(* Title: HOL/Lattice/CompleteLattice.thy Author: Markus Wenzel, TU Muenchen *) header {* Complete lattices *} theory CompleteLattice imports Lattice begin subsection {* Complete lattice operations *} text {* A \emph{complete lattice} is a partial order with general (infinitary) infimum of any set of elements. General supremum exists as well, as a consequence of the connection of infinitary bounds (see \S\ref{sec:connect-bounds}). *} class complete_lattice = assumes ex_Inf: "∃inf. is_Inf A inf" theorem ex_Sup: "∃sup::'a::complete_lattice. is_Sup A sup" proof - from ex_Inf obtain sup where "is_Inf {b. ∀a∈A. a \<sqsubseteq> b} sup" by blast then have "is_Sup A sup" by (rule Inf_Sup) then show ?thesis .. qed text {* The general @{text \<Sqinter>} (meet) and @{text \<Squnion>} (join) operations select such infimum and supremum elements. *} definition Meet :: "'a::complete_lattice set => 'a" where "Meet A = (THE inf. is_Inf A inf)" definition Join :: "'a::complete_lattice set => 'a" where "Join A = (THE sup. is_Sup A sup)" notation (xsymbols) Meet ("\<Sqinter>_" [90] 90) and Join ("\<Squnion>_" [90] 90) text {* Due to unique existence of bounds, the complete lattice operations may be exhibited as follows. *} lemma Meet_equality [elim?]: "is_Inf A inf ==> \<Sqinter>A = inf" proof (unfold Meet_def) assume "is_Inf A inf" then show "(THE inf. is_Inf A inf) = inf" by (rule the_equality) (rule is_Inf_uniq [OF _ `is_Inf A inf`]) qed lemma MeetI [intro?]: "(!!a. a ∈ A ==> inf \<sqsubseteq> a) ==> (!!b. ∀a ∈ A. b \<sqsubseteq> a ==> b \<sqsubseteq> inf) ==> \<Sqinter>A = inf" by (rule Meet_equality, rule is_InfI) blast+ lemma Join_equality [elim?]: "is_Sup A sup ==> \<Squnion>A = sup" proof (unfold Join_def) assume "is_Sup A sup" then show "(THE sup. is_Sup A sup) = sup" by (rule the_equality) (rule is_Sup_uniq [OF _ `is_Sup A sup`]) qed lemma JoinI [intro?]: "(!!a. a ∈ A ==> a \<sqsubseteq> sup) ==> (!!b. ∀a ∈ A. a \<sqsubseteq> b ==> sup \<sqsubseteq> b) ==> \<Squnion>A = sup" by (rule Join_equality, rule is_SupI) blast+ text {* \medskip The @{text \<Sqinter>} and @{text \<Squnion>} operations indeed determine bounds on a complete lattice structure. *} lemma is_Inf_Meet [intro?]: "is_Inf A (\<Sqinter>A)" proof (unfold Meet_def) from ex_Inf obtain inf where "is_Inf A inf" .. then show "is_Inf A (THE inf. is_Inf A inf)" by (rule theI) (rule is_Inf_uniq [OF _ `is_Inf A inf`]) qed lemma Meet_greatest [intro?]: "(!!a. a ∈ A ==> x \<sqsubseteq> a) ==> x \<sqsubseteq> \<Sqinter>A" by (rule is_Inf_greatest, rule is_Inf_Meet) blast lemma Meet_lower [intro?]: "a ∈ A ==> \<Sqinter>A \<sqsubseteq> a" by (rule is_Inf_lower) (rule is_Inf_Meet) lemma is_Sup_Join [intro?]: "is_Sup A (\<Squnion>A)" proof (unfold Join_def) from ex_Sup obtain sup where "is_Sup A sup" .. then show "is_Sup A (THE sup. is_Sup A sup)" by (rule theI) (rule is_Sup_uniq [OF _ `is_Sup A sup`]) qed lemma Join_least [intro?]: "(!!a. a ∈ A ==> a \<sqsubseteq> x) ==> \<Squnion>A \<sqsubseteq> x" by (rule is_Sup_least, rule is_Sup_Join) blast lemma