(* Title: HOL/Algebra/Order.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 Order imports "HOL-Library.FuncSet" Congruence begin section ‹Orders› subsection ‹Partial Orders› record 'a gorder = "'a eq_object" + le :: "['a, 'a] => bool" (infixl "⊑ı" 50) abbreviation inv_gorder :: "_ ⇒ 'a gorder" where "inv_gorder L ≡ ⦇ carrier = carrier L, eq = op .=⇘_{L⇙}, le = (λ x y. y ⊑⇘_{L ⇙}x) ⦈" lemma inv_gorder_inv: "inv_gorder (inv_gorder L) = L" by simp locale weak_partial_order = equivalence L for L (structure) + assumes le_refl [intro, simp]: "x ∈ carrier L ==> x ⊑ x" and weak_le_antisym [intro]: "[| x ⊑ y; y ⊑ x; x ∈ carrier L; y ∈ carrier L |] ==> x .= y" and le_trans [trans]: "[| x ⊑ y; y ⊑ z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x ⊑ z" and le_cong: "⟦ x .= y; z .= w; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L; w ∈ carrier L ⟧ ⟹ x ⊑ z ⟷ y ⊑ w" definition lless :: "[_, 'a, 'a] => bool" (infixl "⊏ı" 50) where "x ⊏⇘_{L⇙}y ⟷ x ⊑⇘_{L⇙}y & x .≠⇘_{L⇙}y" subsubsection ‹The order relation› context weak_partial_order begin lemma le_cong_l [intro, trans]: "⟦ x .= y; y ⊑ z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L ⟧ ⟹ x ⊑ z" by (auto intro: le_cong [THEN iffD2]) lemma le_cong_r [intro, trans]: "⟦ x ⊑ y; y .= z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L ⟧ ⟹ x ⊑ z" by (auto intro: le_cong [THEN iffD1]) lemma weak_refl [intro, simp]: "⟦ x .= y; x ∈ carrier L; y ∈ carrier L ⟧ ⟹ x ⊑ y" by (simp add: le_cong_l) end lemma weak_llessI: fixes R (structure) assumes "x ⊑ y" and "~(x .= y)" shows "x ⊏ y" using assms unfolding lless_def by simp lemma lless_imp_le: fixes R (structure) assumes "x ⊏ y" shows "x ⊑ y" using assms unfolding lless_def by simp lemma weak_lless_imp_not_eq: fixes R (structure) assumes "x ⊏ y" shows "¬ (x .= y)" using assms unfolding lless_def by simp lemma weak_llessE: fixes R (structure) assumes p: "x ⊏ y" and e: "⟦x ⊑ 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' ⊏ y" and carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L" shows "x ⊏ y" using assms unfolding lless_def by (auto intro: trans sym) lemma (in weak_partial_order) lless_cong_r [trans]: assumes xy: "x ⊏ y" and yy': "y .= y'" and carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L" shows "x ⊏ 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 ⊏ b" "b ⊏ a" shows "P" using assms by (elim weak_llessE) auto lemma (in weak_partial_order) lless_trans [trans]: assumes "a ⊏ b" "b ⊏ c" and carr[simp]: "a ∈ carrier L" "b ∈ carrier L" "c ∈ carrier L" shows "a ⊏ c" using assms unfolding lless_def by (blast dest: le_trans intro: sym) lemma weak_partial_order_subset: assumes "weak_partial_order L" "A ⊆ carrier L" shows "weak_partial_order (L⦇ carrier := A ⦈)" proof - interpret L: weak_partial_order L by (simp add: assms) interpret equivalence "(L⦇ carrier := A ⦈)" by (simp add: L.equivalence_axioms assms(2) equivalence_subset) show ?thesis apply (unfold_locales, simp_all) using assms(2) apply auto[1] using assms(2) apply auto[1] apply (meson L.le_trans assms(2) contra_subsetD) apply (meson L.le_cong assms(2) subsetCE) done qed 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 ⊑⇘_{L⇙}u)} ∩ carrier L" definition Lower :: "[_, 'a set] => 