(* Title: HOL/Algebra/Lattice.thy

Author: Clemens Ballarin, started 7 November 2003

Copyright: Clemens Ballarin

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 "\<sqsubseteq>\<index>" 50)

locale weak_partial_order = equivalence L for L (structure) +

assumes le_refl [intro, simp]:

"x ∈ carrier L ==> x \<sqsubseteq> x"

and weak_le_antisym [intro]:

"[| x \<sqsubseteq> y; y \<sqsubseteq> x; x ∈ carrier L; y ∈ carrier L |] ==> x .= y"

and le_trans [trans]:

"[| x \<sqsubseteq> y; y \<sqsubseteq> z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x \<sqsubseteq> z"

and le_cong:

"[| x .= y; z .= w; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L; w ∈ carrier L |] ==>

x \<sqsubseteq> z <-> y \<sqsubseteq> w"

definition

lless :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)

where "x \<sqsubset>⇘_{L⇙}y <-> x \<sqsubseteq>⇘_{L⇙}y & x .≠⇘_{L⇙}y"

subsubsection {* The order relation *}

context weak_partial_order

begin

lemma le_cong_l [intro, trans]:

"[| x .= y; y \<sqsubseteq> z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x \<sqsubseteq> z"

by (auto intro: le_cong [THEN iffD2])

lemma le_cong_r [intro, trans]:

"[| x \<sqsubseteq> y; y .= z; x ∈ carrier L; y ∈ carrier L; z ∈ carrier L |] ==> x \<sqsubseteq> z"

by (auto intro: le_cong [THEN iffD1])

lemma weak_refl [intro, simp]: "[| x .= y; x ∈ carrier L; y ∈ carrier L |] ==> x \<sqsubseteq> y"

by (simp add: le_cong_l)

end

lemma weak_llessI:

fixes R (structure)

assumes "x \<sqsubseteq> y" and "~(x .= y)"

shows "x \<sqsubset> y"

using assms unfolding lless_def by simp

lemma lless_imp_le:

fixes R (structure)

assumes "x \<sqsubset> y"

shows "x \<sqsubseteq> y"

using assms unfolding lless_def by simp

lemma weak_lless_imp_not_eq:

fixes R (structure)

assumes "x \<sqsubset> y"

shows "¬ (x .= y)"

using assms unfolding lless_def by simp

lemma weak_llessE:

fixes R (structure)

assumes p: "x \<sqsubset> y" and e: "[|x \<sqsubseteq> 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' \<sqsubset> y"

and carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L"

shows "x \<sqsubset> y"

using assms unfolding lless_def by (auto intro: trans sym)

lemma (in weak_partial_order) lless_cong_r [trans]:

assumes xy: "x \<sqsubset> y"

and yy': "y .= y'"

and carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L"

shows "x \<sqsubset> 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 \<sqsubset> b" "b \<sqsubset> a"

shows "P"

using assms

by (elim weak_llessE) auto

lemma (in weak_partial_order) lless_trans [trans]:

assumes "a \<sqsubset> b" "b \<sqsubset> c"

and carr[simp]: "a ∈ carrier L" "b ∈ carrier L" "c ∈ carrier L"

shows "a \<sqsubset> 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 \<sqsubseteq>⇘_{L⇙}u)} ∩ carrier L"

definition

Lower :: "[_, 'a set] => 'a set"

where "Lower L A = {l. (ALL x. x ∈ A ∩ carrier L --> l \<sqsubseteq>⇘_{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 \<sqsubseteq> 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 \<sqsubseteq> u"

unfolding Upper_def elem_def

by (blast dest: sym)

lemma Upper_memI:

fixes L (structure)

shows "[| !! y. y ∈ A ==> y \<sqsubseteq> x; x ∈ carrier L |] ==> x ∈ Upper L A"

by (unfold Upper_def) blast

lemma (in weak_partial_order) Upper_elemI:

"[| !! y. y ∈ A ==> y \<sqsubseteq> 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 \<sqsubseteq> a" by (intro Upper_memD[OF aelem, THEN conjunct1] Acarr)

also note aa'

finally

show "y \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> x" by (simp add: aLxCond aA)

finally show "a' \<sqsubseteq> 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' \<sqsubseteq> 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' \<sqsubseteq> x" by (simp add: a'LxCond a'A')

finally show "a \<sqsubseteq> 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 \<sqsubseteq> x ∧ l ∈ carrier L"

by (unfold Lower_def) blast

lemma Lower_memI:

fixes L (structure)

shows "[| !! y. y ∈ A ==> x \<sqsubseteq> 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 \<sqsubseteq> a"

hence aLxCond: "!!a. [|a ∈ A; a ∈ carrier L|] ==> x \<sqsubseteq> 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 \<sqsubseteq> a" by (rule aLxCond)

also note a'a[symmetric]

finally

show "x \<sqsubseteq> 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 \<sqsubseteq> a'"

hence a'LxCond: "!!a'. [|a' ∈ A'; a' ∈ carrier L|] ==> x \<sqsubseteq> 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 \<sqsubseteq> a'" by (rule a'LxCond)

also note aa'[symmetric]

finally show "x \<sqsubseteq> 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 \<sqsubseteq>⇘_{L⇙}x)"

definition

greatest :: "[_, 'a, 'a set] => bool"

where "greatest L g A <-> A ⊆ carrier L & g ∈ A & (ALL x : A. x \<sqsubseteq>⇘_{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 \<sqsubseteq> 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 \<sqsubseteq> s"

and below: "!! y. y ∈ Upper L A ==> s \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> x"

and above: "!! y. y ∈ Lower L A ==> y \<sqsubseteq> 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 \<sqsubseteq> 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 \<sqsubseteq> x"

by (unfold greatest_def) blast

text {* Supremum and infimum *}

definition

sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90)

where "\<Squnion>⇘_{L⇙}A = (SOME x. least L x (Upper L A))"

definition

inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90)

where "\<Sqinter>⇘_{L⇙}A = (SOME x. greatest L x (Lower L A))"

definition

join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)

where "x \<squnion>⇘_{L⇙}y = \<Squnion>⇘_{L⇙}{x, y}"

definition

meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 70)

where "x \<sqinter>⇘_{L⇙}y = \<Sqinter>⇘_{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 \<squnion> 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 \<squnion> 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 \<squnion> y .= x' \<squnion> 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 \<squnion> y .= x \<squnion> 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 ==> \<Squnion>{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 ==> \<Squnion>{x} ∈ carrier L"

unfolding sup_def

by (rule someI2) (auto intro: sup_of_singletonI)

text {* Condition on @{text 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 (\<Squnion>(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 \<sqsubseteq> 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 \<sqsubseteq> y"

proof (rule least_le [OF least_s], rule Upper_memI)

fix z

assume z: "z ∈ {a, x}"

then show "z \<sqsubseteq> 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 (\<Squnion>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 (\<Squnion>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 (\<Squnion> (insert x A))"

proof (cases "A = {}")

case True with P and xA show ?thesis

by (simp add: finite_sup_least)

next

case False with P and xA show ?thesis

by (simp add: sup_insertI finite_sup_least)

qed

lemma (in weak_upper_semilattice) finite_sup_closed [simp]:

"[| finite A; A ⊆ carrier L; A ~= {} |] ==> \<Squnion>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 \<sqsubseteq> x \<squnion> 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 \<sqsubseteq> x \<squnion> 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 (\<Squnion>{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 \<sqsubseteq> z" "y \<sqsubseteq> z"

and x: "x ∈ carrier L" and y: "y ∈ carrier L" and z: "z ∈ carrier L"

shows "x \<squnion> y \<sqsubseteq> z"

proof (rule joinI [OF _ x y])

fix s

assume "least L s (Upper L {x, y})"

with sub z show "s \<sqsubseteq> 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 \<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 weak_le_antisym)

from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"

by (fastforce 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: le_trans join_left join_right join_closed)

qed (simp_all add: L least_closed [OF sup])

qed (simp_all add: L)

text {* Commutativity holds for @{text "="}. *}

lemma join_comm:

fixes L (structure)

shows "x \<squnion> y = y \<squnion> 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 \<squnion> y) \<squnion> z .= x \<squnion> (y \<squnion> z)"

proof -

(* FIXME: could be simplified by improved simp: uniform use of .=,

omit [symmetric] in last step. *)