Join_lower [intro?]: "a ∈ A ==> a \<sqsubseteq> \<Squnion>A" by (rule is_Sup_upper) (rule is_Sup_Join) subsection {* The Knaster-Tarski Theorem *} text {* The Knaster-Tarski Theorem (in its simplest formulation) states that any monotone function on a complete lattice has a least fixed-point (see \cite[pages 93--94]{Davey-Priestley:1990} for example). This is a consequence of the basic boundary properties of the complete lattice operations. *} theorem Knaster_Tarski: assumes mono: "!!x y. x \<sqsubseteq> y ==> f x \<sqsubseteq> f y" obtains a :: "'a::complete_lattice" where "f a = a" and "!!a'. f a' = a' ==> a \<sqsubseteq> a'" proof let ?H = "{u. f u \<sqsubseteq> u}" let ?a = "\<Sqinter>?H" show "f ?a = ?a" proof - have ge: "f ?a \<sqsubseteq> ?a" proof fix x assume x: "x ∈ ?H" then have "?a \<sqsubseteq> x" .. then have "f ?a \<sqsubseteq> f x" by (rule mono) also from x have "... \<sqsubseteq> x" .. finally show "f ?a \<sqsubseteq> x" . qed also have "?a \<sqsubseteq> f ?a" proof from ge have "f (f ?a) \<sqsubseteq> f ?a" by (rule mono) then show "f ?a ∈ ?H" .. qed finally show ?thesis . qed fix a' assume "f a' = a'" then have "f a' \<sqsubseteq> a'" by (simp only: leq_refl) then have "a' ∈ ?H" .. then show "?a \<sqsubseteq> a'" .. qed theorem Knaster_Tarski_dual: assumes mono: "!!x y. x \<sqsubseteq> y ==> f x \<sqsubseteq> f y" obtains a :: "'a::complete_lattice" where "f a = a" and "!!a'. f a' = a' ==> a' \<sqsubseteq> a" proof let ?H = "{u. u \<sqsubseteq> f u}" let ?a = "\<Squnion>?H" show "f ?a = ?a" proof - have le: "?a \<sqsubseteq> f ?a" proof fix x assume x: "x ∈ ?H" then have "x \<sqsubseteq> f x" .. also from x have "x \<sqsubseteq> ?a" .. then have "f x \<sqsubseteq> f ?a" by (rule mono) finally show "x \<sqsubseteq> f ?a" . qed have "f ?a \<sqsubseteq> ?a" proof from le have "f ?a \<sqsubseteq> f (f ?a)" by (rule mono) then show "f ?a ∈ ?H" .. qed from this and le show ?thesis by (rule leq_antisym) qed fix a' assume "f a' = a'" then have "a' \<sqsubseteq> f a'" by (simp only: leq_refl) then have "a' ∈ ?H" .. then show "a' \<sqsubseteq> ?a" .. qed subsection {* Bottom and top elements *} text {* With general bounds available, complete lattices also have least and greatest elements. *} definition bottom :: "'a::complete_lattice" ("⊥") where "⊥ = \<Sqinter>UNIV" definition top :: "'a::complete_lattice" ("\<top>") where "\<top> = \<Squnion>UNIV" lemma bottom_least [intro?]: "⊥ \<sqsubseteq> x" proof (unfold bottom_def) have "x ∈ UNIV" .. then show "\<Sqinter>UNIV \<sqsubseteq> x" .. qed lemma bottomI [intro?]: "(!!a. x \<sqsubseteq> a) ==> ⊥ = x" proof (unfold bottom_def) assume "!!a. x \<sqsubseteq> a" show "\<Sqinter>UNIV = x" proof fix a show "x \<sqsubseteq> a" by fact next fix b :: "'a::complete_lattice" assume b: "∀a ∈ UNIV. b \<sqsubseteq> a" have "x ∈ UNIV" .. with b show "b \<sqsubseteq> x" .. qed qed lemma top_greatest [intro?]: "x \<sqsubseteq> \<top>" proof (unfold top_def) have "x ∈ UNIV" .. then show "x \<sqsubseteq> \<Squnion>UNIV" .. qed lemma topI [intro?]: "(!!a. a \<sqsubseteq> x) ==> \<top> = x" proof (unfold top_def) assume "!!a. a \<sqsubseteq> x" show "\<Squnion>UNIV = x" proof fix a show "a \<sqsubseteq> x" by fact next fix b :: "'a::complete_lattice" assume b: "∀a ∈ UNIV. a \<sqsubseteq> b" have "x ∈ UNIV" .. with b show "x \<sqsubseteq> b" .. qed qed subsection {* Duality *} text {* The class of complete lattices is closed under formation of dual structures. *} instance dual :: (complete_lattice) complete_lattice proof fix A' :: "'a::complete_lattice dual set" show "∃inf'. is_Inf A' inf'" proof - have "∃sup. is_Sup (undual ` A') sup" by (rule ex_Sup) then have "∃sup. is_Inf (dual ` undual ` A') (dual sup)" by (simp only: dual_Inf) then show ?thesis by (simp add: dual_ex [symmetric] image_comp) qed qed text {* Apparently, the @{text \<Sqinter>} and @{text \<Squnion>} operations are dual to each other. *} theorem dual_Meet [intro?]: "dual (\<Sqinter>A) = \<Squnion>(dual ` A)" proof - from is_Inf_Meet have "is_Sup (dual ` A) (dual (\<Sqinter>A))" .. then have "\<Squnion>(dual ` A) = dual (\<Sqinter>A)" .. then show ?thesis .. qed theorem dual_Join [intro?]: "dual (\<Squnion>A) = \<Sqinter>(dual ` A)" proof - from is_Sup_Join have "is_Inf (dual ` A) (dual (\<Squnion>A))" .. then have "\<Sqinter>(dual ` A) = dual (\<Squnion>A)" .. then show ?thesis .. qed text {* Likewise are @{text ⊥} and @{text \<top>} duals of each other. *} theorem dual_bottom [intro?]: "dual ⊥ = \<top>" proof - have "\<top> = dual ⊥" proof fix a' have "⊥ \<sqsubseteq> undual a'" .. then have "dual (undual a') \<sqsubseteq> dual ⊥" .. then show "a' \<sqsubseteq> dual ⊥" by simp qed then show ?thesis .. qed theorem dual_top [intro?]: "dual \<top> = ⊥" proof - have "⊥ = dual \<top>" proof fix a' have "undual a' \<sqsubseteq> \<top>" .. then have "dual \<top> \<sqsubseteq> dual (undual a')" .. then show "dual \<top> \<sqsubseteq> a'" by simp qed then show ?thesis .. qed subsection {* Complete lattices are lattices *} text {* Complete lattices (with general bounds available) are indeed plain lattices as well. This holds due to the connection of general versus binary bounds that has been formally established in \S\ref{sec:gen-bin-bounds}. *} lemma is_inf_binary: "is_inf x y (\<Sqinter>{x, y})" proof - have "is_Inf {x, y} (\<Sqinter>{x, y})" .. then show ?thesis by (simp only: is_Inf_binary) qed lemma is_sup_binary: "is_sup x y (\<Squnion>{x, y})" proof - have "is_Sup {x, y} (\<Squnion>{x, y})" .. then show ?thesis by (simp only: is_Sup_binary) qed instance complete_lattice ⊆ lattice proof fix x y :: "'a::complete_lattice" from is_inf_binary show "∃inf. is_inf x y inf" .. from is_sup_binary show "∃sup. is_sup x y sup" .. qed theorem meet_binary: "x \<sqinter> y = \<Sqinter>{x, y}" by (rule meet_equality) (rule is_inf_binary) theorem join_binary: "x \<squnion> y = \<Squnion>{x, y}" by (rule join_equality) (rule is_sup_binary) subsection {* Complete lattices and set-theory operations *} text {* The complete lattice operations are (anti) monotone wrt.