(* Title: HOL/Algebra/Lattice.thy Author: Clemens Ballarin, started 7 November 2003 Copyright: Clemens Ballarin Most congruence rules by Stephan Hohe. *) theory Lattice imports Congruence begin section {* Orders and Lattices *} subsection {* Partial Orders *} record 'a gorder = "'a eq_object" + le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50) locale weak_partial_order = equivalence L for L (structure) + assumes le_refl [intro, simp]: "x ∈ carrier L ==> x \<sqsubseteq> x" and weak_le_antisym [intro]: "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x ∈ carrier L; y ∈ carrier L |] ==> x .= y" and le_trans [trans]: "[| x \<sqsubseteq> y; y \<sqsubseteq> z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x \<sqsubseteq> z" and le_cong: "[| x .= y; z .= w; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L; w ∈ carrier L |] ==> x \<sqsubseteq> z <-> y \<sqsubseteq> w" definition lless :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50) where "x \<sqsubset>⇘_{L⇙}y <-> x \<sqsubseteq>⇘_{L⇙}y & x .≠⇘_{L⇙}y" subsubsection {* The order relation *} context weak_partial_order begin lemma le_cong_l [intro, trans]: "[| x .= y; y \<sqsubseteq> z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x \<sqsubseteq> z" by (auto intro: le_cong [THEN iffD2]) lemma le_cong_r [intro, trans]: "[| x \<sqsubseteq> y; y .= z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x \<sqsubseteq> z" by (auto intro: le_cong [THEN iffD1]) lemma weak_refl [intro, simp]: "[| x .= y; x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> y" by (simp add: le_cong_l) end lemma weak_llessI: fixes R (structure) assumes "x \<sqsubseteq> y" and "~(x .= y)" shows "x \<sqsubset> y" using assms unfolding lless_def by simp lemma lless_imp_le: fixes R (structure) assumes "x \<sqsubset> y" shows "x \<sqsubseteq> y" using assms unfolding lless_def by simp lemma weak_lless_imp_not_eq: fixes R (structure) assumes "x \<sqsubset> y" shows "¬ (x .= y)" using assms unfolding lless_def by simp lemma weak_llessE: fixes R (structure) assumes p: "x \<sqsubset> y" and e: "[|x \<sqsubseteq> y; ¬ (x .= y)|] ==> P" shows "P" using p by (blast dest: lless_imp_le weak_lless_imp_not_eq e) lemma (in weak_partial_order) lless_cong_l [trans]: assumes xx': "x .= x'" and xy: "x' \<sqsubset> y" and carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L" shows "x \<sqsubset> y" using assms unfolding lless_def by (auto intro: trans sym) lemma (in weak_partial_order) lless_cong_r [trans]: assumes xy: "x \<sqsubset> y" and yy': "y .= y'" and carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L" shows "x \<sqsubset> y'" using assms unfolding lless_def by (auto intro: trans sym) (*slow*) lemma (in weak_partial_order) lless_antisym: assumes "a ∈ carrier L" "b ∈ carrier L" and "a \<sqsubset> b" "b \<sqsubset> a" shows "P" using assms by (elim weak_llessE) auto lemma (in weak_partial_order) lless_trans [trans]: assumes "a \<sqsubset> b" "b \<sqsubset> c" and carr[simp]: "a ∈ carrier L" "b ∈ carrier L" "c ∈ carrier L" shows "a \<sqsubset> c" using assms unfolding lless_def by (blast dest: le_trans intro: sym) subsubsection {* Upper and lower bounds of a set *} definition Upper :: "[_, 'a set] => 'a set" where "Upper L A = {u. (ALL x. x ∈ A ∩ carrier L --> x \<sqsubseteq>⇘_{L⇙}u)} ∩ carrier L" definition Lower :: "[_, 'a set] => 'a set" where "Lower L A = {l. (ALL x. x ∈ A ∩ carrier L --> l \<sqsubseteq>⇘_{L⇙}x)} ∩ carrier L" lemma Upper_closed [intro!, simp]: "Upper L A ⊆ carrier L" by (unfold Upper_def) clarify lemma Upper_memD [dest]: fixes L (structure) shows "[| u ∈ Upper L A; x ∈ A; A ⊆ carrier L |] ==> x \<sqsubseteq> u ∧ u ∈ carrier L" by (unfold Upper_def) blast lemma (in weak_partial_order) Upper_elemD [dest]: "[| u .∈ Upper L A; u ∈ carrier L; x ∈ A; A ⊆ carrier L |] ==> x \<sqsubseteq> u" unfolding Upper_def elem_def by (blast dest: sym) lemma Upper_memI: fixes L (structure) shows "[| !! y. y ∈ A ==> y \<sqsubseteq> x; x ∈ carrier L |] ==> x ∈ Upper L A" by (unfold Upper_def) blast lemma (in weak_partial_order) Upper_elemI: "[| !! y. y ∈ A ==> y \<sqsubseteq> x; x ∈ carrier L |] ==> x .∈ Upper L A" unfolding Upper_def by blast lemma Upper_antimono: "A ⊆ B ==> Upper L B ⊆ Upper L A" by (unfold Upper_def) blast lemma (in weak_partial_order) Upper_is_closed [simp]: "A ⊆ carrier L ==> is_closed (Upper L A)" by (rule is_closedI) (blast intro: Upper_memI)+ lemma (in weak_partial_order) Upper_mem_cong: assumes a'carr: "a' ∈ carrier L" and Acarr: "A ⊆ carrier L" and aa': "a .