓ F2))" by (auto simp: eventually_filtercomap) qed lemma filtercomap_sup: "filtercomap f (sup F1 F2) ≥ sup (filtercomap f F1) (filtercomap f F2)" by (intro sup_least filtercomap_mono inf_sup_ord) lemma filtercomap_INF: "filtercomap f (⨅b∈B. F b) = (⨅b∈B. filtercomap f (F b))" proof - have *: "filtercomap f (⨅b∈B. F b) = (⨅b∈B. filtercomap f (F b))" if "finite B" for B using that by induction (simp_all add: filtercomap_inf) show ?thesis unfolding filter_eq_iff proof fix P have "eventually P (⨅b∈B. filtercomap f (F b)) ⟷ (∃X. (X ⊆ B ∧ finite X) ∧ eventually P (⨅b∈X. filtercomap f (F b)))" by (subst eventually_INF) blast also have "… ⟷ (∃X. (X ⊆ B ∧ finite X) ∧ eventually P (filtercomap f (⨅b∈X. F b)))" by (rule ex_cong) (simp add: *) also have "… ⟷ eventually P (filtercomap f (⨅(F ` B)))" unfolding eventually_filtercomap by (subst eventually_INF) blast finally show "eventually P (filtercomap f (⨅(F ` B))) = eventually P (⨅b∈B. filtercomap f (F b))" .. qed qed lemma filtercomap_SUP: "filtercomap f (⨆b∈B. F b) ≥ (⨆b∈B. filtercomap f (F b))" by (intro SUP_least filtercomap_mono SUP_upper) lemma filtermap_le_iff_le_filtercomap: "filtermap f F ≤ G ⟷ F ≤ filtercomap f G" unfolding le_filter_def eventually_filtermap eventually_filtercomap using eventually_mono by auto lemma filtercomap_neq_bot: assumes "⋀P. eventually P F ⟹ ∃x. P (f x)" shows "filtercomap f F ≠ bot" using assms by (auto simp: trivial_limit_def eventually_filtercomap) lemma filtercomap_neq_bot_surj: assumes "F ≠ bot" and "surj f" shows "filtercomap f F ≠ bot" proof (rule filtercomap_neq_bot) fix P assume *: "eventually P F" show "∃x. P (f x)" proof (rule ccontr) assume **: "¬(∃x. P (f x))" from * have "eventually (λ_. False) F" proof eventually_elim case (elim x) from ‹surj f› obtain y where "x = f y" by auto with elim and ** show False by auto qed with assms show False by (simp add: trivial_limit_def) qed qed lemma eventually_filtercomapI [intro]: assumes "eventually P F" shows "eventually (λx. P (f x)) (filtercomap f F)" using assms by (auto simp: eventually_filtercomap) lemma filtermap_filtercomap: "filtermap f (filtercomap f F) ≤ F" by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap) lemma filtercomap_filtermap: "filtercomap f (filtermap f F) ≥ F" unfolding le_filter_def eventually_filtermap eventually_filtercomap by (auto elim!: eventually_mono) subsubsection ‹Standard filters› definition principal :: "'a set ⇒ 'a filter" where "principal S = Abs_filter (λP. ∀x∈S. P x)" lemma eventually_principal: "eventually P (principal S) ⟷ (∀x∈S. P x)" unfolding principal_def by (rule eventually_Abs_filter, rule is_filter.intro) auto lemma eventually_inf_principal: "eventually P (inf F (principal s)) ⟷ eventually (λx. x ∈ s ⟶ P x) F" unfolding eventually_inf eventually_principal by (auto elim: eventually_mono) lemma principal_UNIV[simp]: "principal UNIV = top" by (auto simp: filter_eq_iff eventually_principal) lemma principal_empty[simp]: "principal {} = bot" by (auto simp: filter_eq_iff eventually_principal) lemma principal_eq_bot_iff: "principal X = bot ⟷ X = {}" by (auto simp add: filter_eq_iff eventually_principal) lemma principal_le_iff[iff]: "principal A ≤ principal B ⟷ A ⊆ B" by (auto simp: le_filter_def eventually_principal) lemma le_principal: "F ≤ principal A ⟷ eventually (λx. x ∈ A) F" unfolding le_filter_def eventually_principal by (force elim: eventually_mono) lemma principal_inject[iff]: "principal A = principal B ⟷ A = B" unfolding eq_iff by simp lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A ∪ B)" unfolding filter_eq_iff eventually_sup eventually_principal by auto lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A ∩ B)" unfolding filter_eq_iff eventually_inf eventually_principal by (auto intro: exI[of _ "λx. x ∈ A"] exI[of _ "λx. x ∈ B"]) lemma SUP_principal[simp]: "(⨆i∈I. principal (A i)) = principal (⋃i∈I. A i)" unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal) lemma INF_principal_finite: "finite X ⟹ (⨅x∈X. principal (f x)) = principal (⋂x∈X. f x)" by (induct X rule: finite_induct) auto lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" unfolding filter_eq_iff eventually_filtermap eventually_principal by simp lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)" unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast subsubsection ‹Order filters› definition at_top :: "('a::order) filter" where "at_top = (⨅k. principal {k ..