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src/HOL/Algebra/Lattice.thy

author | haftmann |

Fri, 17 Jun 2005 16:12:49 +0200 | |

changeset 16417 | 9bc16273c2d4 |

parent 16325 | a6431098a929 |

child 19286 | 83404ccd270a |

permissions | -rw-r--r-- |

migrated theory headers to new format

(* Title: HOL/Algebra/Lattice.thy Id: $Id$ Author: Clemens Ballarin, started 7 November 2003 Copyright: Clemens Ballarin *) header {* Orders and Lattices *} theory Lattice imports Main begin text {* Object with a carrier set. *} record 'a partial_object = carrier :: "'a set" subsection {* Partial Orders *} record 'a order = "'a partial_object" + le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50) locale partial_order = struct L + assumes refl [intro, simp]: "x \<in> carrier L ==> x \<sqsubseteq> x" and anti_sym [intro]: "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y" and trans [trans]: "[| x \<sqsubseteq> y; y \<sqsubseteq> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z" constdefs (structure L) less :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50) "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y" -- {* Upper and lower bounds of a set. *} Upper :: "[_, 'a set] => 'a set" "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter> carrier L" Lower :: "[_, 'a set] => 'a set" "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter> carrier L" -- {* Least and greatest, as predicate. *} least :: "[_, 'a, 'a set] => bool" "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)" greatest :: "[_, 'a, 'a set] => bool" "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)" -- {* Supremum and infimum *} sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90) "\<Squnion>A == THE x. least L x (Upper L A)" inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90) "\<Sqinter>A == THE x. greatest L x (Lower L A)" join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65) "x \<squnion> y == sup L {x, y}" meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 70) "x \<sqinter> y == inf L {x, y}" subsubsection {* Upper *} lemma Upper_closed [intro, simp]: "Upper L A \<subseteq> carrier L" by (unfold Upper_def) clarify lemma UpperD [dest]: includes struct L shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u" by (unfold Upper_def) blast lemma Upper_memI: includes struct L shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A" by (unfold Upper_def) blast lemma Upper_antimono: "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A" by (unfold Upper_def) blast subsubsection {* Lower *} lemma Lower_closed [intro, simp]: "Lower L A \<subseteq> carrier L" by (unfold Lower_def) clarify lemma LowerD [dest]: includes struct L shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x" by (unfold Lower_def) blast lemma Lower_memI: includes struct L shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A" by (unfold Lower_def) blast lemma Lower_antimono: "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A" by (unfold Lower_def) blast subsubsection {* least *} lemma least_carrier [intro, simp]: shows "least L l A ==> l \<in> carrier L" by (unfold least_def) fast lemma least_mem: "least L l A ==> l \<in> A" by (unfold least_def) fast lemma (in partial_order) least_unique: "[| least L x A; least L y A |] ==> x = y" by (unfold least_def) blast lemma least_le: includes struct L shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a" by (unfold least_def) fast lemma least_UpperI: includes struct L assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s" and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y" and L: "A \<subseteq> carrier L" "s \<in> carrier L" shows "least L s (Upper L A)" proof - have "Upper L A \<subseteq> carrier L" by simp moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def) moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast ultimately show ?thesis by (simp add: least_def) qed subsubsection {* greatest *} lemma greatest_carrier [intro, simp]: shows "greatest L l A ==> l \<in> carrier L" by (unfold greatest_def) fast lemma greatest_mem: "greatest L l A ==> l \<in> A" by (unfold greatest_def) fast lemma (in partial_order) greatest_unique: "[| greatest L x A; greatest L y A |] ==> x = y" by (unfold greatest_def) blast lemma greatest_le: includes struct L shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x" by (unfold greatest_def) fast lemma greatest_LowerI: includes struct L assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x" and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i" and L: "A \<subseteq> carrier L" "i \<in> carrier L" shows "greatest L i (Lower L A)" proof - have "Lower L A \<subseteq> carrier L" by simp moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def) moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast ultimately show ?thesis by (simp add: greatest_def) qed subsection {* Lattices *} locale lattice = partial_order + assumes sup_of_two_exists: "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})" and