= a'" and aelem: "a ∈ Upper L A" shows "a' ∈ Upper L A" proof (rule Upper_memI[OF _ a'carr]) fix y assume yA: "y ∈ A" hence "y \<sqsubseteq> a" by (intro Upper_memD[OF aelem, THEN conjunct1] Acarr) also note aa' finally show "y \<sqsubseteq> a'" by (simp add: a'carr subsetD[OF Acarr yA] subsetD[OF Upper_closed aelem]) qed lemma (in weak_partial_order) Upper_cong: assumes Acarr: "A ⊆ carrier L" and A'carr: "A' ⊆ carrier L" and AA': "A {.=} A'" shows "Upper L A = Upper L A'" unfolding Upper_def apply rule apply (rule, clarsimp) defer 1 apply (rule, clarsimp) defer 1 proof - fix x a' assume carr: "x ∈ carrier L" "a' ∈ carrier L" and a'A': "a' ∈ A'" assume aLxCond[rule_format]: "∀a. a ∈ A ∧ a ∈ carrier L --> a \<sqsubseteq> x" from AA' and a'A' have "∃a∈A. a' .= a" by (rule set_eqD2) from this obtain a where aA: "a ∈ A" and a'a: "a' .= a" by auto note [simp] = subsetD[OF Acarr aA] carr note a'a also have "a \<sqsubseteq> x" by (simp add: aLxCond aA) finally show "a' \<sqsubseteq> x" by simp next fix x a assume carr: "x ∈ carrier L" "a ∈ carrier L" and aA: "a ∈ A" assume a'LxCond[rule_format]: "∀a'. a' ∈ A' ∧ a' ∈ carrier L --> a' \<sqsubseteq> x" from AA' and aA have "∃a'∈A'. a .= a'" by (rule set_eqD1) from this obtain a' where a'A': "a' ∈ A'" and aa': "a .= a'" by auto note [simp] = subsetD[OF A'carr a'A'] carr note aa' also have "a' \<sqsubseteq> x" by (simp add: a'LxCond a'A') finally show "a \<sqsubseteq> x" by simp qed lemma Lower_closed [intro!, simp]: "Lower L A ⊆ carrier L" by (unfold Lower_def) clarify lemma Lower_memD [dest]: fixes L (structure) shows "[| l ∈ Lower L A; x ∈ A; A ⊆ carrier L |] ==> l \<sqsubseteq> x ∧ l ∈ carrier L" by (unfold Lower_def) blast lemma Lower_memI: fixes L (structure) shows "[| !! y. y ∈ A ==> x \<sqsubseteq> y; x ∈ carrier L |] ==> x ∈ Lower L A" by (unfold Lower_def) blast lemma Lower_antimono: "A ⊆ B ==> Lower L B ⊆ Lower L A" by (unfold Lower_def) blast lemma (in weak_partial_order) Lower_is_closed [simp]: "A ⊆ carrier L ==> is_closed (Lower L A)" by (rule is_closedI) (blast intro: Lower_memI dest: sym)+ lemma (in weak_partial_order) Lower_mem_cong: assumes a'carr: "a' ∈ carrier L" and Acarr: "A ⊆ carrier L" and aa': "a .= a'" and aelem: "a ∈ Lower L A" shows "a' ∈ Lower L A" using assms Lower_closed[of L A] by (intro Lower_memI) (blast intro: le_cong_l[OF aa'[symmetric]]) lemma (in weak_partial_order) Lower_cong: assumes Acarr: "A ⊆ carrier L" and A'carr: "A' ⊆ carrier L" and AA': "A {.=} A'" shows "Lower L A = Lower L A'" unfolding Lower_def apply rule apply clarsimp defer 1 apply clarsimp defer 1 proof - fix x a' assume carr: "x ∈ carrier L" "a' ∈ carrier L" and a'A': "a' ∈ A'" assume "∀a. a ∈ A ∧ a ∈ carrier L --> x \<sqsubseteq> a" hence aLxCond: "!!a. [|a ∈ A; a ∈ carrier L|] ==> x \<sqsubseteq> a" by fast from AA' and a'A' have "∃a∈A. a' .= a" by (rule set_eqD2) from this obtain a where aA: "a ∈ A" and a'a: "a' .= a" by auto from aA and subsetD[OF Acarr aA] have "x \<sqsubseteq> a" by (rule aLxCond) also note a'a[symmetric] finally show "x \<sqsubseteq> a'" by (simp add: carr subsetD[OF Acarr aA]) next fix x a assume carr: "x ∈ carrier L" "a ∈ carrier L" and aA: "a ∈ A" assume "∀a'. a' ∈ A' ∧ a' ∈ carrier L --> x \<sqsubseteq> a'" hence a'LxCond: "!!a'. [|a' ∈ A'; a' ∈ carrier L|] ==> x \<sqsubseteq> a'" by fast+ from AA' and aA have "∃a'∈A'. a .= a'" by (rule set_eqD1) from this obtain a' where a'A': "a' ∈ A'" and aa': "a .= a'" by auto from a'A' and subsetD[OF A'carr a'A'] have "x \<sqsubseteq> a'" by (rule a'LxCond) also note aa'[symmetric] finally show "x \<sqsubseteq> a" by (simp add: carr subsetD[OF A'carr a'A']) qed subsubsection {* Least and greatest, as predicate *} definition least :: "[_, 'a, 'a set] => bool" where "least L l A <-> A ⊆ carrier L & l ∈ A & (ALL x : A. l \<sqsubseteq>⇘_{L⇙}x)" definition greatest :: "[_, 'a, 'a set] => bool" where "greatest L g A <-> A ⊆ carrier L & g ∈ A & (ALL x : A. x \<sqsubseteq>⇘_{L⇙}g)" text (in weak_partial_order) {* Could weaken these to @{term "l ∈ carrier L ∧ l .∈ A"} and @{term "g ∈ carrier L ∧ g .∈ A"}. *} lemma least_closed [intro, simp]: "least L l A ==> l ∈ carrier L" by (unfold least_def) fast lemma least_mem: "least L l A ==> l ∈ A" by (unfold least_def) fast lemma (in weak_partial_order) weak_least_unique: "[| least L x A; least L y A |] ==> x .