src/HOL/Algebra/Lattice.thy
 author ballarin Wed, 08 Jun 2005 16:11:09 +0200 changeset 16325 a6431098a929 parent 15328 35951e6a7855 child 16417 9bc16273c2d4 permissions -rw-r--r--
Fixed "axiom" generation for mixed locales with and without predicates.

(*
Title:     HOL/Algebra/Lattice.thy
Id:        $Id$
Author:    Clemens Ballarin, started 7 November 2003
*)

header {* Orders and Lattices *}

theory Lattice = Main:

text {* Object with a carrier set. *}

record 'a partial_object =
carrier :: "'a set"

subsection {* Partial Orders *}

record 'a order = "'a partial_object" +
le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)

locale partial_order = struct L +
assumes refl [intro, simp]:
"x \<in> carrier L ==> x \<sqsubseteq> x"
and anti_sym [intro]:
"[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
and trans [trans]:
"[| x \<sqsubseteq> y; y \<sqsubseteq> z;
x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"

constdefs (structure L)
less :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
"x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"

-- {* Upper and lower bounds of a set. *}
Upper :: "[_, 'a set] => 'a set"
"Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter>
carrier L"

Lower :: "[_, 'a set] => 'a set"
"Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter>
carrier L"

-- {* Least and greatest, as predicate. *}
least :: "[_, 'a, 'a set] => bool"
"least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"

greatest :: "[_, 'a, 'a set] => bool"
"greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"

-- {* Supremum and infimum *}
sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90)
"\<Squnion>A == THE x. least L x (Upper L A)"

inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90)
"\<Sqinter>A == THE x. greatest L x (Lower L A)"

join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)
"x \<squnion> y == sup L {x, y}"

meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 70)
"x \<sqinter> y == inf L {x, y}"

subsubsection {* Upper *}

lemma Upper_closed [intro, simp]:
"Upper L A \<subseteq> carrier L"
by (unfold Upper_def) clarify

lemma UpperD [dest]:
includes struct L
shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
by (unfold Upper_def) blast

lemma Upper_memI:
includes struct L
shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
by (unfold Upper_def) blast

lemma Upper_antimono:
"A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
by (unfold Upper_def) blast

subsubsection {* Lower *}

lemma Lower_closed [intro, simp]:
"Lower L A \<subseteq> carrier L"
by (unfold Lower_def) clarify

lemma LowerD [dest]:
includes struct L
shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
by (unfold Lower_def) blast

lemma Lower_memI:
includes struct L
shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
by (unfold Lower_def) blast

lemma Lower_antimono:
"A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
by (unfold Lower_def) blast

subsubsection {* least *}

lemma least_carrier [intro, simp]:
shows "least L l A ==> l \<in> carrier L"
by (unfold least_def) fast

lemma least_mem:
"least L l A ==> l \<in> A"
by (unfold least_def) fast

lemma (in partial_order) least_unique:
"[| least L x A; least L y A |] ==> x = y"
by (unfold least_def) blast

lemma least_le:
includes struct L
shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
by (unfold least_def) fast

lemma least_UpperI:
includes struct L
assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
and L: "A \<subseteq> carrier L"  "s \<in> carrier L"
shows "least L s (Upper L A)"
proof -
have "Upper L A \<subseteq> carrier L" by simp
moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
ultimately show ?thesis by (simp add: least_def)
qed

subsubsection {* greatest *}

lemma greatest_carrier [intro, simp]:
shows "greatest L l A ==> l \<in> carrier L"
by (unfold greatest_def) fast

lemma greatest_mem:
"greatest L l A ==> l \<in> A"
by (unfold greatest_def) fast

lemma (in partial_order) greatest_unique:
"[| greatest L x A; greatest L y A |] ==> x = y"
by (unfold greatest_def) blast

lemma greatest_le:
includes struct L
shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
by (unfold greatest_def) fast

lemma greatest_LowerI:
includes struct L
assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
shows "greatest L i (Lower L A)"
proof -
have "Lower L A \<subseteq> carrier L" by simp
moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
ultimately show ?thesis by (simp add: greatest_def)
qed