inf_of_two_exists: "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})" lemma least_Upper_above: includes struct L shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s" by (unfold least_def) blast lemma greatest_Lower_above: includes struct L shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x" by (unfold greatest_def) blast subsubsection {* Supremum *} lemma (in lattice) joinI: "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |] ==> P (x \<squnion> y)" proof (unfold join_def sup_def) assume L: "x \<in> carrier L" "y \<in> carrier L" and P: "!!l. least L l (Upper L {x, y}) ==> P l" with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast with L show "P (THE l. least L l (Upper L {x, y}))" by (fast intro: theI2 least_unique P) qed lemma (in lattice) join_closed [simp]: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L" by (rule joinI) (rule least_carrier) lemma (in partial_order) sup_of_singletonI: (* only reflexivity needed ? *) "x \<in> carrier L ==> least L x (Upper L {x})" by (rule least_UpperI) fast+ lemma (in partial_order) sup_of_singleton [simp]: includes struct L shows "x \<in> carrier L ==> \<Squnion>{x} = x" by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI) text {* Condition on @{text A}: supremum exists. *} lemma (in lattice) sup_insertI: "[| !!s. least L s (Upper L (insert x A)) ==> P s; least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |] ==> P (\<Squnion>(insert x A))" proof (unfold sup_def) assume L: "x \<in> carrier L" "A \<subseteq> carrier L" and P: "!!l. least L l (Upper L (insert x A)) ==> P l" and least_a: "least L a (Upper L A)" from L least_a have La: "a \<in> carrier L" by simp from L sup_of_two_exists least_a obtain s where least_s: "least L s (Upper L {a, x})" by blast show "P (THE l. least L l (Upper L (insert x A)))" proof (rule theI2) show "least L s (Upper L (insert x A))" proof (rule least_UpperI) fix z assume "z \<in> insert x A" then show "z \<sqsubseteq> s" proof assume "z = x" then show ?thesis by (simp add: least_Upper_above [OF least_s] L La) next assume "z \<in> A" with L least_s least_a show ?thesis by (rule_tac trans [where y = a]) (auto dest: least_Upper_above) qed next fix y assume y: "y \<in> Upper L (insert x A)" show "s \<sqsubseteq> y" proof (rule least_le [OF least_s], rule Upper_memI) fix z assume z: "z \<in> {a, x}" then show "z \<sqsubseteq> y" proof have y': "y \<in> Upper L A" apply (rule subsetD [where A = "Upper L (insert x A)"]) apply (rule Upper_antimono) apply clarify apply assumption done assume "z = a" with y' least_a show ?thesis by (fast dest: least_le) next assume "z \<in> {x}" (* FIXME "z = x"; declare specific elim rule for "insert x {}" (!?) *) with y L show ?thesis by blast qed qed (rule Upper_closed [THEN subsetD]) next from L show "insert x A \<subseteq> carrier L" by simp from least_s show "s \<in> carrier L" by simp qed next fix l assume least_l: "least L l (Upper L (insert x A))" show "l = s" proof (rule least_unique) show "least L s (Upper L (insert x A))" proof (rule least_UpperI) fix z assume "z \<in> insert x A" then show "z \<sqsubseteq> s" proof assume "z = x" then show ?thesis by (simp add: least_Upper_above [OF least_s] L La) next assume "z \<in> A" with L least_s least_a show ?thesis by (rule_tac trans [where y = a]) (auto dest: least_Upper_above) qed next fix y assume y: "y \<in> Upper L (insert x A)" show "s \<sqsubseteq> y" proof (rule least_le [OF least_s], rule Upper_memI) fix z assume z: "z \<in> {a, x}" then show "z \<sqsubseteq> y" proof have y': "y \<in> Upper L A" apply (rule subsetD [where A = "Upper L (insert x A)"]) apply (rule Upper_antimono) apply clarify apply assumption done assume "z = a" with y' least_a show ?thesis by (fast dest: least_le) next assume "z \<in> {x}" with y L show ?thesis by blast qed qed (rule Upper_closed [THEN subsetD]) next from L show "insert x A \<subseteq> carrier L" by simp from least_s show "s \<in> carrier L" by simp qed qed qed qed lemma (in lattice) finite_sup_least: "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)" proof (induct set: Finites) case empty then show ?case by simp next case (insert x A) show ?case proof (cases "A = {}") case True with insert show ?thesis by (simp add: sup_of_singletonI) next case False with insert have "least L (\<Squnion>A) (Upper L A)" by simp with _ show ?thesis by (rule sup_insertI) (simp_all add: insert [simplified]) qed qed lemma (in lattice) finite_sup_insertI: assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l" and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L" shows "P (\<Squnion> (insert x A))" proof (cases "A = {}") case True with P and xA show ?thesis by (simp add: sup_of_singletonI) next case False with P and xA show ?thesis by (simp add: sup_insertI finite_sup_least) qed lemma (in lattice) finite_sup_closed: "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L" proof (induct set: Finites) case empty then show ?case by simp next