= y" by (unfold least_def) blast lemma least_le: fixes L (structure) shows "[| least L x A; a ∈ A |] ==> x \<sqsubseteq> a" by (unfold least_def) fast lemma (in weak_partial_order) least_cong: "[| x .= x'; x ∈ carrier L; x' ∈ carrier L; is_closed A |] ==> least L x A = least L x' A" by (unfold least_def) (auto dest: sym) text (in weak_partial_order) {* @{const least} is not congruent in the second parameter for @{term "A {.=} A'"} *} lemma (in weak_partial_order) least_Upper_cong_l: assumes "x .= x'" and "x ∈ carrier L" "x' ∈ carrier L" and "A ⊆ carrier L" shows "least L x (Upper L A) = least L x' (Upper L A)" apply (rule least_cong) using assms by auto lemma (in weak_partial_order) least_Upper_cong_r: assumes Acarrs: "A ⊆ carrier L" "A' ⊆ carrier L" (* unneccessary with current Upper? *) and AA': "A {.=} A'" shows "least L x (Upper L A) = least L x (Upper L A')" apply (subgoal_tac "Upper L A = Upper L A'", simp) by (rule Upper_cong) fact+ lemma least_UpperI: fixes L (structure) assumes above: "!! x. x ∈ A ==> x \<sqsubseteq> s" and below: "!! y. y ∈ Upper L A ==> s \<sqsubseteq> y" and L: "A ⊆ carrier L" "s ∈ carrier L" shows "least L s (Upper L A)" proof - have "Upper L A ⊆ carrier L" by simp moreover from above L have "s ∈ 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 lemma least_Upper_above: fixes L (structure) shows "[| least L s (Upper L A); x ∈ A; A ⊆ carrier L |] ==> x \<sqsubseteq> s" by (unfold least_def) blast lemma greatest_closed [intro, simp]: "greatest L l A ==> l ∈ carrier L" by (unfold greatest_def) fast lemma greatest_mem: "greatest L l A ==> l ∈ A" by (unfold greatest_def) fast lemma (in weak_partial_order) weak_greatest_unique: "[| greatest L x A; greatest L y A |] ==> x .= y" by (unfold greatest_def) blast lemma greatest_le: fixes L (structure) shows "[| greatest L x A; a ∈ A |] ==> a \<sqsubseteq> x" by (unfold greatest_def) fast lemma (in weak_partial_order) greatest_cong: "[| x .= x'; x ∈ carrier L; x' ∈ carrier L; is_closed A |] ==> greatest L x A = greatest L x' A" by (unfold greatest_def) (auto dest: sym) text (in weak_partial_order) {* @{const greatest} is not congruent in the second parameter for @{term "A {.=} A'"} *} lemma (in weak_partial_order) greatest_Lower_cong_l: assumes "x .= x'" and "x ∈ carrier L" "x' ∈ carrier L" and "A ⊆ carrier L" (* unneccessary with current Lower *) shows "greatest L x (Lower L A) = greatest L x' (Lower L A)" apply (rule greatest_cong) using assms by auto lemma (in weak_partial_order) greatest_Lower_cong_r: assumes Acarrs: "A ⊆ carrier L" "A' ⊆ carrier L" and AA': "A {.=} A'" shows "greatest L x (Lower L A) = greatest L x (Lower L A')" apply (subgoal_tac "Lower L A = Lower L A'", simp) by (rule Lower_cong) fact+ lemma greatest_LowerI: fixes L (structure) assumes below: "!! x. x ∈ A ==> i \<sqsubseteq> x" and above: "!! y. y ∈ Lower L A ==> y \<sqsubseteq> i" and L: "A ⊆ carrier L" "i ∈ carrier L" shows "greatest L i (Lower L A)" proof - have "Lower L A ⊆ carrier L" by simp moreover from below L have "i ∈ 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 lemma greatest_Lower_below: fixes L (structure) shows "[| greatest L i (Lower L A); x ∈ A; A ⊆ carrier L |] ==> i \<sqsubseteq> x" by (unfold greatest_def) blast text {* Supremum and infimum *} definition sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90) where "\<Squnion>⇘_{L⇙}A = (SOME x. least L x (Upper L A))" definition inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90) where "\<Sqinter>⇘_{L⇙}A = (SOME x. greatest L x (Lower L A))" definition join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65) where "x \<squnion>⇘_{L⇙}y = \<Squnion>⇘_{L⇙}{x, y}" definition meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 70) where "x \<sqinter>⇘_{L⇙}y = \<Sqinter>⇘_{L⇙}{x, y}" subsection {* Lattices *} locale weak_upper_semilattice = weak_partial_order + assumes sup_of_two_exists: "[| x ∈ carrier L; y ∈ carrier L |] ==> EX s. least L s (Upper L {x, y})" locale weak_lower_semilattice = weak_partial_order + assumes inf_of_two_exists: "[| x ∈ carrier L; y ∈ carrier L |] ==> EX s. greatest L s (Lower L {x, y})" locale weak_lattice = weak_upper_semilattice + weak_lower_semilattice subsubsection {* Supremum *} lemma (in weak_upper_semilattice) joinI: "[| !!l. least L l (Upper L {x, y}) ==> P l; x ∈ carrier L; y ∈ carrier L |] ==> P (x \<squnion> y)" proof (unfold join_def sup_def) assume L: "x ∈ carrier L" "y ∈ 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 (SOME l. least L l (Upper L {x, y}))" by (fast intro: someI2 P) qed lemma (in weak_upper_semilattice) join_closed [simp]: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<squnion> y ∈ carrier L" by (rule joinI) (rule least_closed) lemma (in weak_upper_semilattice) join_cong_l: assumes carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L" and xx': "x .