})" lemma at_top_sub: "at_top = (⨅k∈{c::'a::linorder..}. principal {k ..})" by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def) lemma eventually_at_top_linorder: "eventually P at_top ⟷ (∃N::'a::linorder. ∀n≥N. P n)" unfolding at_top_def by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) lemma eventually_filtercomap_at_top_linorder: "eventually P (filtercomap f at_top) ⟷ (∃N::'a::linorder. ∀x. f x ≥ N ⟶ P x)" by (auto simp: eventually_filtercomap eventually_at_top_linorder) lemma eventually_at_top_linorderI: fixes c::"'a::linorder" assumes "⋀x. c ≤ x ⟹ P x" shows "eventually P at_top" using assms by (auto simp: eventually_at_top_linorder) lemma eventually_ge_at_top [simp]: "eventually (λx. (c::_::linorder) ≤ x) at_top" unfolding eventually_at_top_linorder by auto lemma eventually_at_top_dense: "eventually P at_top ⟷ (∃N::'a::{no_top, linorder}. ∀n>N. P n)" proof - have "eventually P (⨅k. principal {k <..}) ⟷ (∃N::'a. ∀n>N. P n)" by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) also have "(⨅k. principal {k::'a <..}) = at_top" unfolding at_top_def by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex) finally show ?thesis . qed lemma eventually_filtercomap_at_top_dense: "eventually P (filtercomap f at_top) ⟷ (∃N::'a::{no_top, linorder}. ∀x. f x > N ⟶ P x)" by (auto simp: eventually_filtercomap eventually_at_top_dense) lemma eventually_at_top_not_equal [simp]: "eventually (λx::'a::{no_top, linorder}. x ≠ c) at_top" unfolding eventually_at_top_dense by auto lemma eventually_gt_at_top [simp]: "eventually (λx. (c::_::{no_top, linorder}) < x) at_top" unfolding eventually_at_top_dense by auto lemma eventually_all_ge_at_top: assumes "eventually P (at_top :: ('a :: linorder) filter)" shows "eventually (λx. ∀y≥x. P y) at_top" proof - from assms obtain x where "⋀y. y ≥ x ⟹ P y" by (auto simp: eventually_at_top_linorder) hence "∀z≥y. P z" if "y ≥ x" for y using that by simp thus ?thesis by (auto simp: eventually_at_top_linorder) qed definition at_bot :: "('a::order) filter" where "at_bot = (⨅k. principal {.. k})" lemma at_bot_sub: "at_bot = (⨅k∈{.. c::'a::linorder}. principal {.. k})" by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def) lemma eventually_at_bot_linorder: fixes P :: "'a::linorder ⇒ bool" shows "eventually P at_bot ⟷ (∃N. ∀n≤N. P n)" unfolding at_bot_def by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) lemma eventually_filtercomap_at_bot_linorder: "eventually P (filtercomap f at_bot) ⟷ (∃N::'a::linorder. ∀x. f x ≤ N ⟶ P x)" by (auto simp: eventually_filtercomap eventually_at_bot_linorder) lemma eventually_le_at_bot [simp]: "eventually (λx. x ≤ (c::_::linorder)) at_bot" unfolding eventually_at_bot_linorder by auto lemma eventually_at_bot_dense: "eventually P at_bot ⟷ (∃N::'a::{no_bot, linorder}. ∀n<N. P n)" proof - have "eventually P (⨅k. principal {..< k}) ⟷ (∃N::'a. ∀n<N. P n)" by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) also have "(⨅k. principal {..