case insert then show ?case by - (rule finite_sup_insertI, simp_all) qed lemma (in lattice) join_left: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y" by (rule joinI [folded join_def]) (blast dest: least_mem) lemma (in lattice) join_right: "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y" by (rule joinI [folded join_def]) (blast dest: least_mem) lemma (in lattice) sup_of_two_least: "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})" proof (unfold sup_def) assume L: "x \<in> carrier L" "y \<in> carrier L" with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})" by (fast intro: theI2 least_unique) (* blast fails *) qed lemma (in lattice) join_le: assumes sub: "x \<sqsubseteq> z" "y \<sqsubseteq> z" and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L" shows "x \<squnion> y \<sqsubseteq> z" proof (rule joinI) fix s assume "least L s (Upper L {x, y})" with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI) qed lemma (in lattice) join_assoc_lemma: assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L" shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}" proof (rule finite_sup_insertI) -- {* The textbook argument in Jacobson I, p 457 *} fix s assume sup: "least L s (Upper L {x, y, z})" show "x \<squnion> (y \<squnion> z) = s" proof (rule anti_sym) from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s" by (fastsimp intro!: join_le elim: least_Upper_above) next from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)" by (erule_tac least_le) (blast intro!: Upper_memI intro: trans join_left join_right join_closed) qed (simp_all add: L least_carrier [OF sup]) qed (simp_all add: L) lemma join_comm: includes struct L shows "x \<squnion> y = y \<squnion> x" by (unfold join_def) (simp add: insert_commute) lemma (in lattice) join_assoc: assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L" shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" proof - have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm) also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma) also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute) also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma) finally show ?thesis . qed subsubsection {* Infimum *} lemma (in lattice) meetI: "[| !!i. greatest L i (Lower L {x, y}) ==> P i; x \<in> carrier L; y \<in> carrier L |] ==> P (x \<sqinter> y)" proof (unfold meet_def inf_def) assume L: "x \<in> carrier L" "y \<in> carrier L" and P: "!!g. greatest L g (Lower L {x, y}) ==> P g" with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast with L show "P (THE g. greatest L g (Lower L {x, y}))" by (fast intro: theI2 greatest_unique P) qed lemma (in lattice) meet_closed [simp]: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L" by (rule meetI) (rule greatest_carrier) lemma (in partial_order) inf_of_singletonI: (* only reflexivity needed ? *) "x \<in> carrier L ==> greatest L x (Lower L {x})" by (rule greatest_LowerI) fast+ lemma (in partial_order) inf_of_singleton [simp]: includes struct L shows "x \<in> carrier L ==> \<Sqinter> {x} = x" by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI) text {* Condition on A: infimum exists. *} lemma (in lattice) inf_insertI: "[| !!i. greatest L i (Lower L (insert x A)) ==> P i; greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |] ==> P (\<Sqinter>(insert x A))" proof (unfold inf_def) assume L: "x \<in> carrier L" "A \<subseteq> carrier L" and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g" and greatest_a: "greatest L a (Lower L A)" from L greatest_a have La: "a \<in> carrier L" by simp from L inf_of_two_exists greatest_a obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast show "P (THE g. greatest L g (Lower L (insert x A)))" proof (rule theI2) show "greatest L i (Lower L (insert x A))" proof (rule greatest_LowerI) fix z assume "z \<in> insert x A" then show "i \<sqsubseteq> z" proof assume "z = x" then show ?thesis by (simp add: greatest_Lower_above [OF greatest_i] L La) next assume "z \<in> A" with L greatest_i greatest_a show ?thesis by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above) qed next fix y assume y: "y \<in> Lower L (insert x A)" show "y \<sqsubseteq> i" proof (rule greatest_le [OF greatest_i], rule Lower_memI) fix z assume z: "z \<in> {a, x}" then show "y \<sqsubseteq> z" proof have y': "y \<in> Lower L A" apply (rule subsetD [where A = "Lower L (insert x A)"]) apply (rule Lower_antimono) apply clarify apply assumption done assume "z = a" with y' greatest_a show ?thesis by (fast dest: greatest_le) next assume "z \<in> {x}" with y L show ?thesis by blast qed qed (rule Lower_closed [THEN subsetD]) next from L show "insert x A \<subseteq> carrier L" by simp from greatest_i show "i \<in> carrier L" by simp qed next fix g assume greatest_g: "greatest L g (Lower L (insert x A))" show "g = i" proof (rule greatest_unique) show "greatest L i (Lower L (insert x A))" proof (rule greatest_LowerI) fix z assume "z \<in> insert x A" then