(* Title: HOL/Lattices.thy
Author: Tobias Nipkow
*)
section \<open>Abstract lattices\<close>
theory Lattices
imports Groups
begin
subsection \<open>Abstract semilattice\<close>
text \<open>
These locales provide a basic structure for interpretation into
bigger structures; extensions require careful thinking, otherwise
undesired effects may occur due to interpretation.
\<close>
locale semilattice = abel_semigroup +
assumes idem [simp]: "a \<^bold>* a = a"
begin
lemma left_idem [simp]: "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b"
by (simp add: assoc [symmetric])
lemma right_idem [simp]: "(a \<^bold>* b) \<^bold>* b = a \<^bold>* b"
by (simp add: assoc)
end
locale semilattice_neutr = semilattice + comm_monoid
locale semilattice_order = semilattice +
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold>\<le>" 50)
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold><" 50)
assumes order_iff: "a \<^bold>\<le> b \<longleftrightarrow> a = a \<^bold>* b"
and strict_order_iff: "a \<^bold>< b \<longleftrightarrow> a = a \<^bold>* b \<and> a \<noteq> b"
begin
lemma orderI: "a = a \<^bold>* b \<Longrightarrow> a \<^bold>\<le> b"
by (simp add: order_iff)
lemma orderE:
assumes "a \<^bold>\<le> b"
obtains "a = a \<^bold>* b"
using assms by (unfold order_iff)
sublocale ordering less_eq less
proof
show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b" for a b
by (simp add: order_iff strict_order_iff)
next
show "a \<^bold>\<le> a" for a
by (simp add: order_iff)
next
fix a b
assume "a \<^bold>\<le> b" "b \<^bold>\<le> a"
then have "a = a \<^bold>* b" "a \<^bold>* b = b"
by (simp_all add: order_iff commute)
then show "a = b" by simp
next
fix a b c
assume "a \<^bold>\<le> b" "b \<^bold>\<le> c"
then have "a = a \<^bold>* b" "b = b \<^bold>* c"
by (simp_all add: order_iff commute)
then have "a = a \<^bold>* (b \<^bold>* c)"
by simp
then have "a = (a \<^bold>* b) \<^bold>* c"
by (simp add: assoc)
with \<open>a = a \<^bold>* b\<close> [symmetric] have "a = a \<^bold>* c" by simp
then show "a \<^bold>\<le> c" by (rule orderI)
qed
lemma cobounded1 [simp]: "a \<^bold>* b \<^bold>\<le> a"
by (simp add: order_iff commute)
lemma cobounded2 [simp]: "a \<^bold>* b \<^bold>\<le> b"
by (simp add: order_iff)
lemma boundedI:
assumes "a \<^bold>\<le> b" and "a \<^bold>\<le> c"
shows "a \<^bold>\<le> b \<^bold>* c"
proof (rule orderI)
from assms obtain "a \<^bold>* b = a" and "a \<^bold>* c = a"
by (auto elim!: orderE)
then show "a = a \<^bold>* (b \<^bold>* c)"
by (simp add: assoc [symmetric])
qed
lemma boundedE:
assumes "a \<^bold>\<le> b \<^bold>* c"
obtains "a \<^bold>\<le> b" and "a \<^bold>\<le> c"
using assms by (blast intro: trans cobounded1 cobounded2)
lemma bounded_iff [simp]: "a \<^bold>\<le> b \<^bold>* c \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<^bold>\<le> c"
by (blast intro: boundedI elim: boundedE)
lemma strict_boundedE:
assumes "a \<^bold>< b \<^bold>* c"
obtains "a \<^bold>< b" and "a \<^bold>< c"
using assms by (auto simp add: commute strict_iff_order elim: orderE intro!: that)+
lemma coboundedI1: "a \<^bold>\<le> c \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c"
by (rule trans) auto
lemma coboundedI2: "b \<^bold>\<le> c \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c"
by (rule trans) auto
lemma strict_coboundedI1: "a \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
using irrefl
by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order
elim: strict_boundedE)
lemma strict_coboundedI2: "b \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
using strict_coboundedI1 [of b c a] by (simp add: commute)
lemma mono: "a \<^bold>\<le> c \<Longrightarrow> b \<^bold>\<le> d \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c \<^bold>* d"
by (blast intro: boundedI coboundedI1 coboundedI2)
lemma absorb1: "a \<^bold>\<le> b \<Longrightarrow> a \<^bold>* b = a"
