# Theory Lattice

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

Most congruence rules by Stephan Hohe.
*)

theory Lattice
imports Congruence
begin

section ‹Orders and Lattices›

subsection ‹Partial Orders›

record 'a gorder = "'a eq_object" +
le :: "['a, 'a] => bool" (infixl "⊑ı" 50)

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)

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

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)

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)

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

text ‹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
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}"

subsection ‹Lattices›

locale weak_upper_semilattice = weak_partial_order +
assumes sup_of_two_exists:
"[| x ∈ carrier L; y ∈ carrier L |] ==> EX 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 |] ==> EX s. greatest L s (Lower L {x, y})"

locale weak_lattice = weak_upper_semilattice + weak_lower_semilattice

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"
apply (rule subsetD [where A = "Upper L (insert x A)"])
apply (rule Upper_antimono)
apply blast
apply (rule y)
done
assume "z = a"
with y' least_a show ?thesis by (fast dest: least_le)
next
assume "z ∈ {x}"  (* FIXME "z = x"; declare specific elim rule for "insert 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
next
case False with P and xA show ?thesis
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_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])

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"
apply (rule subsetD [where A = "Lower L (insert x A)"])
apply (rule Lower_antimono)
apply blast
apply (rule y)
done
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
next
case False with P and xA show ?thesis
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_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])

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

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 standard (rule total)

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 "EX 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 "EX 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 ‹Complete Lattices›

locale weak_complete_lattice = weak_lattice +
assumes sup_exists:
"[| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)"
and inf_exists:
"[| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)"

text ‹Introduction rule: the usual definition of complete lattice›

lemma (in weak_partial_order) weak_complete_latticeI:
assumes sup_exists:
"!!A. [| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)"
shows "weak_complete_lattice L"
by standard (auto intro: sup_exists inf_exists)

definition
top :: "_ => 'a" ("⊤ı")
where "⊤⇘L⇙ = sup L (carrier L)"

definition
bottom :: "_ => 'a" ("⊥ı")
where "⊥⇘L⇙ = inf L (carrier L)"

lemma (in weak_complete_lattice) supI:
"[| !!l. least L l (Upper L A) ==> P l; A ⊆ carrier L |]
==> P (⨆A)"
proof (unfold sup_def)
assume L: "A ⊆ carrier L"
and P: "!!l. least L l (Upper L A) ==> P l"
with sup_exists obtain s where "least L s (Upper L A)" by blast
with L show "P (SOME l. least L l (Upper L A))"
by (fast intro: someI2 weak_least_unique P)
qed

lemma (in weak_complete_lattice) sup_closed [simp]:
"A ⊆ carrier L ==> ⨆A ∈ carrier L"
by (rule supI) simp_all

lemma (in weak_complete_lattice) top_closed [simp, intro]:
"⊤ ∈ carrier L"
by (unfold top_def) simp

lemma (in weak_complete_lattice) infI:
"[| !!i. greatest L i (Lower L A) ==> P i; A ⊆ carrier L |]
==> P (⨅A)"
proof (unfold inf_def)
assume L: "A ⊆ carrier L"
and P: "!!l. greatest L l (Lower L A) ==> P l"
with inf_exists obtain s where "greatest L s (Lower L A)" by blast
with L show "P (SOME l. greatest L l (Lower L A))"
by (fast intro: someI2 weak_greatest_unique P)
qed

lemma (in weak_complete_lattice) inf_closed [simp]:
"A ⊆ carrier L ==> ⨅A ∈ carrier L"
by (rule infI) simp_all

lemma (in weak_complete_lattice) bottom_closed [simp, intro]:
"⊥ ∈ carrier L"
by (unfold bottom_def) simp

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

theorem (in weak_partial_order) weak_complete_lattice_criterion1:
assumes top_exists: "EX g. greatest L g (carrier L)"
and inf_exists:
"!!A. [| A ⊆ carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
shows "weak_complete_lattice L"
proof (rule weak_complete_latticeI)
from top_exists obtain top where top: "greatest L top (carrier L)" ..
fix A
assume L: "A ⊆ carrier L"
let ?B = "Upper L A"
from L top have "top ∈ ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B ~= {}" by fast
have B_L: "?B ⊆ carrier L" by simp
from inf_exists [OF B_L B_non_empty]
obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
have "least L b (Upper L A)"
apply (rule least_UpperI)
apply (rule greatest_le [where A = "Lower L ?B"])
apply (rule b_inf_B)
apply (rule Lower_memI)
apply (erule Upper_memD [THEN conjunct1])
apply assumption
apply (rule L)
apply (fast intro: L [THEN subsetD])
apply (erule greatest_Lower_below [OF b_inf_B])
apply simp
apply (rule L)
apply (rule greatest_closed [OF b_inf_B])
done
then show "EX s. least L s (Upper L A)" ..
next
fix A
assume L: "A ⊆ carrier L"
show "EX i. greatest L i (Lower L A)"
proof (cases "A = {}")
case True then show ?thesis
next
case False with L show ?thesis
by (rule inf_exists)
qed
qed

(* TODO: prove dual version *)

subsection ‹Orders and Lattices 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 lless_asym:
assumes "a ∈ carrier L" "b ∈ carrier L"
and "a ⊏ b" "b ⊏ a"
shows "P"
using assms unfolding lless_eq by auto

end

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 .