show "i \<sqsubseteq> z" proof assume "z = x" then show ?thesis by (simp add: greatest_Lower_above [OF greatest_i] L La) next assume "z \<in> A" with L greatest_i greatest_a show ?thesis by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above) qed next fix y assume y: "y \<in> Lower L (insert x A)" show "y \<sqsubseteq> i" proof (rule greatest_le [OF greatest_i], rule Lower_memI) fix z assume z: "z \<in> {a, x}" then show "y \<sqsubseteq> z" proof have y': "y \<in> Lower L A" apply (rule subsetD [where A = "Lower L (insert x A)"]) apply (rule Lower_antimono) apply clarify apply assumption done assume "z = a" with y' greatest_a show ?thesis by (fast dest: greatest_le) next assume "z \<in> {x}" with y L show ?thesis by blast qed qed (rule Lower_closed [THEN subsetD]) next from L show "insert x A \<subseteq> carrier L" by simp from greatest_i show "i \<in> carrier L" by simp qed qed qed qed lemma (in lattice) finite_inf_greatest: "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)" proof (induct set: Finites) case empty then show ?case by simp next case (insert x A) show ?case proof (cases "A = {}") case True with insert show ?thesis by (simp add: inf_of_singletonI) next case False from insert show ?thesis proof (rule_tac inf_insertI) from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp qed simp_all qed qed lemma (in lattice) finite_inf_insertI: assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i" and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L" shows "P (\<Sqinter> (insert x A))" proof (cases "A = {}") case True with P and xA show ?thesis by (simp add: inf_of_singletonI) next case False with P and xA show ?thesis by (simp add: inf_insertI finite_inf_greatest) qed lemma (in lattice) finite_inf_closed: "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L" proof (induct set: Finites) case empty then show ?case by simp next case insert then show ?case by (rule_tac finite_inf_insertI) (simp_all) qed lemma (in lattice) meet_left: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x" by (rule meetI [folded meet_def]) (blast dest: greatest_mem) lemma (in lattice) meet_right: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y" by (rule meetI [folded meet_def]) (blast dest: greatest_mem) lemma (in lattice) inf_of_two_greatest: "[| x \<in> carrier L; y \<in> carrier L |] ==> greatest L (\<Sqinter> {x, y}) (Lower L {x, y})" proof (unfold inf_def) assume L: "x \<in> carrier L" "y \<in> carrier L" with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast with L show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})" by (fast intro: theI2 greatest_unique) (* blast fails *) qed lemma (in lattice) meet_le: assumes sub: "z \<sqsubseteq> x" "z \<sqsubseteq> y" and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L" shows "z \<sqsubseteq> x \<sqinter> y" proof (rule meetI) fix i assume "greatest L i (Lower L {x, y})" with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI) qed lemma (in lattice) meet_assoc_lemma: assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L" shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}" proof (rule finite_inf_insertI) txt {* The textbook argument in Jacobson I, p 457 *} fix i assume inf: "greatest L i (Lower L {x, y, z})" show "x \<sqinter> (y \<sqinter> z) = i" proof (rule anti_sym) from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)" by (fastsimp intro!: meet_le elim: greatest_Lower_above) next from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i" by (erule_tac greatest_le) (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed) qed (simp_all add: L greatest_carrier [OF inf]) qed (simp_all add: L) lemma meet_comm: includes struct L shows "x \<sqinter> y = y \<sqinter> x" by (unfold meet_def) (simp add: insert_commute) lemma (in lattice) meet_assoc: assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L" shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" proof - have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm) also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma) also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute) also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma) finally show ?thesis . qed subsection {* Total Orders *} locale total_order = lattice + assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" text {* Introduction rule: the usual definition of total order *} lemma (in partial_order) total_orderI: assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" shows "total_order L" proof (rule total_order.intro) show "lattice_axioms L" proof (rule lattice_axioms.intro) fix x y assume L: "x \<in> carrier L" "y \<in> carrier L" show "EX s. least L s (Upper L {x, y})" proof - note total L moreover { assume "x \<sqsubseteq> y" with L have "least L y (Upper L {x, y})" by (rule_tac least_UpperI) auto } moreover { assume "y \<sqsubseteq> x" with L have "least L x (Upper L {x, y})" by (rule_tac least_UpperI) auto } ultimately show ?thesis by blast qed