= x'" shows "x \<squnion> y .= x' \<squnion> y" proof (rule joinI, rule joinI) fix a b from xx' carr have seq: "{x, y} {.=} {x', y}" by (rule set_eq_pairI) assume leasta: "least L a (Upper L {x, y})" assume "least L b (Upper L {x', y})" with carr have leastb: "least L b (Upper L {x, y})" by (simp add: least_Upper_cong_r[OF _ _ seq]) from leasta leastb show "a .= b" by (rule weak_least_unique) qed (rule carr)+ lemma (in weak_upper_semilattice) join_cong_r: assumes carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L" and yy': "y .= y'" shows "x \<squnion> y .= x \<squnion> y'" proof (rule joinI, rule joinI) fix a b have "{x, y} = {y, x}" by fast also from carr yy' have "{y, x} {.=} {y', x}" by (intro set_eq_pairI) also have "{y', x} = {x, y'}" by fast finally have seq: "{x, y} {.=} {x, y'}" . assume leasta: "least L a (Upper L {x, y})" assume "least L b (Upper L {x, y'})" with carr have leastb: "least L b (Upper L {x, y})" by (simp add: least_Upper_cong_r[OF _ _ seq]) from leasta leastb show "a .= b" by (rule weak_least_unique) qed (rule carr)+ lemma (in weak_partial_order) sup_of_singletonI: (* only reflexivity needed ? *) "x ∈ carrier L ==> least L x (Upper L {x})" by (rule least_UpperI) auto lemma (in weak_partial_order) weak_sup_of_singleton [simp]: "x ∈ carrier L ==> \<Squnion>{x} .= x" unfolding sup_def by (rule someI2) (auto intro: weak_least_unique sup_of_singletonI) lemma (in weak_partial_order) sup_of_singleton_closed [simp]: "x ∈ carrier L ==> \<Squnion>{x} ∈ carrier L" unfolding sup_def by (rule someI2) (auto intro: sup_of_singletonI) text {* Condition on @{text A}: supremum exists. *} lemma (in weak_upper_semilattice) sup_insertI: "[| !!s. least L s (Upper L (insert x A)) ==> P s; least L a (Upper L A); x ∈ carrier L; A ⊆ carrier L |] ==> P (\<Squnion>(insert x A))" proof (unfold sup_def) assume L: "x ∈ carrier L" "A ⊆ 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 ∈ 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 (SOME l. least L l (Upper L (insert x A)))" proof (rule someI2) show "least L s (Upper L (insert x A))" proof (rule least_UpperI) fix z assume "z ∈ 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 ∈ A" with L least_s least_a show ?thesis by (rule_tac le_trans [where y = a]) (auto dest: least_Upper_above) qed next fix y assume y: "y ∈ Upper L (insert x A)" show "s \<sqsubseteq> y" proof (rule least_le [OF least_s], rule Upper_memI) fix z assume z: "z ∈ {a, x}" then show "z \<sqsubseteq> y" proof have y': "y ∈ Upper L A" apply (rule subsetD [where A = "Upper L (insert x A)"]) apply (rule Upper_antimono) apply blast apply (rule y) done assume "z = a" with y' least_a show ?thesis by (fast dest: least_le) next assume "z ∈ {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, OF y]) next from L show "insert x A ⊆ carrier L" by simp from least_s show "s ∈ carrier L" by simp qed qed (rule P) qed lemma (in weak_upper_semilattice) finite_sup_least: "[| finite A; A ⊆ carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)" proof (induct set: finite) 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 (simp add: least_cong [OF weak_sup_of_singleton] sup_of_singletonI) (* The above step is hairy; least_cong can make simp loop. Would want special version of simp to apply least_cong. *) 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 weak_upper_semilattice) finite_sup_insertI: assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l" and xA: "finite A" "x ∈ carrier L" "A ⊆ carrier L" shows "P (\<Squnion> (insert x A))" proof (cases "A = {}") case True with P and xA show ?thesis by (simp add: finite_sup_least) next case False with P and xA show ?thesis by (simp add: sup_insertI finite_sup_least) qed lemma (in weak_upper_semilattice) finite_sup_closed [simp]: "[| finite A; A ⊆ carrier L; A ~= {} |] ==> \<Squnion>A ∈ carrier L" proof (induct set: finite) case empty then show ?case by simp next case insert then show ?case by - (rule finite_sup_insertI, simp_all) qed lemma (in weak_upper_semilattice) join_left: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> x \<squnion> y" by (rule joinI [folded join_def]) (blast dest: least_mem) lemma (in