(* Title: HOL/Algebra/Lattice.thy Author: Clemens Ballarin, started 7 November 2003 Copyright: Clemens Ballarin Most congruence rules by Stephan Hohe. With additional contributions from Alasdair Armstrong and Simon Foster. *) theory Lattice imports Order begin section \Lattices\ subsection \Supremum and infimum\ definition sup :: "[_, 'a set] => 'a" ("\\_" [90] 90) where "\\<^bsub>L\<^esub>A = (SOME x. least L x (Upper L A))" definition inf :: "[_, 'a set] => 'a" ("\\_" [90] 90) where "\\<^bsub>L\<^esub>A = (SOME x. greatest L x (Lower L A))" definition supr :: "('a, 'b) gorder_scheme \ 'c set \ ('c \ 'a) \ 'a " where "supr L A f = \\<^bsub>L\<^esub>(f ` A)" definition infi :: "('a, 'b) gorder_scheme \ 'c set \ ('c \ 'a) \ 'a " where "infi L A f = \\<^bsub>L\<^esub>(f ` A)" syntax "_inf1" :: "('a, 'b) gorder_scheme \ pttrns \ 'a \ 'a" ("(3IINF\ _./ _)" [0, 10] 10) "_inf" :: "('a, 'b) gorder_scheme \ pttrn \ 'c set \ 'a \ 'a" ("(3IINF\ _:_./ _)" [0, 0, 10] 10) "_sup1" :: "('a, 'b) gorder_scheme \ pttrns \ 'a \ 'a" ("(3SSUP\ _./ _)" [0, 10] 10) "_sup" :: "('a, 'b) gorder_scheme \ pttrn \ 'c set \ 'a \ 'a" ("(3SSUP\ _:_./ _)" [0, 0, 10] 10) translations "IINF\<^bsub>L\<^esub> x. B" == "CONST infi L CONST UNIV (%x. B)" "IINF\<^bsub>L\<^esub> x:A. B" == "CONST infi L A (%x. B)" "SSUP\<^bsub>L\<^esub> x. B" == "CONST supr L CONST UNIV (%x. B)" "SSUP\<^bsub>L\<^esub> x:A. B" == "CONST supr L A (%x. B)" definition join :: "[_, 'a, 'a] => 'a" (infixl "\\" 65) where "x \\<^bsub>L\<^esub> y = \\<^bsub>L\<^esub>{x, y}" definition meet :: "[_, 'a, 'a] => 'a" (infixl "\\" 70) where "x \\<^bsub>L\<^esub> y = \\<^bsub>L\<^esub>{x, y}" definition LEAST_FP :: "('a, 'b) gorder_scheme \ ('a \ 'a) \ 'a" ("LFP\") where "LEAST_FP L f = \\<^bsub>L\<^esub> {u \ carrier L. f u \\<^bsub>L\<^esub> u}" \ \least fixed point\ definition GREATEST_FP:: "('a, 'b) gorder_scheme \ ('a \ 'a) \ 'a" ("GFP\") where "GREATEST_FP L f = \\<^bsub>L\<^esub> {u \ carrier L. u \\<^bsub>L\<^esub> f u}" \ \greatest fixed point\ subsection \Dual operators\ lemma sup_dual [simp]: "\\<^bsub>inv_gorder L\<^esub>A = \\<^bsub>L\<^esub>A" by (simp add: sup_def inf_def) lemma inf_dual [simp]: "\\<^bsub>inv_gorder L\<^esub>A = \\<^bsub>L\<^esub>A" by (simp add: sup_def inf_def) lemma join_dual [simp]: "p \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q" by (simp add:join_def meet_def) lemma meet_dual [simp]: "p \\<^bsub>inv_gorder L\<^esub> q = p \\<^bsub>L\<^esub> q" by (simp add:join_def meet_def) lemma top_dual [simp]: "\\<^bsub>inv_gorder L\<^esub> = \\<^bsub>L\<^esub>" by (simp add: top_def bottom_def) lemma bottom_dual [simp]: "\\<^bsub>inv_gorder L\<^esub> = \\<^bsub>L\<^esub>" by (simp add: top_def bottom_def) lemma LFP_dual [simp]: "LEAST_FP (inv_gorder L) f = GREATEST_FP L f" by (simp add:LEAST_FP_def GREATEST_FP_def) lemma GFP_dual [simp]: "GREATEST_FP (inv_gorder L) f = LEAST_FP L f" by (simp add:LEAST_FP_def GREATEST_FP_def) subsection \Lattices\ locale weak_upper_semilattice = weak_partial_order + assumes sup_of_two_exists: "[| x \ carrier L; y \ carrier L |] ==> \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 |] ==> \s. greatest L s (Lower L {x, y})" locale weak_lattice = weak_upper_semilattice + weak_lower_semilattice lemma (in weak_lattice) dual_weak_lattice: "weak_lattice (inv_gorder L)" proof - interpret dual: weak_partial_order "inv_gorder L" by (metis dual_weak_order) show ?thesis proof qed (simp_all add: inf_of_two_exists sup_of_two_exists) qed 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 \ 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 \ 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 \ y .