(* 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" ("⨆ı_" [90] 90) where "⨆⇘L⇙A = (SOME x. least L x (Upper L A))" definition inf :: "[_, 'a set] => 'a" ("⨅ı_" [90] 90) where "⨅⇘L⇙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 = ⨆⇘L⇙(f ` A)" definition infi :: "('a, 'b) gorder_scheme ⇒ 'c set ⇒ ('c ⇒ 'a) ⇒ 'a " where "infi L A f = ⨅⇘L⇙(f ` A)" syntax "_inf1" :: "('a, 'b) gorder_scheme ⇒ pttrns ⇒ 'a ⇒ 'a" ("(3IINFı _./ _)" [0, 10] 10) "_inf" :: "('a, 'b) gorder_scheme ⇒ pttrn ⇒ 'c set ⇒ 'a ⇒ 'a" ("(3IINFı _:_./ _)" [0, 0, 10] 10) "_sup1" :: "('a, 'b) gorder_scheme ⇒ pttrns ⇒ 'a ⇒ 'a" ("(3SSUPı _./ _)" [0, 10] 10) "_sup" :: "('a, 'b) gorder_scheme ⇒ pttrn ⇒ 'c set ⇒ 'a ⇒ 'a" ("(3SSUPı _:_./ _)" [0, 0, 10] 10) translations "IINF⇘L⇙ x. B" == "CONST infi L CONST UNIV (%x. B)" "IINF⇘L⇙ x:A. B" == "CONST infi L A (%x. B)" "SSUP⇘L⇙ x. B" == "CONST supr L CONST UNIV (%x. B)" "SSUP⇘L⇙ x:A. B" == "CONST supr L A (%x. B)" definition join :: "[_, 'a, 'a] => 'a" (infixl "⊔ı" 65) where "x ⊔⇘L⇙ y = ⨆⇘L⇙{x, y}" definition meet :: "[_, 'a, 'a] => 'a" (infixl "⊓ı" 70) where "x ⊓⇘L⇙ y = ⨅⇘L⇙{x, y}" definition LEAST_FP :: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ 'a" ("LFPı") where "LEAST_FP L f = ⨅⇘L⇙ {u ∈ carrier L. f u ⊑⇘L⇙ u}" ― ‹least fixed point› definition GREATEST_FP:: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ 'a" ("GFPı") where "GREATEST_FP L f = ⨆⇘L⇙ {u ∈ carrier L. u ⊑⇘L⇙ f u}" ― ‹greatest fixed point› subsection ‹Dual operators› lemma sup_dual [simp]: "⨆⇘inv_gorder L⇙A = ⨅⇘L⇙A" by (simp add: sup_def inf_def) lemma inf_dual [simp]: "⨅⇘inv_gorder L⇙A = ⨆⇘L⇙A" by (simp add: sup_def inf_def) lemma join_dual [simp]: "p ⊔⇘inv_gorder L⇙ q = p ⊓⇘L⇙ q" by (simp add:join_def meet_def) lemma meet_dual [simp]: "p ⊓⇘inv_gorder L⇙ q = p ⊔⇘L⇙ q" by (simp add:join_def meet_def) lemma top_dual [simp]: "⊤⇘inv_gorder L⇙ = ⊥⇘L⇙" by (simp add: top_def bottom_def) lemma bottom_dual [simp]: "⊥⇘inv_gorder L⇙ = ⊤⇘L⇙" 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 ⊔⇘X⇙ y) = f x ⊔⇘Y⇙ 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 ⊓⇘X⇙ y) = f x ⊓⇘Y⇙ 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 ⊑⇘X⇙ y⟧ ⟹ f x ⊑⇘Y⇙ 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 ⊑⇘X⇙ y⟧ ⟹ f x ⊑⇘Y⇙ 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