# Theory Lattice

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

Most congruence rules by Stephan Hohe.
With additional contributions from Alasdair Armstrong and Simon Foster.
*)

theory Lattice
imports Order
begin

section ‹Lattices›

subsection ‹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 supr ::
"('a, 'b) gorder_scheme ⇒ 'c set ⇒ ('c ⇒ 'a) ⇒ 'a "
where "supr L A f = ⨆⇘L⇙(f ` A)"

definition infi ::
"('a, 'b) gorder_scheme ⇒ 'c set ⇒ ('c ⇒ 'a) ⇒ 'a "
where "infi L A f = ⨅⇘L⇙(f ` A)"

syntax
"_inf1"     :: "('a, 'b) gorder_scheme ⇒ pttrns ⇒ 'a ⇒ 'a" ("(3IINFı _./ _)" [0, 10] 10)
"_inf"      :: "('a, 'b) gorder_scheme ⇒ pttrn ⇒ 'c set ⇒ 'a ⇒ 'a"  ("(3IINFı _:_./ _)" [0, 0, 10] 10)
"_sup1"     :: "('a, 'b) gorder_scheme ⇒ pttrns ⇒ 'a ⇒ 'a" ("(3SSUPı _./ _)" [0, 10] 10)
"_sup"      :: "('a, 'b) gorder_scheme ⇒ pttrn ⇒ 'c set ⇒ 'a ⇒ 'a"  ("(3SSUPı _:_./ _)" [0, 0, 10] 10)

translations
"IINF⇘L⇙ x. B"     == "CONST infi L CONST UNIV (%x. B)"
"IINF⇘L⇙ x:A. B"   == "CONST infi L A (%x. B)"
"SSUP⇘L⇙ x. B"     == "CONST supr L CONST UNIV (%x. B)"
"SSUP⇘L⇙ x:A. B"   == "CONST supr L A (%x. B)"

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}"

definition
LEAST_FP :: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ 'a" ("LFPı") where
"LEAST_FP L f = ⨅⇘L⇙ {u ∈ carrier L. f u ⊑⇘L⇙ u}"    ― ‹least fixed point›

definition
GREATEST_FP:: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ 'a" ("GFPı") where
"GREATEST_FP L f = ⨆⇘L⇙ {u ∈ carrier L. u ⊑⇘L⇙ f u}"    ― ‹greatest fixed point›

subsection ‹Dual operators›

lemma sup_dual [simp]:
"⨆⇘inv_gorder L⇙A = ⨅⇘L⇙A"

lemma inf_dual [simp]:
"⨅⇘inv_gorder L⇙A = ⨆⇘L⇙A"

lemma join_dual [simp]:
"p ⊔⇘inv_gorder L⇙ q = p ⊓⇘L⇙ q"

lemma meet_dual [simp]:
"p ⊓⇘inv_gorder L⇙ q = p ⊔⇘L⇙ q"

lemma top_dual [simp]:
"⊤⇘inv_gorder L⇙ = ⊥⇘L⇙"

lemma bottom_dual [simp]:
"⊥⇘inv_gorder L⇙ = ⊤⇘L⇙"

lemma LFP_dual [simp]:
"LEAST_FP (inv_gorder L) f = GREATEST_FP L f"

lemma GFP_dual [simp]:
"GREATEST_FP (inv_gorder L) f = LEAST_FP L f"

subsection ‹Lattices›

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

locale weak_lattice = weak_upper_semilattice + weak_lower_semilattice

lemma (in weak_lattice) dual_weak_lattice:
"weak_lattice (inv_gorder L)"
proof -
interpret dual: weak_partial_order "inv_gorder L"
by (metis dual_weak_order)
show ?thesis
proof qed (simp_all add: inf_of_two_exists sup_of_two_exists)
qed

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"
by (meson Upper_antimono in_mono subset_insertI y)
assume "z = a"
with y' least_a show ?thesis by (fast dest: least_le)
next
assume "z ∈ {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_lattice) weak_le_iff_meet:
assumes "x ∈ carrier L" "y ∈ carrier L"
shows "x ⊑ y ⟷ (x ⊔ y) .= y"
by (meson assms(1) assms(2) join_closed join_le join_left join_right le_cong_r local.le_refl weak_le_antisym)

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"
by (meson Lower_antimono in_mono subset_insertI y)
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_lattice) weak_le_iff_join:
assumes "x ∈ carrier L" "y ∈ carrier L"
shows "x ⊑ y ⟷ x .= (x ⊓ y)"
by (meson assms(1) assms(2) local.le_refl local.le_trans meet_closed meet_le meet_left meet_right weak_le_antisym weak_refl)