text ‹Lattices›

locale upper_semilattice = partial_order +
assumes sup_of_two_exists:
"[| x ∈ carrier L; y ∈ carrier L |] ==> EX s. least L s (Upper L {x, y})"

sublocale upper_semilattice < weak?: weak_upper_semilattice
by standard (rule sup_of_two_exists)

locale lower_semilattice = partial_order +
assumes inf_of_two_exists:
"[| x ∈ carrier L; y ∈ carrier L |] ==> EX s. greatest L s (Lower L {x, y})"

sublocale lower_semilattice < weak?: weak_lower_semilattice
by standard (rule inf_of_two_exists)

locale lattice = upper_semilattice + lower_semilattice

text ‹Supremum›

declare (in partial_order) weak_sup_of_singleton [simp del]

lemma (in partial_order) sup_of_singleton [simp]:
"x ∈ carrier L ==> ⨆{x} = x"
using weak_sup_of_singleton unfolding eq_is_equal .

lemma (in upper_semilattice) join_assoc_lemma:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "x ⊔ (y ⊔ z) = ⨆{x, y, z}"
using weak_join_assoc_lemma L unfolding eq_is_equal .

lemma (in upper_semilattice) join_assoc:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "(x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)"
using weak_join_assoc L unfolding eq_is_equal .

text ‹Infimum›

declare (in partial_order) weak_inf_of_singleton [simp del]

lemma (in partial_order) inf_of_singleton [simp]:
"x ∈ carrier L ==> ⨅{x} = x"
using weak_inf_of_singleton unfolding eq_is_equal .

text ‹Condition on ‹A›: infimum exists.›

lemma (in lower_semilattice) meet_assoc_lemma:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "x ⊓ (y ⊓ z) = ⨅{x, y, z}"
using weak_meet_assoc_lemma L unfolding eq_is_equal .

lemma (in lower_semilattice) meet_assoc:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "(x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)"
using weak_meet_assoc L unfolding eq_is_equal .

text ‹Total Orders›

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 standard (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 standard (rule total)

text ‹Total orders are lattices.›

sublocale total_order < weak?: lattice
by standard (auto intro: sup_of_two_exists inf_of_two_exists)

text ‹Complete lattices›

locale complete_lattice = lattice +
assumes sup_exists:
"[| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)"
and inf_exists:
"[| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)"

sublocale complete_lattice < weak?: weak_complete_lattice
by standard (auto intro: sup_exists inf_exists)

text ‹Introduction rule: the usual definition of complete lattice›

lemma (in partial_order) complete_latticeI:
assumes sup_exists:
"!!A. [| A ⊆ carrier L |] ==> EX s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A ⊆ carrier L |] ==> EX i. greatest L i (Lower L A)"
shows "complete_lattice L"
by standard (auto intro: sup_exists inf_exists)

theorem (in partial_order) complete_lattice_criterion1:
assumes top_exists: "EX g. greatest L g (carrier L)"
and inf_exists:
"!!A. [| A ⊆ carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
shows "complete_lattice L"
proof (rule complete_latticeI)
from top_exists obtain top where top: "greatest L top (carrier L)" ..
fix A
assume L: "A ⊆ carrier L"
let ?B = "Upper L A"
from L top have "top ∈ ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B ~= {}" by fast
have B_L: "?B ⊆ carrier L" by simp
from inf_exists [OF B_L B_non_empty]
obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
have "least L b (Upper L A)"
apply (rule least_UpperI)
apply (rule greatest_le [where A = "Lower L ?B"])
apply (rule b_inf_B)
apply (rule Lower_memI)
apply (erule Upper_memD [THEN conjunct1])
apply assumption
apply (rule L)
apply (fast intro: L [THEN subsetD])
apply (erule greatest_Lower_below [OF b_inf_B])
apply simp
apply (rule L)
apply (rule greatest_closed [OF b_inf_B])
done
then show "EX s. least L s (Upper L A)" ..
next
fix A
assume L: "A ⊆ carrier L"
show "EX i. greatest L i (Lower L A)"
proof (cases "A = {}")
case True then show ?thesis
next
case False with L show ?thesis
by (rule inf_exists)
qed
qed

(* TODO: prove dual version *)

subsection ‹Examples›

subsubsection ‹The Powerset of a Set is a Complete Lattice›

theorem powerset_is_complete_lattice:
"complete_lattice ⦇carrier = Pow A, eq = op =, le = op ⊆⦈"
(is "complete_lattice ?L")
proof (rule partial_order.complete_latticeI)
show "partial_order ?L"
by standard auto
next
fix B
assume "B ⊆ carrier ?L"
then have "least ?L (⋃B) (Upper ?L B)"
by (fastforce intro!: least_UpperI simp: Upper_def)
then show "EX s. least ?L s (Upper ?L B)" ..
next
fix B
assume "B ⊆ carrier ?L"
then have "greatest ?L (⋂B ∩ A) (Lower ?L B)"
txt ‹@{term "⋂B"} is not the infimum of @{term B}:
@{term "⋂{} = UNIV"} which is in general bigger than @{term "A"}!›
by (fastforce intro!: greatest_LowerI simp: Lower_def)
then show "EX i. greatest ?L i (Lower ?L B)" ..
qed

text ‹An other example, that of the lattice of subgroups of a group,
can be found in Group theory (Section~\ref{sec:subgroup-lattice}).›

end