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 Lx) "

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 Ly  x Ly  x .≠Ly"

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 Lu)}  carrier L"

definition
  Lower :: "[_, 'a set] => 'a set"
  where "Lower L A = {l. (x. x  A  carrier L  l Lx)}  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 LA'  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  (xA. l Lx)"

definition
  greatest :: "[_, 'a, 'a set] => bool"
  where "greatest L g A  A  carrier L  g  A  (xA. x Lg)"

text (in weak_partial_order) Could weaken these to terml  carrier L  l .∈ A and termg  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) constleast is not congruent in the second parameter for
  termA {.=} 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) constgreatest is not congruent in the second parameter for
  termA {.=} 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" ("(1_.._ı)")
  where "l..uA= {x  carrier A. l Ax  x Au}"

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 
   (xcarrier A. ycarrier A. x Ay  f x Bf 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 L1y 
                    f x L2f 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 Ay 
   f x Af y"
  by (simp add: isotone_def)

lemma use_iso2:
  "isotone A B f; x  carrier A; y  carrier A; x Ay 
   f x Bf 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  xcarrier L. f (f x) .=Lf 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 
                    (xcarrier A. ycarrier A. f x Bf y  x Ay )"

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 = (xcarrier 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