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

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 "∃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 "∃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 ‹Weak Bounded Lattices›

locale weak_bounded_lattice =
weak_lattice +
weak_partial_order_bottom +
weak_partial_order_top
begin

lemma bottom_meet: "x ∈ carrier L ⟹ ⊥ ⊓ x .= ⊥"
by (metis bottom_least least_def meet_closed meet_left weak_le_antisym)

lemma bottom_join: "x ∈ carrier L ⟹ ⊥ ⊔ x .= x"
by (metis bottom_least join_closed join_le join_right le_refl least_def weak_le_antisym)

lemma bottom_weak_eq:
"⟦ b ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ b ⊑ x ⟧ ⟹ b .= ⊥"
by (metis bottom_closed bottom_lower weak_le_antisym)

lemma top_join: "x ∈ carrier L ⟹ ⊤ ⊔ x .= ⊤"
by (metis join_closed join_left top_closed top_higher weak_le_antisym)

lemma top_meet: "x ∈ carrier L ⟹ ⊤ ⊓ x .= x"
by (metis le_refl meet_closed meet_le meet_right top_closed top_higher weak_le_antisym)

lemma top_weak_eq:  "⟦ t ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ x ⊑ t ⟧ ⟹ t .= ⊤"
by (metis top_closed top_higher weak_le_antisym)

end

sublocale weak_bounded_lattice ⊆ weak_partial_order ..

subsection ‹Lattices where ‹eq› is the Equality›

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

sublocale upper_semilattice ⊆ weak?: weak_upper_semilattice
by unfold_locales (rule sup_of_two_exists)

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

sublocale lower_semilattice ⊆ weak?: weak_lower_semilattice
by unfold_locales (rule inf_of_two_exists)

locale lattice = upper_semilattice + lower_semilattice

sublocale lattice ⊆ weak_lattice ..

lemma (in lattice) dual_lattice:
"lattice (inv_gorder L)"
proof -
interpret dual: weak_lattice "inv_gorder L"
by (metis dual_weak_lattice)

show ?thesis
apply (unfold_locales)
apply (rule eq_is_equal)
done
qed

lemma (in lattice) le_iff_join:
assumes "x ∈ carrier L" "y ∈ carrier L"
shows "x ⊑ y ⟷ x = (x ⊓ y)"
by (simp add: assms(1) assms(2) eq_is_equal weak_le_iff_join)

lemma (in lattice) le_iff_meet:
assumes "x ∈ carrier L" "y ∈ carrier L"
shows "x ⊑ y ⟷ (x ⊔ y) = y"
by (simp add: assms eq_is_equal weak_le_iff_meet)

text ‹ Total orders are lattices. ›

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

text ‹Functions that preserve joins and meets›

definition join_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"join_pres X Y f ≡ lattice X ∧ lattice Y ∧ (∀ x ∈ carrier X. ∀ y ∈ carrier X. f (x ⊔⇘X⇙ y) = f x ⊔⇘Y⇙ f y)"

definition meet_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"meet_pres X Y f ≡ lattice X ∧ lattice Y ∧ (∀ x ∈ carrier X. ∀ y ∈ carrier X. f (x ⊓⇘X⇙ y) = f x ⊓⇘Y⇙ f y)"

lemma join_pres_isotone:
assumes "f ∈ carrier X → carrier Y" "join_pres X Y f"
shows "isotone X Y f"
proof (rule isotoneI)
show "weak_partial_order X" "weak_partial_order Y"
using assms unfolding join_pres_def lattice_def upper_semilattice_def lower_semilattice_def
by (meson partial_order.axioms(1))+
show "⋀x y. ⟦x ∈ carrier X; y ∈ carrier X; x ⊑⇘X⇙ y⟧ ⟹ f x ⊑⇘Y⇙ f y"
by (metis (no_types, lifting) PiE assms join_pres_def lattice.le_iff_meet)
qed

lemma meet_pres_isotone:
assumes "f ∈ carrier X → carrier Y" "meet_pres X Y f"
shows "isotone X Y f"
proof (rule isotoneI)
show "weak_partial_order X" "weak_partial_order Y"
using assms unfolding meet_pres_def lattice_def upper_semilattice_def lower_semilattice_def
by (meson partial_order.axioms(1))+
show "⋀x y. ⟦x ∈ carrier X; y ∈ carrier X; x ⊑⇘X⇙ y⟧ ⟹ f x ⊑⇘Y⇙ f y"
by (metis (no_types, lifting) PiE assms lattice.le_iff_join meet_pres_def)
qed

subsection ‹Bounded Lattices›

locale bounded_lattice =
lattice +
weak_partial_order_bottom +
weak_partial_order_top

sublocale bounded_lattice ⊆ weak_bounded_lattice ..

context bounded_lattice
begin

lemma bottom_eq:
"⟦ b ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ b ⊑ x ⟧ ⟹ b = ⊥"
by (metis bottom_closed bottom_lower le_antisym)

lemma top_eq:  "⟦ t ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ x ⊑ t ⟧ ⟹ t = ⊤"
by (metis le_antisym top_closed top_higher)

end

hide_const (open) Lattice.inf
hide_const (open) Lattice.sup

end
```