subsection {* Lattices *}

locale lattice = partial_order +
assumes sup_of_two_exists:
"[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
and inf_of_two_exists:
"[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"

lemma least_Upper_above:
includes struct L
shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
by (unfold least_def) blast

lemma greatest_Lower_above:
includes struct L
shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
by (unfold greatest_def) blast

subsubsection {* Supremum *}

lemma (in lattice) joinI:
"[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
==> P (x \<squnion> y)"
proof (unfold join_def sup_def)
assume L: "x \<in> carrier L"  "y \<in> 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 (THE l. least L l (Upper L {x, y}))"
by (fast intro: theI2 least_unique P)
qed

lemma (in lattice) join_closed [simp]:
"[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
by (rule joinI) (rule least_carrier)

lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
"x \<in> carrier L ==> least L x (Upper L {x})"
by (rule least_UpperI) fast+

lemma (in partial_order) sup_of_singleton [simp]:
includes struct L
shows "x \<in> carrier L ==> \<Squnion>{x} = x"
by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)

text {* Condition on @{text A}: supremum exists. *}

lemma (in lattice) sup_insertI:
"[| !!s. least L s (Upper L (insert x A)) ==> P s;
least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
==> P (\<Squnion>(insert x A))"
proof (unfold sup_def)
assume L: "x \<in> carrier L"  "A \<subseteq> 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 \<in> 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 (THE l. least L l (Upper L (insert x A)))"
proof (rule theI2)
show "least L s (Upper L (insert x A))"
proof (rule least_UpperI)
fix z
assume "z \<in> insert x A"
then show "z \<sqsubseteq> s"
proof
assume "z = x" then show ?thesis
by (simp add: least_Upper_above [OF least_s] L La)
next
assume "z \<in> A"
with L least_s least_a show ?thesis
by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
qed
next
fix y
assume y: "y \<in> Upper L (insert x A)"
show "s \<sqsubseteq> y"
proof (rule least_le [OF least_s], rule Upper_memI)
fix z
assume z: "z \<in> {a, x}"
then show "z \<sqsubseteq> y"
proof
have y': "y \<in> Upper L A"
apply (rule subsetD [where A = "Upper L (insert x A)"])
apply (rule Upper_antimono) apply clarify apply assumption
done
assume "z = a"
with y' least_a show ?thesis by (fast dest: least_le)
next
assume "z \<in> {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])
next
from L show "insert x A \<subseteq> carrier L" by simp
from least_s show "s \<in> carrier L" by simp
qed
next
fix l
assume least_l: "least L l (Upper L (insert x A))"
show "l = s"
proof (rule least_unique)
show "least L s (Upper L (insert x A))"
proof (rule least_UpperI)
fix z
assume "z \<in> insert x A"
then show "z \<sqsubseteq> s"
proof
assume "z = x" then show ?thesis
by (simp add: least_Upper_above [OF least_s] L La)
next
assume "z \<in> A"
with L least_s least_a show ?thesis
by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
qed
next
fix y
assume y: "y \<in> Upper L (insert x A)"
show "s \<sqsubseteq> y"
proof (rule least_le [OF least_s], rule Upper_memI)
fix z
assume z: "z \<in> {a, x}"
then show "z \<sqsubseteq> y"
proof
have y': "y \<in> Upper L A"
apply (rule subsetD [where A = "Upper L (insert x A)"])
apply (rule Upper_antimono) apply clarify apply assumption
done
assume "z = a"
with y' least_a show ?thesis by (fast dest: least_le)
next
assume "z \<in> {x}"
with y L show ?thesis by blast
qed
qed (rule Upper_closed [THEN subsetD])
next
from L show "insert x A \<subseteq> carrier L" by simp
from least_s show "s \<in> carrier L" by simp
qed
qed
qed
qed

lemma (in lattice) finite_sup_least:
"[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
proof (induct set: Finites)
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 add: sup_of_singletonI)
next
case False
with insert have "least L (\<Squnion>A) (Upper L A)" by simp
with _ show ?thesis
by (rule sup_insertI) (simp_all add: insert [simplified])
qed
qed