< k::'a}) = at_bot" unfolding at_bot_def by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex) finally show ?thesis . qed lemma eventually_filtercomap_at_bot_dense: "eventually P (filtercomap f at_bot) ⟷ (∃N::'a::{no_bot, linorder}. ∀x. f x < N ⟶ P x)" by (auto simp: eventually_filtercomap eventually_at_bot_dense) lemma eventually_at_bot_not_equal [simp]: "eventually (λx::'a::{no_bot, linorder}. x ≠ c) at_bot" unfolding eventually_at_bot_dense by auto lemma eventually_gt_at_bot [simp]: "eventually (λx. x < (c::_::unbounded_dense_linorder)) at_bot" unfolding eventually_at_bot_dense by auto lemma trivial_limit_at_bot_linorder [simp]: "¬ trivial_limit (at_bot ::('a::linorder) filter)" unfolding trivial_limit_def by (metis eventually_at_bot_linorder order_refl) lemma trivial_limit_at_top_linorder [simp]: "¬ trivial_limit (at_top ::('a::linorder) filter)" unfolding trivial_limit_def by (metis eventually_at_top_linorder order_refl) subsection ‹Sequentially› abbreviation sequentially :: "nat filter" where "sequentially ≡ at_top" lemma eventually_sequentially: "eventually P sequentially ⟷ (∃N. ∀n≥N. P n)" by (rule eventually_at_top_linorder) lemma sequentially_bot [simp, intro]: "sequentially ≠ bot" unfolding filter_eq_iff eventually_sequentially by auto lemmas trivial_limit_sequentially = sequentially_bot lemma eventually_False_sequentially [simp]: "¬ eventually (λn. False) sequentially" by (simp add: eventually_False) lemma le_sequentially: "F ≤ sequentially ⟷ (∀N. eventually (λn. N ≤ n) F)" by (simp add: at_top_def le_INF_iff le_principal) lemma eventually_sequentiallyI [intro?]: assumes "⋀x. c ≤ x ⟹ P x" shows "eventually P sequentially" using assms by (auto simp: eventually_sequentially) lemma eventually_sequentially_Suc [simp]: "eventually (λi. P (Suc i)) sequentially ⟷ eventually P sequentially" unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq) lemma eventually_sequentially_seg [simp]: "eventually (λn. P (n + k)) sequentially ⟷ eventually P sequentially" using eventually_sequentially_Suc[of "λn. P (n + k)" for k] by (induction k) auto lemma filtermap_sequentually_ne_bot: "filtermap f sequentially ≠ bot" by (simp add: filtermap_bot_iff) subsection ‹Increasing finite subsets› definition finite_subsets_at_top where "finite_subsets_at_top A = (⨅ X∈{X. finite X ∧ X ⊆ A}. principal {Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A})" lemma eventually_finite_subsets_at_top: "eventually P (finite_subsets_at_top A) ⟷ (∃X. finite X ∧ X ⊆ A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ P Y))" unfolding finite_subsets_at_top_def proof (subst eventually_INF_base, goal_cases) show "{X. finite X ∧ X ⊆ A} ≠ {}" by auto next case (2 B C) thus ?case by (intro bexI[of _ "B ∪ C"]) auto qed (simp_all add: eventually_principal) lemma eventually_finite_subsets_at_top_weakI [intro]: assumes "⋀X. finite X ⟹ X ⊆ A ⟹ P X" shows "eventually P (finite_subsets_at_top A)" proof - have "eventually (λX. finite X ∧ X ⊆ A) (finite_subsets_at_top A)" by (auto simp: eventually_finite_subsets_at_top) thus ?thesis by eventually_elim (use assms in auto) qed lemma finite_subsets_at_top_neq_bot [simp]: "finite_subsets_at_top A ≠ bot" proof - have "¬eventually (λx. False) (finite_subsets_at_top A)" by (auto simp: eventually_finite_subsets_at_top) thus ?thesis by auto qed lemma filtermap_image_finite_subsets_at_top: assumes "inj_on f A" shows "filtermap ((`) f) (finite_subsets_at_top A) = finite_subsets_at_top (f ` A)" unfolding filter_eq_iff eventually_filtermap proof (safe, goal_cases) case (1 P) then obtain X where X: "finite X" "X ⊆ A" "⋀Y. finite Y ⟹ X ⊆ Y ⟹ Y ⊆ A ⟹ P (f ` Y)" unfolding eventually_finite_subsets_at_top by force show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap proof (rule exI[of _ "f ` X"], intro conjI allI impI, goal_cases) case (3 Y) with assms and X(1,2) have "P (f ` (f -` Y ∩ A))" using X(1,2) by (intro X(3) finite_vimage_IntI) auto also have "f ` (f -` Y ∩ A) = Y" using assms 3 by blast finally show ?case . qed (insert assms X(1,2), auto intro!: finite_vimage_IntI) next case (2 P) then obtain X where X: "finite X" "X ⊆ f ` A" "⋀Y. finite Y ⟹ X ⊆ Y ⟹ Y ⊆ f ` A ⟹ P Y" unfolding eventually_finite_subsets_at_top by force