by (rule antisym) (auto simp: refl)
lemma absorb2: "b \<^bold>\<le> a \<Longrightarrow> a \<^bold>* b = b"
by (rule antisym) (auto simp: refl)
lemma absorb_iff1: "a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>* b = a"
using order_iff by auto
lemma absorb_iff2: "b \<^bold>\<le> a \<longleftrightarrow> a \<^bold>* b = b"
using order_iff by (auto simp add: commute)
end
locale semilattice_neutr_order = semilattice_neutr + semilattice_order
begin
sublocale ordering_top less_eq less "\<^bold>1"
by standard (simp add: order_iff)
end
text \<open>Passive interpretations for boolean operators\<close>
lemma semilattice_neutr_and:
"semilattice_neutr HOL.conj True"
by standard auto
lemma semilattice_neutr_or:
"semilattice_neutr HOL.disj False"
by standard auto
subsection \<open>Syntactic infimum and supremum operations\<close>
class inf =
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
class sup =
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
subsection \<open>Concrete lattices\<close>
class semilattice_inf = order + inf +
assumes inf_le1 [simp]: "x \<sqinter> y \<le> x"
and inf_le2 [simp]: "x \<sqinter> y \<le> y"
and inf_greatest: "x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<sqinter> z"
class semilattice_sup = order + sup +
assumes sup_ge1 [simp]: "x \<le> x \<squnion> y"
and sup_ge2 [simp]: "y \<le> x \<squnion> y"
and sup_least: "y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<squnion> z \<le> x"
begin
text \<open>Dual lattice.\<close>
lemma dual_semilattice: "class.semilattice_inf sup greater_eq greater"
by (rule class.semilattice_inf.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
end
class lattice = semilattice_inf + semilattice_sup
subsubsection \<open>Intro and elim rules\<close>
context semilattice_inf
begin
lemma le_infI1: "a \<le> x \<Longrightarrow> a \<sqinter> b \<le> x"
by (rule order_trans) auto
lemma le_infI2: "b \<le> x \<Longrightarrow> a \<sqinter> b \<le> x"
by (rule order_trans) auto
lemma le_infI: "x \<le> a \<Longrightarrow> x \<le> b \<Longrightarrow> x \<le> a \<sqinter> b"
by (fact inf_greatest) (* FIXME: duplicate lemma *)
lemma le_infE: "x \<le> a \<sqinter> b \<Longrightarrow> (x \<le> a \<Longrightarrow> x \<le> b \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans inf_le1 inf_le2)
lemma le_inf_iff: "x \<le> y \<sqinter> z \<longleftrightarrow> x \<le> y \<and> x \<le> z"
by (blast intro: le_infI elim: le_infE)
lemma le_iff_inf: "x \<le> y \<longleftrightarrow> x \<sqinter> y = x"
by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1] simp add: le_inf_iff)
lemma inf_mono: "a \<le> c \<Longrightarrow> b \<le> d \<Longrightarrow> a \<sqinter> b \<le> c \<sqinter> d"
by (fast intro: inf_greatest le_infI1 le_infI2)
lemma mono_inf: "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B" for f :: "'a \<Rightarrow> 'b::semilattice_inf"
by (auto simp add: mono_def intro: Lattices.inf_greatest)
end
context semilattice_sup
begin
lemma le_supI1: "x \<le> a \<Longrightarrow> x \<le> a \<squnion> b"
by (rule order_trans) auto
lemma le_supI2: "x \<le> b \<Longrightarrow> x \<le> a \<squnion> b"
by (rule order_trans) auto
lemma le_supI: "a \<le> x \<Longrightarrow> b \<le> x \<Longrightarrow> a \<squnion> b \<le> x"
by (fact sup_least) (* FIXME: duplicate lemma *)
lemma le_supE: "a \<squnion> b \<le> x \<Longrightarrow> (a \<le> x \<Longrightarrow> b \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans sup_ge1 sup_ge2)
lemma le_sup_iff: "x \<squnion> y \<le> z \<longleftrightarrow> x \<le> z \<and> y \<le> z"
by (blast intro: le_supI elim: le_supE)
lemma le_iff_sup: "x \<le> y \<longleftrightarrow> x \<squnion> y = y"
by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1] simp add: le_sup_iff)
lemma sup_mono: "a \<le> c \<Longrightarrow> b \<le> d \<Longrightarrow> a \<squnion> b \<le> c \<squnion> d"
by (fast intro: sup_least le_supI1 le_supI2)