= x' \ 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 \ y .= x \ 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 ==> \{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 \ \{x} \ carrier L" unfolding sup_def by (rule someI2) (auto intro: sup_of_singletonI) text \Condition on \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 (\(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 \ 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 \ y" proof (rule least_le [OF least_s], rule Upper_memI) fix z assume z: "z \ {a, x}" then show "z \ y" proof have y': "y \ Upper L A" by (meson Upper_antimono in_mono subset_insertI y) assume "z = a" with y' least_a show ?thesis by (fast dest: least_le) next assume "z \ {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 (\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 (\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 (\ (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 \ {} |] ==> \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 \ x \ 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 \ x \ 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 (\{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 \ z" "y \ z" and x: "x \ carrier L" and y: "y \ carrier L" and z: "z \ carrier L" shows "x \ y \ z" proof (rule joinI [OF _ x y]) fix s assume "least L s (Upper L {x, y})" with sub z show "s \ z" by (fast elim: least_le intro: Upper_memI) qed lemma (in weak_lattice) weak_le_iff_meet: assumes "x \ carrier L" "y \ carrier L" shows "x \ y \ (x \ y) .= y" by (meson assms(1) assms(2) join_closed join_le join_left join_right le_cong_r local.le_refl weak_le_antisym) lemma (in weak_upper_semilattice) weak_join_assoc_lemma: assumes L: "x \ carrier L" "y \ carrier L" "z \ carrier L" shows "x \ (y \ z) .= \{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 \ (y \ z) .= s" proof (rule weak_le_antisym) from sup L show "x \ (y \ z) \ s" by (fastforce intro!: join_le elim: least_Upper_above) next from sup L show "s \ x \ (y \ 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 \=\.\ lemma join_comm: fixes L (structure) shows "x \ y = y \ 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 \ y) \ z .= x \ (y \ z)" proof - (* FIXME: could be simplified by improved simp: uniform use of .=, omit [symmetric] in last step. *) have "(x \ y) \ z = z \ (x \ y)" by (simp only: join_comm) also from L have "... .= \{z, x, y}" by (simp add: weak_join_assoc_lemma) also from L have "... = \{x, y, z}" by (simp add: insert_commute) also from L have "... .= x \ (y \ 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 \ 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 \ 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 \ y .= x' \ 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 \ y .