'a set" where "Lower L A = {l. (ALL x. x ∈ A ∩ carrier L --> l ⊑⇘_{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 ⊑ 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 ⊑ u" unfolding Upper_def elem_def by (blast dest: sym) lemma Upper_memI: fixes L (structure) shows "[| !! y. y ∈ A ==> y ⊑ x; x ∈ carrier L |] ==> x ∈ Upper L A" by (unfold Upper_def) blast lemma (in weak_partial_order) Upper_elemI: "[| !! y. y ∈ A ==> y ⊑ 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 ⊑ a" by (intro Upper_memD[OF aelem, THEN conjunct1] Acarr) also note aa' finally show "y ⊑ 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 ⊑ 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 ⊑ x" by (simp add: aLxCond aA) finally show "a' ⊑ 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' ⊑ 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' ⊑ x" by (simp add: a'LxCond a'A') finally show "a ⊑ 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 ⊑ x ∧ l ∈ carrier L" by (unfold Lower_def) blast lemma Lower_memI: fixes L (structure) shows "[| !! y. y ∈ A ==> x ⊑ 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 ⊑ a" hence aLxCond: "⋀a. ⟦a ∈ A; a ∈ carrier L⟧ ⟹ x ⊑ 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 ⊑ a" by (rule aLxCond) also note a'a[symmetric] finally show "x ⊑ 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 ⊑ a'" hence a'LxCond: "⋀a'. ⟦a' ∈ A'; a' ∈ carrier L⟧ ⟹ x ⊑ 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 ⊑ a'" by (rule a'LxCond) also note aa'[symmetric] finally show "x ⊑ a" by (simp add: carr subsetD[OF A'carr a'A']) qed 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 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 ⊑⇘_{L⇙}x)" definition greatest :: "[_, 'a, 'a set] => bool" where "greatest L g A ⟷ A ⊆ carrier L & g ∈ A & (ALL x : A. x ⊑⇘_{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 ⊑ 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) abbreviation is_lub :: "[_, 'a, 'a set] => bool" where "is_lub L x A ≡ least L x (Upper L A)" 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 ⊑ s" and below: "!! y. y ∈ Upper L A ==> s ⊑ 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 ⊑ 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 ⊑ 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 ⊑ 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) abbreviation is_glb :: "[_, 'a, 'a set] => bool" where "is_glb L x A ≡ greatest L x (Lower L A)" 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 ⊑ x" and above: "!! y. y ∈ Lower L A ==> y ⊑ 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 ⊑ 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 ⊑ x" by (unfold greatest_def) blast lemma Lower_dual [simp]: "Lower (inv_gorder L) A = Upper L A" by (simp add:Upper_def Lower_def) lemma Upper_dual [simp]: "Upper (inv_gorder L) A = Lower L A" by (simp add:Upper_def Lower_def) lemma least_dual [simp]: "least (inv_gorder L) x A = greatest L x A" by (simp add:least_def greatest_def) lemma greatest_dual [simp]: "greatest (inv_gorder L) x A = least L x A" by (simp add:least_def greatest_def) lemma (in weak_partial_order) dual_weak_order: "weak_partial_order (inv_gorder L)" apply (unfold_locales) apply (simp_all) apply (metis sym) apply (metis trans) apply (metis weak_le_antisym) apply (metis le_trans) apply (metis le_cong_l le_cong_r sym) done lemma dual_weak_order_iff: "weak_partial_order (inv_gorder A) ⟷ weak_partial_order A" proof assume "weak_partial_order (inv_gorder A)" then interpret dpo: weak_partial_order "inv_gorder A" rewrites "carrier (inv_gorder A) = carrier A" and "le (inv_gorder A) = (λ x y. le A y x)" and "eq (inv_gorder A) = eq A" by (simp_all) show "weak_partial_order A" by (unfold_locales, auto intro: dpo.sym dpo.trans dpo.le_trans) next assume "weak_partial_order A" thus "weak_partial_order (inv_gorder A)" by (metis weak_partial_order.dual_weak_order) qed subsubsection ‹Intervals› definition at_least_at_most :: "('a, 'c) gorder_scheme ⇒ 'a => 'a => 'a set" ("(1⦃_.._⦄ı)") where "⦃l..u⦄⇘_{A⇙}= {x ∈ carrier A. l ⊑⇘_{A⇙}x ∧ x ⊑⇘_{A⇙}u}" context weak_partial_order begin lemma at_least_at_most_upper [dest]: "x ∈ ⦃a..b⦄ ⟹ x ⊑ b" by (simp add: at_least_at_most_def) lemma at_least_at_most_lower [dest]: "x ∈ ⦃a..b⦄ ⟹ a ⊑ x" by (simp add: at_least_at_most_def) lemma at_least_at_most_closed: "⦃a..b⦄ ⊆ carrier L" by (auto simp add: at_least_at_most_def) lemma at_least_at_most_member [intro]: "⟦ x ∈ carrier L; a ⊑ x; x ⊑ b ⟧ ⟹ x ∈ ⦃a..b⦄" by (simp add: at_least_at_most_def) end subsubsection ‹Isotone functions› definition isotone :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where "isotone A B f ≡ weak_partial_order A ∧ weak_partial_order B ∧ (∀x∈carrier A. ∀y∈carrier A. x ⊑⇘_{A⇙}y ⟶ f x ⊑⇘_{B⇙}f y)" lemma isotoneI [intro?]: fixes f :: "'a ⇒ 'b" assumes "weak_partial_order L1" "weak_partial_order L2" "(⋀x y. ⟦ x ∈ carrier L1; y ∈ carrier L1; x ⊑⇘_{L1⇙}y ⟧ ⟹ f x ⊑⇘_{L2⇙}f y)" shows "isotone L1 L2 f" using assms by (auto simp add:isotone_def) abbreviation Monotone :: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ bool" ("Monoı") where "Monotone L f ≡ isotone L L f" lemma use_iso1: "⟦isotone A A f; x ∈ carrier A; y ∈ carrier A; x ⊑⇘_{A⇙}y⟧ ⟹ f x ⊑⇘_{A⇙}f y" by (simp add: isotone_def) lemma use_iso2: "⟦isotone A B f; x ∈ carrier A; y ∈ carrier A; x ⊑⇘_{A⇙}y⟧ ⟹ f x ⊑⇘_{B⇙}f y" by (simp add: isotone_def) lemma iso_compose: "⟦f ∈ carrier A → carrier B; isotone A B f; g ∈ carrier B → carrier C; isotone B C g⟧ ⟹ isotone A C (g ∘ f)" by (simp add: isotone_def, safe, metis Pi_iff) lemma (in weak_partial_order) inv_isotone [simp]: "isotone (inv_gorder A) (inv_gorder B) f = isotone A B f" by (auto simp add:isotone_def dual_weak_order dual_weak_order_iff) subsubsection ‹Idempotent functions› definition idempotent :: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ bool" ("Idemı") where "idempotent L f ≡ ∀x∈carrier L. f (f x) .=⇘_{L⇙}f x" lemma (in weak_partial_order) idempotent: "⟦ Idem f; x ∈ carrier L ⟧ ⟹ f (f x) .= f x" by (auto simp add: idempotent_def) subsubsection ‹Order embeddings› definition order_emb :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where "order_emb A B f ≡ weak_partial_order A ∧ weak_partial_order B ∧ (∀x∈carrier A. ∀y∈carrier A. f x ⊑⇘_{B⇙}f y ⟷ x ⊑⇘_{A⇙}y )" lemma order_emb_isotone: "order_emb A B f ⟹ isotone A B f" by (auto simp add: isotone_def order_emb_def) subsubsection ‹Commuting functions› definition commuting :: "('a, 'c) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ ('a ⇒ 'a) ⇒ bool" where "commuting A f g = (∀x∈carrier A. (f ∘ g) x .