lemma mono_sup: "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)" for f :: "'a \<Rightarrow> 'b::semilattice_sup"
by (auto simp add: mono_def intro: Lattices.sup_least)
end
subsubsection \<open>Equational laws\<close>
context semilattice_inf
begin
sublocale inf: semilattice inf
proof
fix a b c
show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
by (rule antisym) (auto intro: le_infI1 le_infI2 simp add: le_inf_iff)
show "a \<sqinter> b = b \<sqinter> a"
by (rule antisym) (auto simp add: le_inf_iff)
show "a \<sqinter> a = a"
by (rule antisym) (auto simp add: le_inf_iff)
qed
sublocale inf: semilattice_order inf less_eq less
by standard (auto simp add: le_iff_inf less_le)
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
by (fact inf.assoc)
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
by (fact inf.commute)
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
by (fact inf.left_commute)
lemma inf_idem: "x \<sqinter> x = x"
by (fact inf.idem) (* already simp *)
lemma inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
by (fact inf.left_idem) (* already simp *)
lemma inf_right_idem: "(x \<sqinter> y) \<sqinter> y = x \<sqinter> y"
by (fact inf.right_idem) (* already simp *)
lemma inf_absorb1: "x \<le> y \<Longrightarrow> x \<sqinter> y = x"
by (rule antisym) auto
lemma inf_absorb2: "y \<le> x \<Longrightarrow> x \<sqinter> y = y"
by (rule antisym) auto
lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
end
context semilattice_sup
begin
sublocale sup: semilattice sup
proof
fix a b c
show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
by (rule antisym) (auto intro: le_supI1 le_supI2 simp add: le_sup_iff)
show "a \<squnion> b = b \<squnion> a"
by (rule antisym) (auto simp add: le_sup_iff)
show "a \<squnion> a = a"
by (rule antisym) (auto simp add: le_sup_iff)
qed
sublocale sup: semilattice_order sup greater_eq greater
by standard (auto simp add: le_iff_sup sup.commute less_le)
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
by (fact sup.assoc)
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
by (fact sup.commute)
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
by (fact sup.left_commute)
lemma sup_idem: "x \<squnion> x = x"
by (fact sup.idem) (* already simp *)
lemma sup_left_idem [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
by (fact sup.left_idem)
lemma sup_absorb1: "y \<le> x \<Longrightarrow> x \<squnion> y = x"
by (rule antisym) auto
lemma sup_absorb2: "x \<le> y \<Longrightarrow> x \<squnion> y = y"
by (rule antisym) auto
lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
end
context lattice
begin
lemma dual_lattice: "class.lattice sup (\<ge>) (>) inf"
by (rule class.lattice.intro,
rule dual_semilattice,
rule class.semilattice_sup.intro,
rule dual_order)
(unfold_locales, auto)
lemma inf_sup_absorb [simp]: "x \<sqinter> (x \<squnion> y) = x"
by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
lemma sup_inf_absorb [simp]: "x \<squnion> (x \<sqinter> y) = x"
by (blast intro: antisym sup_ge1 sup_least inf_le1)
lemmas inf_sup_aci = inf_aci sup_aci
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
text \<open>Towards distributivity.\<close>
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<le> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<le> x \<sqinter> (y \<squnion> z)"
by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
text \<open>If you have one of them, you have them all.\<close>
lemma distrib_imp1:
assumes distrib: "\<And>x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
proof-
have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)"
by simp
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))"
by (simp add: distrib inf_commute sup_assoc del: sup_inf_absorb)
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
by (simp add: inf_commute)
also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:distrib)
finally show ?thesis .