\ set inclusion. *} theorem Meet_subset_antimono: "A ⊆ B ==> \<Sqinter>B \<sqsubseteq> \<Sqinter>A" proof (rule Meet_greatest) fix a assume "a ∈ A" also assume "A ⊆ B" finally have "a ∈ B" . then show "\<Sqinter>B \<sqsubseteq> a" .. qed theorem Join_subset_mono: "A ⊆ B ==> \<Squnion>A \<sqsubseteq> \<Squnion>B" proof - assume "A ⊆ B" then have "dual ` A ⊆ dual ` B" by blast then have "\<Sqinter>(dual ` B) \<sqsubseteq> \<Sqinter>(dual ` A)" by (rule Meet_subset_antimono) then have "dual (\<Squnion>B) \<sqsubseteq> dual (\<Squnion>A)" by (simp only: dual_Join) then show ?thesis by (simp only: dual_leq) qed text {* Bounds over unions of sets may be obtained separately. *} theorem Meet_Un: "\<Sqinter>(A ∪ B) = \<Sqinter>A \<sqinter> \<Sqinter>B" proof fix a assume "a ∈ A ∪ B" then show "\<Sqinter>A \<sqinter> \<Sqinter>B \<sqsubseteq> a" proof assume a: "a ∈ A" have "\<Sqinter>A \<sqinter> \<Sqinter>B \<sqsubseteq> \<Sqinter>A" .. also from a have "… \<sqsubseteq> a" .. finally show ?thesis . next assume a: "a ∈ B" have "\<Sqinter>A \<sqinter> \<Sqinter>B \<sqsubseteq> \<Sqinter>B" .. also from a have "… \<sqsubseteq> a" .. finally show ?thesis . qed next fix b assume b: "∀a ∈ A ∪ B. b \<sqsubseteq> a" show "b \<sqsubseteq> \<Sqinter>A \<sqinter> \<Sqinter>B" proof show "b \<sqsubseteq> \<Sqinter>A" proof fix a assume "a ∈ A" then have "a ∈ A ∪ B" .. with b show "b \<sqsubseteq> a" .. qed show "b \<sqsubseteq> \<Sqinter>B" proof fix a assume "a ∈ B" then have "a ∈ A ∪ B" .. with b show "b \<sqsubseteq> a" .. qed qed qed theorem Join_Un: "\<Squnion>(A ∪ B) = \<Squnion>A \<squnion> \<Squnion>B" proof - have "dual (\<Squnion>(A ∪ B)) = \<Sqinter>(dual ` A ∪ dual ` B)" by (simp only: dual_Join image_Un) also have "… = \<Sqinter>(dual ` A) \<sqinter> \<Sqinter>(dual ` B)" by (rule Meet_Un) also have "… = dual (\<Squnion>A \<squnion> \<Squnion>B)" by (simp only: dual_join dual_Join) finally show ?thesis .. qed text {* Bounds over singleton sets are trivial. *} theorem Meet_singleton: "\<Sqinter>{x} = x" proof fix a assume "a ∈ {x}" then have "a = x" by simp then show "x \<sqsubseteq> a" by (simp only: leq_refl) next fix b assume "∀a ∈ {x}. b \<sqsubseteq> a" then show "b \<sqsubseteq> x" by simp qed theorem Join_singleton: "\<Squnion>{x} = x" proof - have "dual (\<Squnion>{x}) = \<Sqinter>{dual x}" by (simp add: dual_Join) also have "… = dual x" by (rule Meet_singleton) finally show ?thesis .. qed text {* Bounds over the empty and universal set correspond to each other. *} theorem Meet_empty: "\<Sqinter>{} = \<Squnion>UNIV" proof fix a :: "'a::complete_lattice" assume "a ∈ {}" then have False by simp then show "\<Squnion>UNIV \<sqsubseteq> a" .. next fix b :: "'a::complete_lattice" have "b ∈ UNIV" .. then show "b \<sqsubseteq> \<Squnion>UNIV" .. qed theorem Join_empty: "\<Squnion>{} = \<Sqinter>UNIV" proof - have "dual (\<Squnion>{}) = \<Sqinter>{}" by (simp add: dual_Join) also have "… = \<Squnion>UNIV" by (rule Meet_empty) also have "… = dual (\<Sqinter>UNIV)" by (simp add: dual_Meet) finally show ?thesis .. qed end