next fix x y assume L: "x \<in> carrier L" "y \<in> carrier L" show "EX i. greatest L i (Lower L {x, y})" proof - note total L moreover { assume "y \<sqsubseteq> x" with L have "greatest L y (Lower L {x, y})" by (rule_tac greatest_LowerI) auto } moreover { assume "x \<sqsubseteq> y" with L have "greatest L x (Lower L {x, y})" by (rule_tac greatest_LowerI) auto } ultimately show ?thesis by blast qed qed qed (assumption | rule total_order_axioms.intro)+ subsection {* Complete lattices *} locale complete_lattice = lattice + assumes sup_exists: "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)" and inf_exists: "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)" text {* Introduction rule: the usual definition of complete lattice *} lemma (in partial_order) complete_latticeI: assumes sup_exists: "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)" and inf_exists: "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)" shows "complete_lattice L" proof (rule complete_lattice.intro) show "lattice_axioms L" by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+ qed (assumption | rule complete_lattice_axioms.intro)+ constdefs (structure L) top :: "_ => 'a" ("\<top>\<index>") "\<top> == sup L (carrier L)" bottom :: "_ => 'a" ("\<bottom>\<index>") "\<bottom> == inf L (carrier L)" lemma (in complete_lattice) supI: "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |] ==> P (\<Squnion>A)" proof (unfold sup_def) assume L: "A \<subseteq> carrier L" and P: "!!l. least L l (Upper L A) ==> P l" with sup_exists obtain s where "least L s (Upper L A)" by blast with L show "P (THE l. least L l (Upper L A))" by (fast intro: theI2 least_unique P) qed lemma (in complete_lattice) sup_closed [simp]: "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L" by (rule supI) simp_all lemma (in complete_lattice) top_closed [simp, intro]: "\<top> \<in> carrier L" by (unfold top_def) simp lemma (in complete_lattice) infI: "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |] ==> P (\<Sqinter>A)" proof (unfold inf_def) assume L: "A \<subseteq> carrier L" and P: "!!l. greatest L l (Lower L A) ==> P l" with inf_exists obtain s where "greatest L s (Lower L A)" by blast with L show "P (THE l. greatest L l (Lower L A))" by (fast intro: theI2 greatest_unique P) qed lemma (in complete_lattice) inf_closed [simp]: "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L" by (rule infI) simp_all lemma (in complete_lattice) bottom_closed [simp, intro]: "\<bottom> \<in> carrier L" by (unfold bottom_def) simp text {* Jacobson: Theorem 8.1 *} lemma Lower_empty [simp]: "Lower L {} = carrier L" by (unfold Lower_def) simp lemma Upper_empty [simp]: "Upper L {} = carrier L" by (unfold Upper_def) simp theorem (in partial_order) complete_lattice_criterion1: assumes top_exists: "EX g. greatest L g (carrier L)" and inf_exists: "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)" shows "complete_lattice L" proof (rule complete_latticeI) from top_exists obtain top where top: "greatest L top (carrier L)" .. fix A assume L: "A \<subseteq> carrier L" let ?B = "Upper L A" from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le) then have B_non_empty: "?B ~= {}" by fast have B_L: "?B \<subseteq> carrier L" by simp from inf_exists [OF B_L B_non_empty] obtain b where b_inf_B: "greatest L b (Lower L ?B)" .. have "least L b (Upper L A)" apply (rule least_UpperI) apply (rule greatest_le [where A = "Lower L ?B"]) apply (rule b_inf_B) apply (rule Lower_memI) apply (erule UpperD) apply assumption apply (rule L) apply (fast intro: L [THEN subsetD]) apply (erule greatest_Lower_above [OF b_inf_B]) apply simp apply (rule L) apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *) done then show "EX s. least L s (Upper L A)" .. next fix A assume L: "A \<subseteq> carrier L" show "EX i. greatest L i (Lower L A)" proof (cases "A = {}") case True then show ?thesis by (simp add: top_exists) next case False with L show ?thesis by (rule inf_exists) qed qed (* TODO: prove dual version *) subsection {* Examples *} subsubsection {* Powerset of a set is a complete lattice *} theorem powerset_is_complete_lattice: "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)" (is "complete_lattice ?L") proof (rule partial_order.complete_latticeI) show "partial_order ?L" by (rule partial_order.intro) auto next fix B assume "B \<subseteq> carrier ?L" then have "least ?L (\<Union> B) (Upper ?L B)" by (fastsimp intro!: least_UpperI simp: Upper_def) then show "EX s. least ?L s (Upper ?L B)" .. next fix B assume "B \<subseteq> carrier ?L" then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)" txt {* @{term "\<Inter> B"} is not the infimum of @{term B}: @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *} by (fastsimp intro!: greatest_LowerI simp: Lower_def) then show "EX i. greatest ?L i (Lower ?L B)" .. qed text {* An other example, that of the lattice of subgroups of a group, can be found in Group theory (Section~\ref{sec:subgroup-lattice}). *} end