show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap proof (rule exI[of _ "f -` X ∩ A"], intro conjI allI impI, goal_cases) case (3 Y) with X(1,2) and assms show ?case by (intro X(3)) force+ qed (insert assms X(1), auto intro!: finite_vimage_IntI) qed lemma eventually_finite_subsets_at_top_finite: assumes "finite A" shows "eventually P (finite_subsets_at_top A) ⟷ P A" unfolding eventually_finite_subsets_at_top using assms by force lemma finite_subsets_at_top_finite: "finite A ⟹ finite_subsets_at_top A = principal {A}" by (auto simp: filter_eq_iff eventually_finite_subsets_at_top_finite eventually_principal) subsection ‹The cofinite filter› definition "cofinite = Abs_filter (λP. finite {x. ¬ P x})" abbreviation Inf_many :: "('a ⇒ bool) ⇒ bool" (binder "∃⇩_{∞}" 10) where "Inf_many P ≡ frequently P cofinite" abbreviation Alm_all :: "('a ⇒ bool) ⇒ bool" (binder "∀⇩_{∞}" 10) where "Alm_all P ≡ eventually P cofinite" notation (ASCII) Inf_many (binder "INFM " 10) and Alm_all (binder "MOST " 10) lemma eventually_cofinite: "eventually P cofinite ⟷ finite {x. ¬ P x}" unfolding cofinite_def proof (rule eventually_Abs_filter, rule is_filter.intro) fix P Q :: "'a ⇒ bool" assume "finite {x. ¬ P x}" "finite {x. ¬ Q x}" from finite_UnI[OF this] show "finite {x. ¬ (P x ∧ Q x)}" by (rule rev_finite_subset) auto next fix P Q :: "'a ⇒ bool" assume P: "finite {x. ¬ P x}" and *: "∀x. P x ⟶ Q x" from * show "finite {x. ¬ Q x}" by (intro finite_subset[OF _ P]) auto qed simp lemma frequently_cofinite: "frequently P cofinite ⟷ ¬ finite {x. P x}" by (simp add: frequently_def eventually_cofinite) lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) ⟷ finite (UNIV :: 'a set)" unfolding trivial_limit_def eventually_cofinite by simp lemma cofinite_eq_sequentially: "cofinite = sequentially" unfolding filter_eq_iff eventually_sequentially eventually_cofinite proof safe fix P :: "nat ⇒ bool" assume [simp]: "finite {x. ¬ P x}" show "∃N. ∀n≥N. P n" proof cases assume "{x. ¬ P x} ≠ {}" then show ?thesis by (intro exI[of _ "Suc (Max {x. ¬ P x})"]) (auto simp: Suc_le_eq) qed auto next fix P :: "nat ⇒ bool" and N :: nat assume "∀n≥N. P n" then have "{x. ¬ P x} ⊆ {..< N}" by (auto simp: not_le) then show "finite {x. ¬ P x}" by (blast intro: finite_subset) qed subsubsection ‹Product of filters› definition prod_filter :: "'a filter ⇒ 'b filter ⇒ ('a × 'b) filter" (infixr "×⇩_{F}" 80) where "prod_filter F G = (⨅(P, Q)∈{(P, Q). eventually P F ∧ eventually Q G}. principal {(x, y). P x ∧ Q y})" lemma eventually_prod_filter: "eventually P (F ×⇩_{F}G) ⟷ (∃Pf Pg. eventually Pf F ∧ eventually Pg G ∧ (∀x y. Pf x ⟶ Pg y ⟶ P (x, y)))" unfolding prod_filter_def proof (subst eventually_INF_base, goal_cases) case 2 moreover have "eventually Pf F ⟹ eventually Qf F ⟹ eventually Pg G ⟹ eventually Qg G ⟹ ∃P Q. eventually P F ∧ eventually Q G ∧ Collect P × Collect Q ⊆ Collect Pf × Collect Pg ∩ Collect Qf × Collect Qg" for Pf Pg Qf Qg by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"]) (auto simp: inf_fun_def eventually_conj) ultimately show ?case by auto qed (auto simp: eventually_principal intro: eventually_True) lemma eventually_prod1: assumes "B ≠ bot" shows "(∀⇩_{F}(x, y) in A ×⇩_{F}B. P x) ⟷ (∀⇩_{F}x in A. P x)" unfolding eventually_prod_filter proof safe fix R Q assume *: "∀⇩_{F}x in A. R x" "∀⇩_{F}x in B. Q x" "∀x y. R x ⟶ Q y ⟶ P x" with ‹B ≠ bot› obtain y where "Q y" by (auto dest: eventually_happens) with * show "eventually P A" by (force elim: eventually_mono) next assume "eventually P A" then show "∃Pf Pg. eventually Pf A ∧ eventually Pg B ∧ (∀x y. Pf x ⟶ Pg y ⟶ P x)" by (intro exI[of _ P] exI[of _ "λx. True"]) auto qed lemma eventually_prod2: assumes "A ≠ bot" shows "(∀⇩_{F}(x, y) in A ×⇩_{F}B. P y) ⟷ (∀⇩_{F}y in B. P y)" unfolding eventually_prod_filter proof safe fix R Q assume *: "∀⇩_{F}x in A. R x" "∀⇩_{F}x in B. Q x" "∀x