weak_upper_semilattice) join_right: "[| x ∈ carrier L; y ∈ carrier L |] ==> y \<sqsubseteq> x \<squnion> y" by (rule joinI [folded join_def]) (blast dest: least_mem) lemma (in weak_upper_semilattice) sup_of_two_least: "[| x ∈ carrier L; y ∈ carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})" proof (unfold sup_def) assume L: "x ∈ carrier L" "y ∈ carrier L" with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast with L show "least L (SOME z. least L z (Upper L {x, y})) (Upper L {x, y})" by (fast intro: someI2 weak_least_unique) (* blast fails *) qed lemma (in weak_upper_semilattice) join_le: assumes sub: "x \<sqsubseteq> z" "y \<sqsubseteq> z" and x: "x ∈ carrier L" and y: "y ∈ carrier L" and z: "z ∈ carrier L" shows "x \<squnion> y \<sqsubseteq> z" proof (rule joinI [OF _ x y]) fix s assume "least L s (Upper L {x, y})" with sub z show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI) qed lemma (in weak_upper_semilattice) weak_join_assoc_lemma: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ 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 weak_le_antisym) from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s" by (fastforce 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: le_trans join_left join_right join_closed) qed (simp_all add: L least_closed [OF sup]) qed (simp_all add: L) text {* Commutativity holds for @{text "="}. *} lemma join_comm: fixes L (structure) shows "x \<squnion> y = y \<squnion> x" by (unfold join_def) (simp add: insert_commute) lemma (in weak_upper_semilattice) weak_join_assoc: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L" shows "(x \<squnion> y) \<squnion> z .= x \<squnion> (y \<squnion> z)" proof - (* FIXME: could be simplified by improved simp: uniform use of .=, omit [symmetric] in last step. *) 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: weak_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: weak_join_assoc_lemma [symmetric]) finally show ?thesis by (simp add: L) qed subsubsection {* Infimum *} lemma (in weak_lower_semilattice) meetI: "[| !!i. greatest L i (Lower L {x, y}) ==> P i; x ∈ carrier L; y ∈ carrier L |] ==> P (x \<sqinter> y)" proof (unfold meet_def inf_def) assume L: "x ∈ carrier L" "y ∈ 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 (SOME g. greatest L g (Lower L {x, y}))" by (fast intro: someI2 weak_greatest_unique P) qed lemma (in weak_lower_semilattice) meet_closed [simp]: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqinter> y ∈ carrier L" by (rule meetI) (rule greatest_closed) lemma (in weak_lower_semilattice) meet_cong_l: assumes carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L" and xx': "x .= x'" shows "x \<sqinter> y .= x' \<sqinter> y" proof (rule meetI, rule meetI) fix a b from xx' carr have seq: "{x, y} {.=} {x', y}" by (rule set_eq_pairI) assume greatesta: "greatest L a (Lower L {x, y})" assume "greatest L b (Lower L {x', y})" with carr have greatestb: "greatest L b (Lower L {x, y})" by (simp add: greatest_Lower_cong_r[OF _ _ seq]) from greatesta greatestb show "a .= b" by (rule weak_greatest_unique) qed (rule carr)+ lemma (in weak_lower_semilattice) meet_cong_r: assumes carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L" and yy': "y .= y'" shows "x \<sqinter> y .= x \<sqinter> y'" proof (rule meetI, rule meetI) fix a b have "{x, y} = {y, x}" by fast also from carr yy' have "{y, x} {.=} {y', x}" by (intro set_eq_pairI) also have "{y', x} = {x, y'}" by fast finally have seq: "{x, y} {.=} {x, y'}" . assume greatesta: "greatest L a (Lower L {x, y})" assume "greatest L b (Lower L {x, y'})" with carr have greatestb: "greatest L b (Lower L {x, y})" by (simp add: greatest_Lower_cong_r[OF _ _ seq]) from greatesta greatestb show "a .= b" by (rule weak_greatest_unique) qed (rule carr)+ lemma (in weak_partial_order) inf_of_singletonI: (* only reflexivity needed ? *) "x ∈ carrier L ==> greatest L x (Lower L {x})" by (rule greatest_LowerI) auto lemma (in weak_partial_order) weak_inf_of_singleton [simp]: "x ∈ carrier L ==> \<Sqinter>{x} .