(* Title: HOL/Lattice/Bounds.thy Author: Markus Wenzel, TU Muenchen *) section ‹Bounds› theory Bounds imports Orders begin hide_const (open) inf sup subsection ‹Infimum and supremum› text ‹ Given a partial order, we define infimum (greatest lower bound) and supremum (least upper bound) wrt.\ ‹⊑› for two and for any number of elements. › definition is_inf :: "'a::partial_order ⇒ 'a ⇒ 'a ⇒ bool" where "is_inf x y inf = (inf ⊑ x ∧ inf ⊑ y ∧ (∀z. z ⊑ x ∧ z ⊑ y ⟶ z ⊑ inf))" definition is_sup :: "'a::partial_order ⇒ 'a ⇒ 'a ⇒ bool" where "is_sup x y sup = (x ⊑ sup ∧ y ⊑ sup ∧ (∀z. x ⊑ z ∧ y ⊑ z ⟶ sup ⊑ z))" definition is_Inf :: "'a::partial_order set ⇒ 'a ⇒ bool" where "is_Inf A inf = ((∀x ∈ A. inf ⊑ x) ∧ (∀z. (∀x ∈ A. z ⊑ x) ⟶ z ⊑ inf))" definition is_Sup :: "'a::partial_order set ⇒ 'a ⇒ bool" where "is_Sup A sup = ((∀x ∈ A. x ⊑ sup) ∧ (∀z. (∀x ∈ A. x ⊑ z) ⟶ sup ⊑ z))" text ‹ These definitions entail the following basic properties of boundary elements. › lemma is_infI [intro?]: "inf ⊑ x ⟹ inf ⊑ y ⟹ (⋀z. z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ inf) ⟹ is_inf x y inf" by (unfold is_inf_def) blast lemma is_inf_greatest [elim?]: "is_inf x y inf ⟹ z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ inf" by (unfold is_inf_def) blast lemma is_inf_lower [elim?]: "is_inf x y inf ⟹ (inf ⊑ x ⟹ inf ⊑ y ⟹ C) ⟹ C" by (unfold is_inf_def) blast lemma is_supI [intro?]: "x ⊑ sup ⟹ y ⊑ sup ⟹ (⋀z. x ⊑ z ⟹ y ⊑ z ⟹ sup ⊑ z) ⟹ is_sup x y sup" by (unfold is_sup_def) blast lemma is_sup_least [elim?]: "is_sup x y sup ⟹ x ⊑ z ⟹ y ⊑ z ⟹ sup ⊑ z" by (unfold is_sup_def) blast lemma is_sup_upper [elim?]: "is_sup x y sup ⟹ (x ⊑ sup ⟹ y ⊑ sup ⟹ C) ⟹ C" by (unfold is_sup_def) blast lemma is_InfI [intro?]: "(⋀x. x ∈ A ⟹ inf ⊑ x) ⟹ (⋀z. (∀x ∈ A. z ⊑ x) ⟹ z ⊑ inf) ⟹ is_Inf A inf" by (unfold is_Inf_def) blast lemma is_Inf_greatest [elim?]: "is_Inf A inf ⟹ (⋀x. x ∈ A ⟹ z ⊑ x) ⟹ z ⊑ inf" by (unfold is_Inf_def) blast lemma is_Inf_lower [dest?]: "is_Inf A inf ⟹ x ∈ A ⟹ inf ⊑ x" by (unfold is_Inf_def) blast lemma is_SupI [intro?]: "(⋀x. x ∈ A ⟹ x ⊑ sup) ⟹ (⋀z. (∀x ∈ A. x ⊑ z) ⟹ sup ⊑ z) ⟹ is_Sup A sup" by (unfold is_Sup_def) blast lemma is_Sup_least [elim?]: "is_Sup A sup ⟹ (⋀x. x ∈ A ⟹ x ⊑ z) ⟹ sup ⊑ z" by (unfold is_Sup_def) blast lemma is_Sup_upper [dest?]: "is_Sup A sup ⟹ x ∈ A ⟹ x ⊑ sup" by (unfold is_Sup_def) blast subsection ‹Duality› text ‹ Infimum and supremum are dual to each other. › theorem dual_inf [iff?]: "is_inf (dual x) (dual y) (dual sup) = is_sup x y sup" by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq) theorem dual_sup [iff?]: "is_sup (dual x) (dual y) (dual inf) = is_inf x y inf" by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq) theorem dual_Inf [iff?]: "is_Inf (dual ` A) (dual sup) = is_Sup A sup" by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq) theorem dual_Sup [iff?]: "is_Sup (dual ` A) (dual inf) = is_Inf A inf" by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq) subsection ‹Uniqueness› text ‹ Infima and suprema on partial orders are unique; this is mainly due to anti-symmetry of the underlying relation. › theorem is_inf_uniq: "is_inf x y inf ⟹ is_inf x y inf' ⟹ inf = inf'" proof - assume inf: "is_inf x y inf" assume inf': "is_inf x y inf'" show ?thesis proof (rule leq_antisym) from inf' show "inf ⊑ inf'" proof (rule is_inf_greatest) from inf show "inf ⊑ x" .. from inf show "inf ⊑ y" .. qed from inf show "inf' ⊑ inf" proof (rule is_inf_greatest) from inf' show "inf' ⊑ x" .. from inf' show "inf' ⊑ y" .. qed qed qed theorem is_sup_uniq: "is_sup x y sup ⟹ is_sup x y sup' ⟹ sup = sup'" proof - assume sup: "is_sup x y sup" and sup': "is_sup x y sup'" have "dual sup = dual sup'" proof (rule is_inf_uniq) from sup show "is_inf (dual x) (dual y) (dual sup)" .. from sup' show "is_inf (dual x) (dual y) (dual sup')" .. qed then show "sup = sup'" .. qed theorem is_Inf_uniq: "is_Inf A inf ⟹ is_Inf A inf' ⟹ inf = inf'" proof - assume inf: "is_Inf A inf" assume inf': "is_Inf A inf'" show ?thesis proof (rule leq_antisym) from inf' show "inf ⊑ inf'" proof (rule is_Inf_greatest) fix x assume "x ∈ A" with inf show "inf ⊑ x" .. qed from inf show "inf' ⊑ inf" proof (rule is_Inf_greatest) fix x assume "x ∈ A" with inf' show "inf' ⊑ x" .. qed qed qed theorem is_Sup_uniq: "is_Sup A sup ⟹ is_Sup A sup' ⟹ sup = sup'" proof - assume sup: "is_Sup A sup" and sup': "is_Sup A sup'" have "dual sup = dual sup'" proof (rule is_Inf_uniq) from sup show "is_Inf (dual ` A) (dual sup)" .. from sup' show "is_Inf (dual ` A) (dual sup')" .. qed then show "sup = sup'" .. qed subsection ‹Related elements› text ‹ The binary bound of related elements is either one of the argument. › theorem [elim?]: "x ⊑ y ⟹ is_inf x y x" proof - assume "x ⊑ y" show ?thesis proof show "x ⊑ x" .. show "x ⊑ y" by fact fix z assume "z ⊑ x" and "z ⊑ y" show "z ⊑ x" by fact qed qed theorem [elim?]: "x ⊑ y ⟹ is_sup x y y" proof - assume "x ⊑ y" show ?thesis proof show "x ⊑ y" by fact show "y ⊑ y" .. fix z assume "x ⊑ z" and "y ⊑ z" show "y ⊑ z" by fact qed qed subsection ‹General versus binary bounds \label{sec:gen-bin-bounds}› text ‹ General bounds of two-element sets coincide with binary bounds. › theorem is_Inf_binary: "is_Inf {x, y} inf = is_inf x y inf" proof - let ?A = "{x, y}" show ?thesis proof assume is_Inf: "is_Inf ?A inf" show "is_inf x y inf" proof have "x ∈ ?A" by simp with is_Inf show "inf ⊑ x" .. have "y ∈ ?A" by simp with is_Inf show "inf ⊑ y" .. fix z assume zx: "z ⊑ x" and zy: "z ⊑ y" from is_Inf show "z ⊑ inf" proof (rule is_Inf_greatest) fix a assume "a ∈ ?A" then have "a = x ∨ a = y" by blast then show "z ⊑ a" proof assume "a = x" with zx show ?thesis by simp next assume "a = y" with zy show ?thesis by simp qed qed qed next assume is_inf: "is_inf x y inf" show "is_Inf {x, y} inf" proof fix a assume "a ∈ ?A" then have "a = x ∨ a = y" by blast then show "inf ⊑ a" proof assume "a = x" also from is_inf have "inf ⊑ x" .. finally show ?thesis . next assume "a = y" also from is_inf have "inf ⊑ y" .. finally show ?thesis . qed next fix z assume z: "∀a ∈ ?A. z ⊑ a" from is_inf show "z ⊑ inf" proof (rule is_inf_greatest) from z show "z ⊑ x" by blast from z show "z ⊑ y" by blast qed qed qed qed theorem is_Sup_binary: "is_Sup {x, y} sup = is_sup x y sup" proof - have "is_Sup {x, y} sup = is_Inf (dual ` {x, y}) (dual sup)" by (simp only: dual_Inf) also have "dual ` {x, y} = {dual x, dual y}" by simp also have "is_Inf … (dual sup) = is_inf (dual x) (dual y) (dual sup)" by (rule is_Inf_binary) also have "… = is_sup x y sup" by (simp only: dual_inf) finally show ?thesis . qed subsection ‹Connecting general bounds \label{sec:connect-bounds}› text ‹ Either kind of general bounds is sufficient to express the other. The least upper bound (supremum) is the same as the the greatest lower bound of the set of all upper bounds; the dual statements holds as well; the dual statement holds as well. › theorem Inf_Sup: "is_Inf {b. ∀a ∈ A. a ⊑ b} sup ⟹ is_Sup A sup" proof - let ?B = "{b. ∀a ∈ A. a ⊑ b}" assume is_Inf: "is_Inf ?B sup" show "is_Sup A sup" proof fix x assume x: "x ∈ A" from is_Inf show "x ⊑ sup" proof (rule is_Inf_greatest) fix y assume "y ∈ ?B" then have "∀a ∈ A. a ⊑ y" .. from this x show "x ⊑ y" .. qed next fix z assume "∀x ∈ A. x ⊑ z" then have "z ∈ ?B" .. with is_Inf show "sup ⊑ z" .. qed qed theorem Sup_Inf: "is_Sup {b. ∀a ∈ A. b ⊑ a} inf ⟹ is_Inf A inf" proof - assume "is_Sup {b. ∀a ∈ A. b ⊑ a} inf" then have "is_Inf (dual ` {b. ∀a ∈ A. dual a ⊑ dual b}) (dual inf)" by (simp only: dual_Inf dual_leq) also have "dual ` {b. ∀a ∈ A. dual a ⊑ dual b} = {b'. ∀a' ∈ dual ` A. a' ⊑ b'}" by (auto iff: dual_ball dual_Collect simp add: image_Collect) (* FIXME !? *) finally have "is_Inf … (dual inf)" . then have "is_Sup (dual ` A) (dual inf)" by (rule Inf_Sup) then show ?thesis .. qed end