Theory Order

(*  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
    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 = (.=⇘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. (∀x. x ∈ A ∩ carrier L ⟶ x ⊑⇘L⇙ u)} ∩ carrier L"

definition
  Lower :: "[_, 'a set] => 'a set"
  where "Lower L A = {l. (∀x. x ∈ A ∩ carrier L ⟶ l ⊑⇘L⇙ x)} ∩ carrier L"

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 (in weak_partial_order) equivalence_dual: "equivalence (inv_gorder L)"
  by (rule equivalence.intro) (auto simp: intro: sym trans)

lemma  (in weak_partial_order) dual_weak_order: "weak_partial_order (inv_gorder L)"
  by intro_locales (auto simp add: weak_partial_order_axioms_def le_cong intro: equivalence_dual le_trans)

lemma (in weak_partial_order) dual_eq_iff [simp]: "A {.=}⇘inv_gorder L⇙ A' ⟷ A {.=} A'"
  by (auto simp: set_eq_def elem_def)

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

lemma Upper_closed [iff]:
  "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' ∈ carrier L" "A ⊆ carrier L" "a .= a'" "a ∈ Upper L A"
  shows "a' ∈ Upper L A"
  by (metis assms Upper_closed Upper_is_closed closure_of_eq complete_classes)

lemma (in weak_partial_order) Upper_semi_cong:
  assumes "A ⊆ carrier L" "A {.=} A'"
  shows "Upper L A ⊆ Upper L A'"
  unfolding Upper_def
   by clarsimp (meson assms equivalence.refl equivalence_axioms le_cong set_eqD2 subset_eq)

lemma (in weak_partial_order) Upper_cong:
  assumes "A ⊆ carrier L" "A' ⊆ carrier L" "A {.=} A'"
  shows "Upper L A = Upper L A'"
  using assms by (simp add: Upper_semi_cong set_eq_sym subset_antisym)

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' ∈ carrier L"  "A ⊆ carrier L" "a .= a'" "a ∈ Lower L A"
  shows "a' ∈ Lower L A"
  by (meson assms Lower_closed Lower_is_closed is_closed_eq subsetCE)

lemma (in weak_partial_order) Lower_cong:
  assumes "A ⊆ carrier L" "A' ⊆ carrier L" "A {.=} A'"
  shows "Lower L A = Lower L A'"
  unfolding Upper_dual [symmetric]
  by (rule weak_partial_order.Upper_cong [OF dual_weak_order]) (simp_all add: assms)

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 ∧ (∀x∈A. l ⊑⇘L⇙ x)"

definition
  greatest :: "[_, 'a, 'a set] => bool"
  where "greatest L g A ⟷ A ⊆ carrier L ∧ g ∈ A ∧ (∀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_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 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"
  unfolding least_def
  by (meson is_closed_eq is_closed_eq_rev le_cong local.refl subset_iff)

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 "A ⊆ carrier L" "A' ⊆ carrier L" "A {.=} A'"
  shows "least L x (Upper L A) = least L x (Upper L A')"
  using Upper_cong assms by auto

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 "∀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"
  unfolding greatest_def
  by (meson is_closed_eq_rev le_cong_r local.sym subset_eq)

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"
  shows "greatest L x (Lower L A) = greatest L x' (Lower L A)"
proof -
  have "∀A. is_closed (Lower L (A ∩ carrier L))"
    by simp
  then show ?thesis
    by (simp add: Lower_def assms greatest_cong)
qed

lemma (in weak_partial_order) greatest_Lower_cong_r:
  assumes "A ⊆ carrier L" "A' ⊆ carrier L" "A {.=} A'"
  shows "greatest L x (Lower L A) = greatest L x (Lower L A')"
  using Lower_cong assms by auto

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 "∀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


subsubsection ‹Intervals›

definition
  at_least_at_most :: "('a, 'c) gorder_scheme ⇒ 'a => 'a => 'a set"
    (‹(‹indent=1 notation=‹mixfix interval››⦃_.._⦄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"
    (‹(‹open_block notation=‹prefix Mono››Monoı)›)
  where "Mono⇘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"
    (‹(‹open_block notation=‹prefix Idem››Idemı)›)
  where "Idem⇘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: "(.=) = (=)"
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 add: po.sym)
    apply (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