y. R x ⟶ Q y ⟶ P y" with ‹A ≠ bot› obtain x where "R x" by (auto dest: eventually_happens) with * show "eventually P B" by (force elim: eventually_mono) next assume "eventually P B" then show "∃Pf Pg. eventually Pf A ∧ eventually Pg B ∧ (∀x y. Pf x ⟶ Pg y ⟶ P y)" by (intro exI[of _ P] exI[of _ "λx. True"]) auto qed lemma INF_filter_bot_base: fixes F :: "'a ⇒ 'b filter" assumes *: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ ∃k∈I. F k ≤ F i ⊓ F j" shows "(⨅i∈I. F i) = bot ⟷ (∃i∈I. F i = bot)" proof (cases "∃i∈I. F i = bot") case True then have "(⨅i∈I. F i) ≤ bot" by (auto intro: INF_lower2) with True show ?thesis by (auto simp: bot_unique) next case False moreover have "(⨅i∈I. F i) ≠ bot" proof (cases "I = {}") case True then show ?thesis by (auto simp add: filter_eq_iff) next case False': False show ?thesis proof (rule INF_filter_not_bot) fix J assume "finite J" "J ⊆ I" then have "∃k∈I. F k ≤ (⨅i∈J. F i)" proof (induct J) case empty then show ?case using ‹I ≠ {}› by auto next case (insert i J) then obtain k where "k ∈ I" "F k ≤ (⨅i∈J. F i)" by auto with insert *[of i k] show ?case by auto qed with False show "(⨅i∈J. F i) ≠ ⊥" by (auto simp: bot_unique) qed qed ultimately show ?thesis by auto qed lemma Collect_empty_eq_bot: "Collect P = {} ⟷ P = ⊥" by auto lemma prod_filter_eq_bot: "A ×⇩_{F}B = bot ⟷ A = bot ∨ B = bot" unfolding trivial_limit_def proof assume "∀⇩_{F}x in A ×⇩_{F}B. False" then obtain Pf Pg where Pf: "eventually (λx. Pf x) A" and Pg: "eventually (λy. Pg y) B" and *: "∀x y. Pf x ⟶ Pg y ⟶ False" unfolding eventually_prod_filter by fast from * have "(∀x. ¬ Pf x) ∨ (∀y. ¬ Pg y)" by fast with Pf Pg show "(∀⇩_{F}x in A. False) ∨ (∀⇩_{F}x in B. False)" by auto next assume "(∀⇩_{F}x in A. False) ∨ (∀⇩_{F}x in B. False)" then show "∀⇩_{F}x in A ×⇩_{F}B. False" unfolding eventually_prod_filter by (force intro: eventually_True) qed lemma prod_filter_mono: "F ≤ F' ⟹ G ≤ G' ⟹ F ×⇩_{F}G ≤ F' ×⇩_{F}G'" by (auto simp: le_filter_def eventually_prod_filter) lemma prod_filter_mono_iff: assumes nAB: "A ≠ bot" "B ≠ bot" shows "A ×⇩_{F}B ≤ C ×⇩_{F}D ⟷ A ≤ C ∧ B ≤ D" proof safe assume *: "A ×⇩_{F}B ≤ C ×⇩_{F}D" with assms have "A ×⇩_{F}B ≠ bot" by (auto simp: bot_unique prod_filter_eq_bot) with * have "C ×⇩_{F}D ≠ bot" by (auto simp: bot_unique) then have nCD: "C ≠ bot" "D ≠ bot" by (auto simp: prod_filter_eq_bot) show "A ≤ C" proof (rule filter_leI) fix P assume "eventually P C" with *[THEN filter_leD, of "λ(x, y). P x"] show "eventually P A" using nAB nCD by (simp add: eventually_prod1 eventually_prod2) qed show "B ≤ D" proof (rule filter_leI) fix P assume "eventually P D" with *[THEN filter_leD, of "λ(x, y). P y"] show "eventually P B" using nAB nCD by (simp add: eventually_prod1 eventually_prod2) qed qed (intro prod_filter_mono) lemma eventually_prod_same: "eventually P (F ×⇩_{F}F) ⟷ (∃Q. eventually Q F ∧ (∀x y. Q x ⟶ Q y ⟶ P (x, y)))" unfolding eventually_prod_filter by (blast intro!: eventually_conj) lemma eventually_prod_sequentially: "eventually P (sequentially ×⇩_{F}sequentially) ⟷ (∃N. ∀m ≥ N. ∀n ≥ N. P (n, m))" unfolding eventually_prod_same eventually_sequentially by auto lemma principal_prod_principal: "principal A ×⇩_{F}principal B = principal (A × B)" unfolding filter_eq_iff eventually_prod_filter eventually_principal by (fast intro: exI[of _ "λx. x ∈ A"] exI[of _ "λx. x ∈ B"]) lemma le_prod_filterI: "filtermap fst F ≤ A ⟹ filtermap snd F ≤ B ⟹ F ≤ A ×⇩_{F}B" unfolding le_filter_def eventually_filtermap eventually_prod_filter by (force elim: eventually_elim2) lemma filtermap_fst_prod_filter: "filtermap fst (A ×⇩_{F}B) ≤ A" unfolding le_filter_def eventually_filtermap eventually_prod_filter by (force intro: eventually_True) lemma filtermap_snd_prod_filter: "filtermap snd (A ×⇩_{F}B) ≤ B" unfolding le_filter_def eventually_filtermap eventually_prod_filter by (force intro: eventually_True) lemma prod_filter_INF: assumes "I ≠ {}" and "J ≠ {}" shows "(⨅i∈I. A i) ×⇩_{F}(⨅j∈J. B j) = (⨅i∈I. ⨅j∈J. A i ×⇩_{F}B j)" proof (rule antisym) from ‹I ≠ {}› obtain i where "i ∈ I" by auto from ‹J ≠ {}› obtain j where "j ∈ J" by auto show "(⨅i∈I. ⨅j∈J. A i ×⇩_{F}B j) ≤ (⨅i∈I. A i) ×⇩_{F}(⨅j∈J. B j)" by (fast intro: le_prod_filterI INF_greatest INF_lower2 order_trans[OF filtermap_INF] ‹i ∈ I› ‹j ∈ J› filtermap_fst_prod_filter filtermap_snd_prod_filter) show "(⨅i∈I. A i) ×⇩_{F}(⨅j∈J. B j) ≤ (⨅i∈I. ⨅j∈J. A i ×⇩_{F}B j)" by (intro INF_greatest prod_filter_mono INF_lower) qed lemma filtermap_Pair: "filtermap (λx. (f x, g x)) F ≤ filtermap f F ×⇩_{F}filtermap g F" by (rule le_prod_filterI, simp_all add: filtermap_filtermap) lemma eventually_prodI: "eventually P F ⟹ eventually Q G ⟹ eventually (λx. P (fst x) ∧ Q (snd x)) (F ×⇩_{F}G)" unfolding eventually_prod_filter by auto lemma prod_filter_INF1: "I ≠ {} ⟹ (⨅i∈I. A i) ×⇩_{F}B = (⨅i∈I. A i ×⇩_{F}B)" using prod_filter_INF[of I "{B}" A "λx. x"] by simp lemma prod_filter_INF2: "J ≠ {} ⟹ A ×⇩_{F}(⨅i∈J. B i) = (⨅i∈J. A ×⇩_{F}B i)" using prod_filter_INF[of "{A}" J "λx. x" B] by simp lemma prod_filtermap1: "prod_filter (filtermap f F) G = filtermap (apfst f) (prod_filter F G)" unfolding filter_eq_iff eventually_filtermap eventually_prod_filter apply safe subgoal by auto subgoal for P Q R by(rule exI[where x="λy. ∃x. y = f x ∧ Q x"])(auto intro: eventually_mono) done lemma prod_filtermap2: "prod_filter F (filtermap g G) = filtermap (apsnd g) (prod_filter F G)" unfolding filter_eq_iff eventually_filtermap eventually_prod_filter apply safe subgoal by auto subgoal for P Q R by(auto intro: exI[where x="λy. ∃x. y = g x ∧ R x"] eventually_mono) done lemma prod_filter_assoc: "prod_filter (prod_filter F G) H = filtermap (λ(x, y, z). ((x, y), z)) (prod_filter F (prod_filter G H))" apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) subgoal for P Q R S T by(auto 4 4 intro: exI[where x="λ(a, b). T a ∧ S b"]) subgoal for P Q R S T by(auto 4 3 intro: exI[where x="λ(a, b). Q a ∧ S b"]) done lemma prod_filter_principal_singleton: "prod_filter (principal {x}) F = filtermap (Pair x) F" by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="λa. a = x"]) lemma prod_filter_principal_singleton2: "prod_filter F (principal {x}) = filtermap (λa. (a, x)) F" by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="λa. a = x"]) lemma prod_filter_commute: "prod_filter F G = filtermap prod.swap (prod_filter G F)" by(auto simp add: filter_eq_iff eventually_prod_filter eventually_filtermap) subsection ‹Limits› definition filterlim :: "('a ⇒ 'b) ⇒ 'b filter ⇒ 'a filter ⇒ bool" where "filterlim f F2 F1 ⟷ filtermap f F1 ≤ F2" syntax "_LIM" :: "pttrns ⇒ 'a ⇒ 'b ⇒ 'a ⇒ bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) translations "LIM x F1. f :> F2" == "CONST filterlim (λx. f) F2 F1" lemma filterlim_filtercomapI: "filterlim f F G ⟹ filterlim (λx. f (g x)) F (filtercomap g G)" unfolding filterlim_def by (metis order_trans filtermap_filtercomap filtermap_filtermap filtermap_mono) lemma filterlim_top [simp]: "filterlim f top F" by (simp add: filterlim_def) lemma filterlim_iff: "(LIM x F1. f x :> F2) ⟷ (∀P. eventually P F2 ⟶ eventually (λx. P (f x)) F1)" unfolding filterlim_def le_filter_def eventually_filtermap .. lemma filterlim_compose: "filterlim g F3 F2 ⟹ filterlim f F2 F1 ⟹ filterlim (λx. g (f x)) F3 F1" unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) lemma filterlim_mono: "filterlim f F2 F1 ⟹ F2 ≤ F2' ⟹ F1' ≤ F1 ⟹ filterlim f F2' F1'" unfolding filterlim_def by (metis filtermap_mono order_trans) lemma filterlim_ident: "LIM x F. x :> F" by (simp add: filterlim_def filtermap_ident) lemma filterlim_cong: "F1 = F1' ⟹ F2 = F2' ⟹ eventually (λx. f x = g x) F2 ⟹ filterlim f F1 F2 = filterlim g F1' F2'" by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) lemma