lemma (in lattice) finite_sup_insertI:
assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
shows "P (\<Squnion> (insert x A))"
proof (cases "A = {}")
case True with P and xA show ?thesis
by (simp add: sup_of_singletonI)
next
case False with P and xA show ?thesis
by (simp add: sup_insertI finite_sup_least)
qed

lemma (in lattice) finite_sup_closed:
"[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
proof (induct set: Finites)
case empty then show ?case by simp
next
case insert then show ?case
by - (rule finite_sup_insertI, simp_all)
qed

lemma (in lattice) join_left:
"[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
by (rule joinI [folded join_def]) (blast dest: least_mem)

lemma (in lattice) join_right:
"[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
by (rule joinI [folded join_def]) (blast dest: least_mem)

lemma (in lattice) sup_of_two_least:
"[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
proof (unfold sup_def)
assume L: "x \<in> carrier L"  "y \<in> carrier L"
with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
by (fast intro: theI2 least_unique)  (* blast fails *)
qed

lemma (in lattice) join_le:
assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
shows "x \<squnion> y \<sqsubseteq> z"
proof (rule joinI)
fix s
assume "least L s (Upper L {x, y})"
with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
qed

lemma (in lattice) join_assoc_lemma:
assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
shows "x \<squnion> (y \<squnion> z) = \<Squnion>{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 \<squnion> (y \<squnion> z) = s"
proof (rule anti_sym)
from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
by (fastsimp intro!: join_le elim: least_Upper_above)
next
from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
by (erule_tac least_le)
(blast intro!: Upper_memI intro: trans join_left join_right join_closed)
qed (simp_all add: L least_carrier [OF sup])
qed (simp_all add: L)

lemma join_comm:
includes struct L
shows "x \<squnion> y = y \<squnion> x"
by (unfold join_def) (simp add: insert_commute)

lemma (in lattice) join_assoc:
assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
proof -
have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma)
also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute)
also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
finally show ?thesis .
qed

subsubsection {* Infimum *}

lemma (in lattice) meetI:
"[| !!i. greatest L i (Lower L {x, y}) ==> P i;
x \<in> carrier L; y \<in> carrier L |]
==> P (x \<sqinter> y)"
proof (unfold meet_def inf_def)
assume L: "x \<in> carrier L"  "y \<in> 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 (THE g. greatest L g (Lower L {x, y}))"
by (fast intro: theI2 greatest_unique P)
qed

lemma (in lattice) meet_closed [simp]:
"[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
by (rule meetI) (rule greatest_carrier)

lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
"x \<in> carrier L ==> greatest L x (Lower L {x})"
by (rule greatest_LowerI) fast+

lemma (in partial_order) inf_of_singleton [simp]:
includes struct L
shows "x \<in> carrier L ==> \<Sqinter> {x} = x"
by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)