= x" unfolding inf_def by (rule someI2) (auto intro: weak_greatest_unique inf_of_singletonI) lemma (in weak_partial_order) inf_of_singleton_closed: "x ∈ carrier L ==> \<Sqinter>{x} ∈ carrier L" unfolding inf_def by (rule someI2) (auto intro: inf_of_singletonI) text {* Condition on @{text A}: infimum exists. *} lemma (in weak_lower_semilattice) inf_insertI: "[| !!i. greatest L i (Lower L (insert x A)) ==> P i; greatest L a (Lower L A); x ∈ carrier L; A ⊆ carrier L |] ==> P (\<Sqinter>(insert x A))" proof (unfold inf_def) assume L: "x ∈ carrier L" "A ⊆ 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 ∈ 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 (SOME g. greatest L g (Lower L (insert x A)))" proof (rule someI2) show "greatest L i (Lower L (insert x A))" proof (rule greatest_LowerI) fix z assume "z ∈ insert x A" then show "i \<sqsubseteq> z" proof assume "z = x" then show ?thesis by (simp add: greatest_Lower_below [OF greatest_i] L La) next assume "z ∈ A" with L greatest_i greatest_a show ?thesis by (rule_tac le_trans [where y = a]) (auto dest: greatest_Lower_below) qed next fix y assume y: "y ∈ Lower L (insert x A)" show "y \<sqsubseteq> i" proof (rule greatest_le [OF greatest_i], rule Lower_memI) fix z assume z: "z ∈ {a, x}" then show "y \<sqsubseteq> z" proof have y': "y ∈ Lower L A" apply (rule subsetD [where A = "Lower L (insert x A)"]) apply (rule Lower_antimono) apply blast apply (rule y) done assume "z = a" with y' greatest_a show ?thesis by (fast dest: greatest_le) next assume "z ∈ {x}" with y L show ?thesis by blast qed qed (rule Lower_closed [THEN subsetD, OF y]) next from L show "insert x A ⊆ carrier L" by simp from greatest_i show "i ∈ carrier L" by simp qed qed (rule P) qed lemma (in weak_lower_semilattice) finite_inf_greatest: "[| finite A; A ⊆ carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)" proof (induct set: finite) 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 (simp add: greatest_cong [OF weak_inf_of_singleton] inf_of_singleton_closed 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 weak_lower_semilattice) finite_inf_insertI: assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i" and xA: "finite A" "x ∈ carrier L" "A ⊆ carrier L" shows "P (\<Sqinter> (insert x A))" proof (cases "A = {}") case True with P and xA show ?thesis by (simp add: finite_inf_greatest) next case False with P and xA show ?thesis by (simp add: inf_insertI finite_inf_greatest) qed lemma (in weak_lower_semilattice) finite_inf_closed [simp]: "[| finite A; A ⊆ carrier L; A ~= {} |] ==> \<Sqinter>A ∈ carrier L" proof (induct set: finite) case empty then show ?case by simp next case insert then show ?case by (rule_tac finite_inf_insertI) (simp_all) qed lemma (in weak_lower_semilattice) meet_left: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqinter> y \<sqsubseteq> x" by (rule meetI [folded meet_def]) (blast dest: greatest_mem) lemma (in weak_lower_semilattice) meet_right: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqinter> y \<sqsubseteq> y" by (rule meetI [folded meet_def]) (blast dest: greatest_mem) lemma (in weak_lower_semilattice) inf_of_two_greatest: "[| x ∈ carrier L; y ∈ carrier L |] ==> greatest L (\<Sqinter> {x, y}) (Lower L {x, y})" proof (unfold inf_def) assume L: "x ∈ carrier L" "y ∈ carrier L" with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast with L show "greatest L (SOME z. greatest L z (Lower L {x, y})) (Lower L {x, y})" by (fast intro: someI2 weak_greatest_unique) (* blast fails *) qed lemma (in weak_lower_semilattice) meet_le: assumes sub: "z \<sqsubseteq> x" "z \<sqsubseteq> y" and x: "x ∈ carrier L" and y: "y ∈ carrier L" and z: "z ∈ carrier L" shows "z \<sqsubseteq> x \<sqinter> y" proof (rule meetI [OF _ x y]) fix i assume "greatest L i (Lower L {x, y})" with sub z show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI) qed lemma (in weak_lower_semilattice) weak_meet_assoc_lemma: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ 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 weak_le_antisym) from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)" by (fastforce intro!: meet_le elim: greatest_Lower_below) next from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i" by (erule_tac greatest_le) (blast intro!: Lower_memI intro: le_trans meet_left meet_right meet_closed) qed (simp_all add: L greatest_closed [OF inf]) qed (simp_all add: L) lemma meet_comm: fixes L (structure) shows "x \<sqinter> y = y \<sqinter> x" by (unfold meet_def) (simp add: insert_commute) lemma (in weak_lower_semilattice) weak_meet_assoc: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L" shows "(x \<sqinter> y) \<sqinter> z .