filterlim_mono_eventually: assumes "filterlim f F G" and ord: "F ≤ F'" "G' ≤ G" assumes eq: "eventually (λx. f x = f' x) G'" shows "filterlim f' F' G'" proof - have "filterlim f F' G'" by (simp add: filterlim_mono[OF _ ord] assms) then show ?thesis by (rule filterlim_cong[OF refl refl eq, THEN iffD1]) qed lemma filtermap_mono_strong: "inj f ⟹ filtermap f F ≤ filtermap f G ⟷ F ≤ G" apply (safe intro!: filtermap_mono) apply (auto simp: le_filter_def eventually_filtermap) apply (erule_tac x="λx. P (inv f x)" in allE) apply auto done lemma eventually_compose_filterlim: assumes "eventually P F" "filterlim f F G" shows "eventually (λx. P (f x)) G" using assms by (simp add: filterlim_iff) lemma filtermap_eq_strong: "inj f ⟹ filtermap f F = filtermap f G ⟷ F = G" by (simp add: filtermap_mono_strong eq_iff) lemma filtermap_fun_inverse: assumes g: "filterlim g F G" assumes f: "filterlim f G F" assumes ev: "eventually (λx. f (g x) = x) G" shows "filtermap f F = G" proof (rule antisym) show "filtermap f F ≤ G" using f unfolding filterlim_def . have "G = filtermap f (filtermap g G)" using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap) also have "… ≤ filtermap f F" using g by (intro filtermap_mono) (simp add: filterlim_def) finally show "G ≤ filtermap f F" . qed lemma filterlim_principal: "(LIM x F. f x :> principal S) ⟷ (eventually (λx. f x ∈ S) F)" unfolding filterlim_def eventually_filtermap le_principal .. lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)" unfolding filterlim_def by (rule filtermap_filtercomap) lemma filterlim_inf: "(LIM x F1. f x :> inf F2 F3) ⟷ ((LIM x F1. f x :> F2) ∧ (LIM x F1. f x :> F3))" unfolding filterlim_def by simp lemma filterlim_INF: "(LIM x F. f x :> (⨅b∈B. G b)) ⟷ (∀b∈B. LIM x F. f x :> G b)" unfolding filterlim_def le_INF_iff .. lemma filterlim_INF_INF: "(⋀m. m ∈ J ⟹ ∃i∈I. filtermap f (F i) ≤ G m) ⟹ LIM x (⨅i∈I. F i). f x :> (⨅j∈J. G j)" unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono]) lemma filterlim_INF': "x ∈ A ⟹ filterlim f F (G x) ⟹ filterlim f F (⨅ x∈A. G x)" unfolding filterlim_def by (rule order.trans[OF filtermap_mono[OF INF_lower]]) lemma filterlim_filtercomap_iff: "filterlim f (filtercomap g G) F ⟷ filterlim (g ∘ f) G F" by (simp add: filterlim_def filtermap_le_iff_le_filtercomap filtercomap_filtercomap o_def) lemma filterlim_iff_le_filtercomap: "filterlim f F G ⟷ G ≤ filtercomap f F" by (simp add: filterlim_def filtermap_le_iff_le_filtercomap) lemma filterlim_base: "(⋀m x. m ∈ J ⟹ i m ∈ I) ⟹ (⋀m x. m ∈ J ⟹ x ∈ F (i m) ⟹ f x ∈ G m) ⟹ LIM x (⨅i∈I. principal (F i)). f x :> (⨅j∈J. principal (G j))" by (force intro!: filterlim_INF_INF simp: image_subset_iff) lemma filterlim_base_iff: assumes "I ≠ {}" and chain: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ F i ⊆ F j ∨ F j ⊆ F i" shows "(LIM x (⨅i∈I. principal (F i)). f x :> ⨅j∈J. principal (G j)) ⟷ (∀j∈J. ∃i∈I. ∀x∈F i. f x ∈ G j)" unfolding filterlim_INF filterlim_principal proof (subst eventually_INF_base) fix i j assume "i ∈ I" "j ∈ I" with chain[OF this] show "∃x∈I. principal (F x) ≤ inf (principal (F i)) (principal (F j))" by auto qed (auto simp: eventually_principal ‹I ≠ {}›) lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (λx. f (g x)) F1 F2" unfolding filterlim_def filtermap_filtermap .. lemma filterlim_sup: "filterlim f F F1 ⟹ filterlim f F F2 ⟹ filterlim f F (sup F1 F2)" unfolding filterlim_def filtermap_sup by auto lemma filterlim_sequentially_Suc: "(LIM x sequentially. f (Suc x) :> F) ⟷ (LIM x sequentially. f x :> F)" unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp lemma filterlim_Suc: "filterlim Suc sequentially sequentially" by (simp add: filterlim_iff eventually_sequentially) lemma filterlim_If: "LIM x inf F (principal {x. P x}). f x :> G ⟹ LIM x inf F (principal {x. ¬ P x}). g x :> G ⟹ LIM x F. if P x then f x else g x :> G" unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff) lemma filterlim_Pair: "LIM x F. f x :> G ⟹ LIM x F. g x :> H ⟹ LIM x F. (f x, g x) :> G ×⇩_{F}H" unfolding filterlim_def by (rule order_trans[OF filtermap_Pair prod_filter_mono]) subsection ‹Limits to \<^const>‹at_top› and \<^const>‹at_bot›› lemma filterlim_at_top: fixes f :: "'a ⇒ ('b::linorder)" shows "(LIM x F. f x :> at_top) ⟷ (∀Z. eventually (λx. Z ≤ f x) F)" by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono) lemma filterlim_at_top_mono: "LIM x F. f x :> at_top ⟹ eventually (λx. f x ≤ (g x::'a::linorder)) F ⟹ LIM x F. g x :> at_top" by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans) lemma filterlim_at_top_dense: fixes f :: "'a ⇒ ('b::unbounded_dense_linorder)" shows "(LIM x F. f x :> at_top) ⟷ (∀Z. eventually (λx. Z < f x) F)" by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le filterlim_at_top[of f F] filterlim_iff[of f at_top F]) lemma filterlim_at_top_ge: fixes f :: "'a ⇒ ('b::linorder)" and c :: "'b" shows "(LIM x F. f x :> at_top) ⟷ (∀Z≥c. eventually (λx. Z ≤ f x) F)" unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal) lemma filterlim_at_top_at_top: fixes f :: "'a::linorder ⇒ 'b::linorder" assumes mono: "⋀x y. Q x ⟹ Q y ⟹ x ≤ y ⟹ f x ≤ f y" assumes bij: "⋀x. P x ⟹ f (g x) = x" "⋀x. P x ⟹ Q (g x)" assumes Q: "eventually Q at_top" assumes P: "eventually P at_top" shows "filterlim f at_top at_top" proof - from P obtain x where x: "⋀y. x ≤ y ⟹ P y" unfolding eventually_at_top_linorder by auto show ?thesis proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) fix z assume "x ≤ z" with x have "P z" by auto have "eventually (λx. g z ≤ x) at_top" by (rule eventually_ge_at_top) with Q show "eventually (λx. z ≤ f x) at_top" by eventually_elim (metis mono bij ‹P z›) qed qed lemma filterlim_at_top_gt: fixes f :: "'a ⇒ ('b::unbounded_dense_linorder)" and c :: "'b" shows "(LIM x F. f x :> at_top) ⟷ (∀Z>c. eventually (λx. Z ≤ f x) F)" by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) lemma filterlim_at_bot: fixes f :: "'a ⇒ ('b::linorder)" shows "(LIM x F. f x :> at_bot) ⟷ (∀Z. eventually (λx. f x ≤ Z) F)" by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono) lemma filterlim_at_bot_dense: fixes f :: "'a ⇒ ('b::{dense_linorder, no_bot})" shows "(LIM x F. f x :> at_bot) ⟷ (∀Z. eventually (λx. f x < Z) F)" proof (auto simp add: filterlim_at_bot[of f F]) fix Z :: 'b from lt_ex [of Z] obtain Z' where 1: "Z' < Z" .. assume "∀Z. eventually (λx. f x ≤ Z) F" hence "eventually (λx. f x ≤ Z') F" by auto thus "eventually (λx. f x < Z) F" by (rule eventually_mono) (use 1 in auto) next fix Z :: 'b show "∀Z. eventually (λx. f x < Z) F ⟹ eventually (λx. f x ≤ Z) F" by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le) qed lemma filterlim_at_bot_le: fixes f :: "'a ⇒ ('b::linorder)" and c :: "'b" shows "(LIM x F. f x :> at_bot) ⟷ (∀Z≤c. eventually (λx. Z ≥ f x) F)" unfolding filterlim_at_bot proof safe fix Z assume *: "∀Z≤c. eventually (λx. Z ≥ f x) F" with *[THEN spec, of "min Z c"] show "eventually (λx. Z ≥ f x) F" by (auto elim!: eventually_mono) qed simp lemma filterlim_at_bot_lt: fixes f :: "'a ⇒ ('b::unbounded_dense_linorder)" and c :: "'b" shows "(LIM x F. f x :> at_bot) ⟷ (∀Z<c. eventually (λx. Z ≥ f x) F)" by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) lemma filterlim_at_top_div_const_nat: assumes "c > 0" shows "filterlim (λx::nat. x div c) at_top at_top" unfolding filterlim_at_top proof fix C :: nat have *: "n div c ≥ C" if "n ≥ C * c" for n using assms that by (metis div_le_mono div_mult_self_is_m) have "eventually (λn. n ≥ C * c) at_top" by (rule eventually_ge_at_top) thus "eventually (λn. n div c ≥ C) at_top" by eventually_elim (use * in auto) qed lemma filterlim_finite_subsets_at_top: "filterlim f (finite_subsets_at_top A) F ⟷ (∀X. finite X ∧ X ⊆ A ⟶ eventually (λy. finite (f y) ∧ X ⊆ f y ∧ f y ⊆ A) F)" (is