text {* Condition on A: infimum exists. *}

lemma (in lattice) inf_insertI:
"[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
==> P (\<Sqinter>(insert x A))"
proof (unfold inf_def)
assume L: "x \<in> carrier L"  "A \<subseteq> 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 \<in> 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 (THE g. greatest L g (Lower L (insert x A)))"
proof (rule theI2)
show "greatest L i (Lower L (insert x A))"
proof (rule greatest_LowerI)
fix z
assume "z \<in> insert x A"
then show "i \<sqsubseteq> z"
proof
assume "z = x" then show ?thesis
by (simp add: greatest_Lower_above [OF greatest_i] L La)
next
assume "z \<in> A"
with L greatest_i greatest_a show ?thesis
by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
qed
next
fix y
assume y: "y \<in> Lower L (insert x A)"
show "y \<sqsubseteq> i"
proof (rule greatest_le [OF greatest_i], rule Lower_memI)
fix z
assume z: "z \<in> {a, x}"
then show "y \<sqsubseteq> z"
proof
have y': "y \<in> Lower L A"
apply (rule subsetD [where A = "Lower L (insert x A)"])
apply (rule Lower_antimono) apply clarify apply assumption
done
assume "z = a"
with y' greatest_a show ?thesis by (fast dest: greatest_le)
next
assume "z \<in> {x}"
with y L show ?thesis by blast
qed
qed (rule Lower_closed [THEN subsetD])
next
from L show "insert x A \<subseteq> carrier L" by simp
from greatest_i show "i \<in> carrier L" by simp
qed
next
fix g
assume greatest_g: "greatest L g (Lower L (insert x A))"
show "g = i"
proof (rule greatest_unique)
show "greatest L i (Lower L (insert x A))"
proof (rule greatest_LowerI)
fix z
assume "z \<in> insert x A"
then show "i \<sqsubseteq> z"
proof
assume "z = x" then show ?thesis
by (simp add: greatest_Lower_above [OF greatest_i] L La)
next
assume "z \<in> A"
with L greatest_i greatest_a show ?thesis
by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
qed
next
fix y
assume y: "y \<in> Lower L (insert x A)"
show "y \<sqsubseteq> i"
proof (rule greatest_le [OF greatest_i], rule Lower_memI)
fix z
assume z: "z \<in> {a, x}"
then show "y \<sqsubseteq> z"
proof
have y': "y \<in> Lower L A"
apply (rule subsetD [where A = "Lower L (insert x A)"])
apply (rule Lower_antimono) apply clarify apply assumption
done
assume "z = a"
with y' greatest_a show ?thesis by (fast dest: greatest_le)
next
assume "z \<in> {x}"
with y L show ?thesis by blast
qed
qed (rule Lower_closed [THEN subsetD])
next
from L show "insert x A \<subseteq> carrier L" by simp
from greatest_i show "i \<in> carrier L" by simp
qed
qed
qed
qed

lemma (in lattice) finite_inf_greatest:
"[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
proof (induct set: Finites)
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 add: inf_of_singletonI)
next
case False
from insert show ?thesis
proof (rule_tac inf_insertI)
from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
qed simp_all
qed
qed

lemma (in lattice) finite_inf_insertI:
assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
shows "P (\<Sqinter> (insert x A))"
proof (cases "A = {}")
case True with P and xA show ?thesis
by (simp add: inf_of_singletonI)
next
case False with P and xA show ?thesis
by (simp add: inf_insertI finite_inf_greatest)
qed

lemma (in lattice) finite_inf_closed:
"[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
proof (induct set: Finites)
case empty then show ?case by simp
next
case insert then show ?case
by (rule_tac finite_inf_insertI) (simp_all)
qed

lemma (in lattice) meet_left:
"[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
by (rule meetI [folded meet_def]) (blast dest: greatest_mem)

lemma (in lattice) meet_right:
"[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
by (rule meetI [folded meet_def]) (blast dest: greatest_mem)

lemma (in lattice) inf_of_two_greatest:
"[| x \<in> carrier L; y \<in> carrier L |] ==>
greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
proof (unfold inf_def)
assume L: "x \<in> carrier L"  "y \<in> carrier L"
with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
with L
show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
by (fast intro: theI2 greatest_unique)  (* blast fails *)
qed

lemma (in lattice) meet_le:
assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
shows "z \<sqsubseteq> x \<sqinter> y"
proof (rule meetI)
fix i
assume "greatest L i (Lower L {x, y})"
with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
qed

lemma (in lattice) meet_assoc_lemma:
assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{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 \<sqinter> (y \<sqinter> z) = i"
proof (rule anti_sym)
from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
by (fastsimp intro!: meet_le elim: greatest_Lower_above)
next
from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
by (erule_tac greatest_le)
(blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
qed (simp_all add: L greatest_carrier [OF inf])
qed (simp_all add: L)

lemma meet_comm:
includes struct L
shows "x \<sqinter> y = y \<sqinter> x"
by (unfold meet_def) (simp add: insert_commute)

lemma (in lattice) meet_assoc:
assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
proof -
have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma)
finally show ?thesis .
qed

subsection {* Total Orders *}

locale total_order = lattice +
assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"