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: weak_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: 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 \<sqinter> 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 \<sqinter> 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 \<sqinter> y .= x' \<sqinter> 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 \<sqinter> y .= x \<sqinter> 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 ==> \<Sqinter>{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 ==> \<Sqinter>{x} ∈ carrier L"

unfolding inf_def

by (rule someI2) (auto intro: inf_of_singletonI)

text {* Condition on @{text 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 (\<Sqinter>(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 \<sqsubseteq> 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 \<sqsubseteq> i"

proof (rule greatest_le [OF greatest_i], rule Lower_memI)

fix z

assume z: "z ∈ {a, x}"

then show "y \<sqsubseteq> 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 (\<Sqinter>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 (\<Sqinter>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 (\<Sqinter> (insert x A))"

proof (cases "A = {}")

case True with P and xA show ?thesis

by (simp add: finite_inf_greatest)

next

case False with P and xA show ?thesis

by (simp add: inf_insertI finite_inf_greatest)

qed

lemma (in weak_lower_semilattice) finite_inf_closed [simp]:

"[| finite A; A ⊆ carrier L; A ~= {} |] ==> \<Sqinter>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 \<sqinter> y \<sqsubseteq> 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 \<sqinter> y \<sqsubseteq> 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 (\<Sqinter> {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 \<sqsubseteq> x" "z \<sqsubseteq> y"

and x: "x ∈ carrier L" and y: "y ∈ carrier L" and z: "z ∈ carrier L"

shows "z \<sqsubseteq> x \<sqinter> y"

proof (rule meetI [OF _ x y])

fix i

assume "greatest L i (Lower L {x, y})"

with sub z show "z \<sqsubseteq> 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 \<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 weak_le_antisym)

from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"

by (fastforce intro!: meet_le elim: greatest_Lower_below)

next

from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> 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])

qed (simp_all add: L)

lemma meet_comm:

fixes L (structure)

shows "x \<sqinter> y = y \<sqinter> 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 \<sqinter> y) \<sqinter> z .= x \<sqinter> (y \<sqinter> z)"

proof -

(* FIXME: improved simp, see weak_join_assoc above *)

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: weak_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: 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 \<sqsubseteq> y | y \<sqsubseteq> 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 \<sqsubseteq> y | y \<sqsubseteq> x"

shows "weak_total_order L"

by default (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 \<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 ∈ carrier L" "y ∈ 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

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 default (auto intro: sup_exists inf_exists)

definition

top :: "_ => 'a" ("\<top>\<index>")

where "\<top>⇘_{L⇙}= sup L (carrier L)"

definition

bottom :: "_ => 'a" ("⊥\<index>")

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 (\<Squnion>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 ==> \<Squnion>A ∈ carrier L"

by (rule supI) simp_all

lemma (in weak_complete_lattice) top_closed [simp, intro]:

"\<top> ∈ 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 (\<Sqinter>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 ==> \<Sqinter>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

by (simp add: top_exists)

next

case False with L show ?thesis

by (rule inf_exists)

qed

qed

(* TODO: prove dual version *)

subsection {* Orders and Lattices where @{text 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 \<sqsubseteq> y; y \<sqsubseteq> x; x ∈ carrier L; y ∈ carrier L |] ==> x = y"

using weak_le_antisym unfolding eq_is_equal .

lemma lless_eq:

"x \<sqsubset> y <-> x \<sqsubseteq> y & x ≠ y"

unfolding lless_def by (simp add: eq_is_equal)

lemma lless_asym:

assumes "a ∈ carrier L" "b ∈ carrier L"

and "a \<sqsubset> b" "b \<sqsubset> 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 default (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 default (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 ==> \<Squnion>{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 \<squnion> (y \<squnion> z) = \<Squnion>{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 \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> 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 ==> \<Sqinter>{x} = x"

using weak_inf_of_singleton unfolding eq_is_equal .

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

lemma (in lower_semilattice) meet_assoc_lemma:

assumes L: "x ∈ carrier L" "y ∈ carrier L" "z ∈ carrier L"

shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{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 \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> 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 \<sqsubseteq> y | y \<sqsubseteq> x"

sublocale total_order < weak: weak_total_order

by default (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 \<sqsubseteq> y | y \<sqsubseteq> x"

shows "total_order L"

by default (rule total)

text {* Total orders are lattices. *}

sublocale total_order < weak: lattice

by default (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 default (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 default (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

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 {* 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 default auto

next

fix B

assume "B ⊆ carrier ?L"

then have "least ?L (\<Union> 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 (\<Inter> B ∩ 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 (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