= x \<sqinter> (y \<sqinter> z)" proof - (* FIXME: improved simp, see weak_join_assoc above *) 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: weak_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: weak_meet_assoc_lemma [symmetric]) finally show ?thesis by (simp add: L) qed subsection {* Total Orders *} locale weak_total_order = weak_partial_order + assumes total: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" text {* Introduction rule: the usual definition of total order *} lemma (in weak_partial_order) weak_total_orderI: assumes total: "!!x y. [| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" shows "weak_total_order L" by default (rule total) text {* Total orders are lattices. *} sublocale weak_total_order < weak: weak_lattice proof fix x y assume L: "x ∈ carrier L" "y ∈ 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 ∈ carrier L" "y ∈ 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 subsection {* Complete Lattices *} locale weak_complete_lattice = weak_lattice + assumes sup_exists: "[| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)" and inf_exists: "[| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)" text {* Introduction rule: the usual definition of complete lattice *} lemma (in weak_partial_order) weak_complete_latticeI: assumes sup_exists: "!!A. [| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)" and inf_exists: "!!A. [| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)" shows "weak_complete_lattice L" by default (auto intro: sup_exists inf_exists) definition top :: "_ => 'a" ("\<top>\<index>") where "\<top>⇘_{L⇙}= sup L (carrier L)" definition bottom :: "_ => 'a" ("⊥\<index>") where "⊥⇘_{L⇙}= inf L (carrier L)" lemma (in weak_complete_lattice) supI: "[| !!l. least L l (Upper L A) ==> P l; A ⊆ carrier L |] ==> P (\<Squnion>A)" proof (unfold sup_def) assume L: "A ⊆ 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 (SOME l. least L l (Upper L A))" by (fast intro: someI2 weak_least_unique P) qed lemma (in weak_complete_lattice) sup_closed [simp]: "A ⊆ carrier L ==> \<Squnion>A ∈ carrier L" by (rule supI) simp_all lemma (in weak_complete_lattice) top_closed [simp, intro]: "\<top> ∈ carrier L" by (unfold top_def) simp lemma (in weak_complete_lattice) infI: "[| !!i. greatest L i (Lower L A) ==> P i; A ⊆ carrier L |] ==> P (\<Sqinter>A)" proof (unfold inf_def) assume L: "A ⊆ 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 (SOME l. greatest L l (Lower L A))" by (fast intro: someI2 weak_greatest_unique P) qed lemma (in weak_complete_lattice) inf_closed [simp]: "A ⊆ carrier L ==> \<Sqinter>A ∈ carrier L" by (rule infI) simp_all lemma (in weak_complete_lattice) bottom_closed [simp, intro]: "⊥ ∈ 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 weak_partial_order) weak_complete_lattice_criterion1: assumes top_exists: "EX g. greatest L g (carrier L)" and inf_exists: "!!A. [| A ⊆ carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)" shows "weak_complete_lattice L" proof (rule weak_complete_latticeI) from top_exists obtain top where top: "greatest L top (carrier L)" .. fix A assume L: "A ⊆ carrier L" let ?B = "Upper L A" from L top have "top ∈ ?B" by (fast intro!: Upper_memI intro: greatest_le) then have B_non_empty: "?B ~= {}" by fast have B_L: "?B ⊆ 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 Upper_memD [THEN conjunct1]) apply assumption apply (rule L) apply (fast intro: L [THEN subsetD]) apply (erule greatest_Lower_below [OF b_inf_B]) apply simp apply (rule L) apply (rule greatest_closed [OF b_inf_B]) done then show "EX s. least L s (Upper L A)" .. next fix A assume L: "A ⊆ 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 {* Orders and Lattices where @{text eq} is the Equality *} locale partial_order = weak_partial_order + assumes eq_is_equal: "op .= = op =" begin declare weak_le_antisym [rule del] lemma le_antisym [intro]: "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x ∈ carrier L; y ∈ carrier L |] ==> x = y" using weak_le_antisym unfolding eq_is_equal . lemma lless_eq: "x \<sqsubset> y <-> x \<sqsubseteq> y & x ≠ y" unfolding lless_def by (simp add: eq_is_equal) lemma lless_asym: assumes "a ∈ carrier L" "b ∈ carrier L" and "a \<sqsubset> b" "b \<sqsubset> a" shows "P" using assms unfolding lless_eq by auto end text {* Least and greatest, as predicate *} lemma (in partial_order) least_unique: "[| least L x A; least L y A |] ==> x = y" using weak_least_unique unfolding eq_is_equal . lemma (in partial_order) greatest_unique: "[| greatest L x A; greatest L y A |] ==> x = y" using weak_greatest_unique unfolding eq_is_equal . text {* Lattices *} locale upper_semilattice = partial_order + assumes