# Theory Order

theory Order
imports Congruence
```(*  Title:      HOL/Algebra/Order.thy
Author:     Clemens Ballarin, started 7 November 2003

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"

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
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"

lemma Upper_dual [simp]:
"Upper (inv_gorder L) A = Lower L A"

lemma least_dual [simp]:
"least (inv_gorder L) x A = greatest L x A"

lemma greatest_dual [simp]:
"greatest (inv_gorder L) x A = least L x A"

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"

lemma at_least_at_most_lower [dest]:
"x ∈ ⦃a..b⦄ ⟹ a ⊑ x"

lemma at_least_at_most_closed: "⦃a..b⦄ ⊆ carrier L"

lemma at_least_at_most_member [intro]:
"⟦ x ∈ carrier L; a ⊑ x; x ⊑ b ⟧ ⟹ x ∈ ⦃a..b⦄"

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"

lemma use_iso2:
"⟦isotone A B f; x ∈ carrier A; y ∈ carrier A; x ⊑⇘A⇙ y⟧ ⟹
f x ⊑⇘B⇙ f y"

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"

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
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
```