=⇘_{A⇙}(g ∘ f) x)" subsection ‹Partial orders where ‹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 ⊑ y; y ⊑ x; x ∈ carrier L; y ∈ carrier L |] ==> x = y" using weak_le_antisym unfolding eq_is_equal . lemma lless_eq: "x ⊏ y ⟷ x ⊑ y & x ≠ y" unfolding lless_def by (simp add: eq_is_equal) lemma set_eq_is_eq: "A {.=} B ⟷ A = B" by (auto simp add: set_eq_def elem_def eq_is_equal) end lemma (in partial_order) dual_order: "partial_order (inv_gorder L)" proof - interpret dwo: weak_partial_order "inv_gorder L" by (metis dual_weak_order) show ?thesis by (unfold_locales, simp add:eq_is_equal) qed lemma dual_order_iff: "partial_order (inv_gorder A) ⟷ partial_order A" proof assume assm:"partial_order (inv_gorder A)" then interpret po: partial_order "inv_gorder A" rewrites "carrier (inv_gorder A) = carrier A" and "le (inv_gorder A) = (λ x y. le A y x)" and "eq (inv_gorder A) = eq A" by (simp_all) show "partial_order A" apply (unfold_locales, simp_all) apply (metis po.sym, metis po.trans) apply (metis po.weak_le_antisym, metis po.le_trans) apply (metis (full_types) po.eq_is_equal, metis po.eq_is_equal) done next assume "partial_order A" thus "partial_order (inv_gorder A)" by (metis partial_order.dual_order) qed 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 . subsection ‹Bounded Orders› definition top :: "_ => 'a" ("⊤ı") where "⊤⇘_{L⇙}= (SOME x. greatest L x (carrier L))" definition bottom :: "_ => 'a" ("⊥ı") where "⊥⇘_{L⇙}= (SOME x. least L x (carrier L))" locale weak_partial_order_bottom = weak_partial_order L for L (structure) + assumes bottom_exists: "∃ x. least L x (carrier L)" begin lemma bottom_least: "least L ⊥ (carrier L)" proof - obtain x where "least L x (carrier L)" by (metis bottom_exists) thus ?thesis by (auto intro:someI2 simp add: bottom_def) qed lemma bottom_closed [simp, intro]: "⊥ ∈ carrier L" by (metis bottom_least least_mem) lemma bottom_lower [simp, intro]: "x ∈ carrier L ⟹ ⊥ ⊑ x" by (metis bottom_least least_le) end locale weak_partial_order_top = weak_partial_order L for L (structure) + assumes top_exists: "∃ x. greatest L x (carrier L)" begin lemma top_greatest: "greatest L ⊤ (carrier L)" proof - obtain x where "greatest L x (carrier L)" by (metis top_exists) thus ?thesis by (auto intro:someI2 simp add: top_def) qed lemma top_closed [simp, intro]: "⊤ ∈ carrier L" by (metis greatest_mem top_greatest) lemma top_higher [simp, intro]: "x ∈ carrier L ⟹ x ⊑ ⊤" by (metis greatest_le top_greatest) end subsection ‹Total Orders› locale weak_total_order = weak_partial_order + assumes total: "⟦ x ∈ carrier L; y ∈ carrier L ⟧ ⟹ x ⊑ y ∨ y ⊑ 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 ⊑ y ∨ y ⊑ x" shows "weak_total_order L" by unfold_locales (rule total) subsection ‹Total orders where ‹eq› is the Equality› locale total_order = partial_order + assumes total_order_total: "⟦ x ∈ carrier L; y ∈ carrier L ⟧ ⟹ x ⊑ y ∨ y ⊑ x" sublocale total_order < weak?: weak_total_order by unfold_locales (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 ⊑ y ∨ y ⊑ x" shows "total_order L" by unfold_locales (rule total) end