sup_of_two_exists: "[| x ∈ carrier L; y ∈ carrier L |] ==> EX s. least L s (Upper L {x, y})" sublocale upper_semilattice < weak: weak_upper_semilattice by default (rule sup_of_two_exists) locale lower_semilattice = partial_order + assumes inf_of_two_exists: "[| x ∈ carrier L; y ∈ carrier L |] ==> EX s. greatest L s (Lower L {x, y})" sublocale lower_semilattice < weak: weak_lower_semilattice by default (rule inf_of_two_exists) locale lattice = upper_semilattice + lower_semilattice text {* Supremum *} declare (in partial_order) weak_sup_of_singleton [simp del] lemma (in partial_order) sup_of_singleton [simp]: "x ∈ carrier L ==> \<Squnion>{x} = x" using weak_sup_of_singleton unfolding eq_is_equal . lemma (in upper_semilattice) join_assoc_lemma: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L" shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}" using weak_join_assoc_lemma L unfolding eq_is_equal . lemma (in upper_semilattice) join_assoc: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L" shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" using weak_join_assoc L unfolding eq_is_equal . text {* Infimum *} declare (in partial_order) weak_inf_of_singleton [simp del] lemma (in partial_order) inf_of_singleton [simp]: "x ∈ carrier L ==> \<Sqinter>{x} = x" using weak_inf_of_singleton unfolding eq_is_equal . text {* Condition on @{text A}: infimum exists. *} lemma (in lower_semilattice) meet_assoc_lemma: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L" shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}" using weak_meet_assoc_lemma L unfolding eq_is_equal . lemma (in lower_semilattice) meet_assoc: assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L" shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" using weak_meet_assoc L unfolding eq_is_equal . text {* Total Orders *} locale total_order = partial_order + assumes total_order_total: "[| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" sublocale total_order < weak: weak_total_order by default (rule total_order_total) text {* Introduction rule: the usual definition of total order *} lemma (in partial_order) total_orderI: assumes total: "!!x y. [| x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" shows "total_order L" by default (rule total) text {* Total orders are lattices. *} sublocale total_order < weak: lattice by default (auto intro: sup_of_two_exists inf_of_two_exists) text {* Complete lattices *} locale complete_lattice = lattice + assumes sup_exists: "[| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)" and inf_exists: "[| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)" sublocale complete_lattice < weak: weak_complete_lattice by default (auto intro: sup_exists inf_exists) text {* Introduction rule: the usual definition of complete lattice *} lemma (in partial_order) complete_latticeI: assumes sup_exists: "!!A. [| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)" and inf_exists: "!!A. [| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)" shows "complete_lattice L" by default (auto intro: sup_exists inf_exists) theorem (in partial_order) complete_lattice_criterion1: assumes top_exists: "EX g. greatest L g (carrier L)" and inf_exists: "!!A. [| A ⊆ 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 ⊆ carrier L" let ?B = "Upper L A" from L top have "top ∈ ?B" by (fast intro!: Upper_memI intro: greatest_le) then have B_non_empty: "?B ~= {}" by fast have B_L: "?B ⊆ 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 Upper_memD [THEN conjunct1]) apply assumption apply (rule L) apply (fast intro: L [THEN subsetD]) apply (erule greatest_Lower_below [OF b_inf_B]) apply simp apply (rule L) apply (rule greatest_closed [OF b_inf_B]) done then show "EX s. least L s (Upper L A)" .. next fix A assume L: "A ⊆ 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 {* The Powerset of a Set is a Complete Lattice *} theorem powerset_is_complete_lattice: "complete_lattice (|carrier = Pow A, eq = op =, le = op ⊆|))," (is "complete_lattice ?L") proof (rule partial_order.complete_latticeI) show "partial_order ?L" by default auto next fix B assume "B ⊆ carrier ?L" then have "least ?L (\<Union> B) (Upper ?L B)" by (fastforce intro!: least_UpperI simp: Upper_def) then show "EX s. least ?L s (Upper ?L B)" .. next fix B assume "B ⊆ carrier ?L" then have "greatest ?L (\<Inter> B ∩ 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 (fastforce 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