text {* Introduction rule: the usual definition of total order *}

lemma (in partial_order) total_orderI:
assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
shows "total_order L"
proof (rule total_order.intro)
show "lattice_axioms L"
proof (rule lattice_axioms.intro)
fix x y
assume L: "x \<in> carrier L"  "y \<in> carrier L"
show "EX s. least L s (Upper L {x, y})"
proof -
note total L
moreover
{
assume "x \<sqsubseteq> y"
with L have "least L y (Upper L {x, y})"
by (rule_tac least_UpperI) auto
}
moreover
{
assume "y \<sqsubseteq> 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 \<in> carrier L"  "y \<in> carrier L"
show "EX i. greatest L i (Lower L {x, y})"
proof -
note total L
moreover
{
assume "y \<sqsubseteq> x"
with L have "greatest L y (Lower L {x, y})"
by (rule_tac greatest_LowerI) auto
}
moreover
{
assume "x \<sqsubseteq> y"
with L have "greatest L x (Lower L {x, y})"
by (rule_tac greatest_LowerI) auto
}
ultimately show ?thesis by blast
qed
qed
qed (assumption | rule total_order_axioms.intro)+

subsection {* Complete lattices *}

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

text {* Introduction rule: the usual definition of complete lattice *}

lemma (in partial_order) complete_latticeI:
assumes sup_exists:
"!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
shows "complete_lattice L"
proof (rule complete_lattice.intro)
show "lattice_axioms L"
by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
qed (assumption | rule complete_lattice_axioms.intro)+

constdefs (structure L)
top :: "_ => 'a" ("\<top>\<index>")
"\<top> == sup L (carrier L)"

bottom :: "_ => 'a" ("\<bottom>\<index>")
"\<bottom> == inf L (carrier L)"

lemma (in complete_lattice) supI:
"[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
==> P (\<Squnion>A)"
proof (unfold sup_def)
assume L: "A \<subseteq> 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 (THE l. least L l (Upper L A))"
by (fast intro: theI2 least_unique P)
qed

lemma (in complete_lattice) sup_closed [simp]:
"A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
by (rule supI) simp_all

lemma (in complete_lattice) top_closed [simp, intro]:
"\<top> \<in> carrier L"
by (unfold top_def) simp

lemma (in complete_lattice) infI:
"[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
==> P (\<Sqinter>A)"
proof (unfold inf_def)
assume L: "A \<subseteq> 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 (THE l. greatest L l (Lower L A))"
by (fast intro: theI2 greatest_unique P)
qed

lemma (in complete_lattice) inf_closed [simp]:
"A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
by (rule infI) simp_all

lemma (in complete_lattice) bottom_closed [simp, intro]:
"\<bottom> \<in> 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 partial_order) complete_lattice_criterion1:
assumes top_exists: "EX g. greatest L g (carrier L)"
and inf_exists:
"!!A. [| A \<subseteq> 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 \<subseteq> carrier L"
let ?B = "Upper L A"
from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B ~= {}" by fast
have B_L: "?B \<subseteq> 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 UpperD)
apply assumption
apply (rule L)
apply (fast intro: L [THEN subsetD])
apply (erule greatest_Lower_above [OF b_inf_B])
apply simp
apply (rule L)
apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
done
then show "EX s. least L s (Upper L A)" ..
next
fix A
assume L: "A \<subseteq> carrier L"
show "EX i. greatest L i (Lower L A)"
proof (cases "A = {}")
case True then show ?thesis
by (simp add: top_exists)
next
case False with L show ?thesis
by (rule inf_exists)
qed
qed

(* TODO: prove dual version *)

subsection {* Examples *}

subsubsection {* Powerset of a set is a complete lattice *}

theorem powerset_is_complete_lattice:
"complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
(is "complete_lattice ?L")
proof (rule partial_order.complete_latticeI)
show "partial_order ?L"
by (rule partial_order.intro) auto
next
fix B
assume "B \<subseteq> carrier ?L"
then have "least ?L (\<Union> B) (Upper ?L B)"
by (fastsimp intro!: least_UpperI simp: Upper_def)
then show "EX s. least ?L s (Upper ?L B)" ..
next
fix B
assume "B \<subseteq> carrier ?L"
then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
@{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
by (fastsimp 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