= x \ 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 ==> \{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 ==> \{x} \ carrier L" unfolding inf_def by (rule someI2) (auto intro: inf_of_singletonI) text \Condition on \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 (\(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 \ 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 \ i" proof (rule greatest_le [OF greatest_i], rule Lower_memI) fix z assume z: "z \ {a, x}" then show "y \ z" proof have y': "y \ Lower L A" by (meson Lower_antimono in_mono subset_insertI y) 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 (\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 (\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 (\ (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 \ {} |] ==> \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 \ y \ 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 \ y \ 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 (\{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 \ x" "z \ y" and x: "x \ carrier L" and y: "y \ carrier L" and z: "z \ carrier L" shows "z \ x \ y" proof (rule meetI [OF _ x y]) fix i assume "greatest L i (Lower L {x, y})" with sub z show "z \ i" by (fast elim: greatest_le intro: Lower_memI) qed lemma (in weak_lattice) weak_le_iff_join: assumes "x \ carrier L" "y \ carrier L" shows "x \ y \ x .= (x \ y)" by (meson assms(1) assms(2) local.le_refl local.le_trans meet_closed meet_le meet_left meet_right weak_le_antisym weak_refl) lemma (in weak_lower_semilattice) weak_meet_assoc_lemma: assumes L: "x \ carrier L" "y \ carrier L" "z \ carrier L" shows "x \ (y \ z) .= \{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 \ (y \ z) .= i" proof (rule weak_le_antisym) from inf L show "i \ x \ (y \ z)" by (fastforce intro!: meet_le elim: greatest_Lower_below) next from inf L show "x \ (y \ z) \ 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 \ y = y \ 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 \ y) \ z .= x \ (y \ z)" proof - (* FIXME: improved simp, see weak_join_assoc above *) have "(x \ y) \ z = z \ (x \ y)" by (simp only: meet_comm) also from L have "... .= \ {z, x, y}" by (simp add: weak_meet_assoc_lemma) also from L have "... = \ {x, y, z}" by (simp add: insert_commute) also from L have "... .= x \ (y \ z)" by (simp add: weak_meet_assoc_lemma [symmetric]) finally show ?thesis by (simp add: L) qed 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 "\s. least L s (Upper L {x, y})" proof - note total L moreover { assume "x \ y" with L have "least L y (Upper L {x, y})" by (rule_tac least_UpperI) auto } moreover { assume "y \ 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 "\i. greatest L i (Lower L {x, y})" proof - note total L moreover { assume "y \ x" with L have "greatest L y (Lower L {x, y})" by (rule_tac greatest_LowerI) auto } moreover { assume "x \ 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 \Weak Bounded Lattices\ locale weak_bounded_lattice = weak_lattice + weak_partial_order_bottom + weak_partial_order_top begin lemma bottom_meet: "x \ carrier L \ \ \ x .= \" by (metis bottom_least least_def meet_closed meet_left weak_le_antisym) lemma bottom_join: "x \ carrier L \ \ \ x .= x" by (metis bottom_least join_closed join_le join_right le_refl least_def weak_le_antisym) lemma bottom_weak_eq: "\ b \ carrier L; \ x. x \ carrier L \ b \ x \ \ b .= \" by (metis bottom_closed bottom_lower weak_le_antisym) lemma top_join: "x \ carrier L \ \ \ x .= \" by (metis join_closed join_left top_closed top_higher weak_le_antisym) lemma top_meet: "x \ carrier L \ \ \ x .= x" by (metis le_refl meet_closed meet_le meet_right top_closed top_higher weak_le_antisym) lemma top_weak_eq: "\ t \ carrier L; \ x. x \ carrier L \ x \ t \ \ t .= \" by (metis top_closed top_higher weak_le_antisym) end sublocale weak_bounded_lattice \ weak_partial_order .. subsection \Lattices where \eq\ is the Equality\ locale upper_semilattice = partial_order + assumes sup_of_two_exists: "[| x \ carrier L; y \ carrier L |] ==> \s. least L s (Upper L {x, y})" sublocale upper_semilattice \ weak?: weak_upper_semilattice by unfold_locales (rule sup_of_two_exists) locale lower_semilattice = partial_order + assumes inf_of_two_exists: "[| x \ carrier L; y \ carrier L |] ==> \s. greatest L s (Lower L {x, y})" sublocale lower_semilattice \ weak?: weak_lower_semilattice by unfold_locales (rule inf_of_two_exists) locale lattice = upper_semilattice + lower_semilattice sublocale lattice \ weak_lattice .. lemma (in lattice) dual_lattice: "lattice (inv_gorder L)" proof - interpret dual: weak_lattice "inv_gorder L" by (metis dual_weak_lattice) show ?thesis apply (unfold_locales) apply (simp_all add: inf_of_two_exists sup_of_two_exists) apply (rule eq_is_equal) done qed lemma (in lattice) le_iff_join: assumes "x \ carrier L" "y \ carrier L" shows "x \ y \ x = (x \ y)" by (simp add: assms(1) assms(2) eq_is_equal weak_le_iff_join) lemma (in lattice) le_iff_meet: assumes "x \ carrier L" "y \ carrier L" shows "x \ y \ (x \ y) = y" by (simp add: assms eq_is_equal weak_le_iff_meet) text \ Total orders are lattices. \ sublocale total_order \ weak?: lattice by standard (auto intro: weak.weak.sup_of_two_exists weak.weak.inf_of_two_exists) text \Functions that preserve joins and meets\ definition join_pres :: "('a, 'c) gorder_scheme \ ('b, 'd) gorder_scheme \ ('a \ 'b) \ bool" where "join_pres X Y f \ lattice X \ lattice Y \ (\ x \ carrier X. \ y \ carrier X. f (x \\<^bsub>X\<^esub> y) = f x \\<^bsub>Y\<^esub> f y)" definition meet_pres :: "('a, 'c) gorder_scheme \ ('b, 'd) gorder_scheme \ ('a \ 'b) \ bool" where "meet_pres X Y f \ lattice X \ lattice Y \ (\ x \ carrier X. \ y \ carrier X. f (x \\<^bsub>X\<^esub> y) = f x \\<^bsub>Y\<^esub> f y)" lemma join_pres_isotone: assumes "f \ carrier X \ carrier Y" "join_pres X Y f" shows "isotone X Y f" proof (rule isotoneI) show "weak_partial_order X" "weak_partial_order Y" using assms unfolding join_pres_def lattice_def upper_semilattice_def lower_semilattice_def by (meson partial_order.axioms(1))+ show "\x y. \x \ carrier X; y \ carrier X; x \\<^bsub>X\<^esub> y\ \ f x \\<^bsub>Y\<^esub> f y" by (metis (no_types, lifting) PiE assms join_pres_def lattice.le_iff_meet) qed lemma meet_pres_isotone: assumes "f \ carrier X \ carrier Y" "meet_pres X Y f" shows "isotone X Y f" proof (rule isotoneI) show "weak_partial_order X" "weak_partial_order Y" using assms unfolding meet_pres_def lattice_def upper_semilattice_def lower_semilattice_def by (meson partial_order.axioms(1))+ show "\x y. \x \ carrier X; y \ carrier X; x \\<^bsub>X\<^esub> y\ \ f x \\<^bsub>Y\<^esub> f y" by (metis (no_types, lifting) PiE assms lattice.le_iff_join meet_pres_def) qed subsection \Bounded Lattices\ locale bounded_lattice = lattice + weak_partial_order_bottom + weak_partial_order_top sublocale bounded_lattice \ weak_bounded_lattice .. context bounded_lattice begin lemma bottom_eq: "\ b \ carrier L; \ x. x \ carrier L \ b \ x \ \ b = \" by (metis bottom_closed bottom_lower le_antisym) lemma top_eq: "\ t \ carrier L; \ x. x \ carrier L \ x \ t \ \ t = \" by (metis le_antisym top_closed top_higher) end hide_const (open) Lattice.inf hide_const (open) Lattice.sup end