qed
lemma distrib_imp2:
assumes distrib: "\<And>x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
proof-
have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)"
by simp
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))"
by (simp add: distrib sup_commute inf_assoc del: inf_sup_absorb)
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
by (simp add: sup_commute)
also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by (simp add:distrib)
finally show ?thesis .
qed
end
subsubsection \<open>Strict order\<close>
context semilattice_inf
begin
lemma less_infI1: "a < x \<Longrightarrow> a \<sqinter> b < x"
by (auto simp add: less_le inf_absorb1 intro: le_infI1)
lemma less_infI2: "b < x \<Longrightarrow> a \<sqinter> b < x"
by (auto simp add: less_le inf_absorb2 intro: le_infI2)
end
context semilattice_sup
begin
lemma less_supI1: "x < a \<Longrightarrow> x < a \<squnion> b"
using dual_semilattice
by (rule semilattice_inf.less_infI1)
lemma less_supI2: "x < b \<Longrightarrow> x < a \<squnion> b"
using dual_semilattice
by (rule semilattice_inf.less_infI2)
end
subsection \<open>Distributive lattices\<close>
class distrib_lattice = lattice +
assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
context distrib_lattice
begin
lemma sup_inf_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
by (simp add: sup_commute sup_inf_distrib1)
lemma inf_sup_distrib1: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
by (rule distrib_imp2 [OF sup_inf_distrib1])
lemma inf_sup_distrib2: "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
by (simp add: inf_commute inf_sup_distrib1)
lemma dual_distrib_lattice: "class.distrib_lattice sup (\<ge>) (>) inf"
by (rule class.distrib_lattice.intro, rule dual_lattice)
(unfold_locales, fact inf_sup_distrib1)
lemmas sup_inf_distrib = sup_inf_distrib1 sup_inf_distrib2
lemmas inf_sup_distrib = inf_sup_distrib1 inf_sup_distrib2
lemmas distrib = sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
end
subsection \<open>Bounded lattices and boolean algebras\<close>
class bounded_semilattice_inf_top = semilattice_inf + order_top
begin
sublocale inf_top: semilattice_neutr inf top
+ inf_top: semilattice_neutr_order inf top less_eq less
proof
show "x \<sqinter> \<top> = x" for x
by (rule inf_absorb1) simp
qed
end
class bounded_semilattice_sup_bot = semilattice_sup + order_bot
begin
sublocale sup_bot: semilattice_neutr sup bot
+ sup_bot: semilattice_neutr_order sup bot greater_eq greater
proof
show "x \<squnion> \<bottom> = x" for x
by (rule sup_absorb1) simp
qed
end
class bounded_lattice_bot = lattice + order_bot
begin
subclass bounded_semilattice_sup_bot ..
lemma inf_bot_left [simp]: "\<bottom> \<sqinter> x = \<bottom>"
by (rule inf_absorb1) simp
lemma inf_bot_right [simp]: "x \<sqinter> \<bottom> = \<bottom>"
by (rule inf_absorb2) simp
lemma sup_bot_left: "\<bottom> \<squnion> x = x"
by (fact sup_bot.left_neutral)
lemma sup_bot_right: "x \<squnion> \<bottom> = x"
by (fact sup_bot.right_neutral)
lemma sup_eq_bot_iff [simp]: "x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
by (simp add: eq_iff)
lemma bot_eq_sup_iff [simp]: "\<bottom> = x \<squnion> y \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
by (simp add: eq_iff)
end
class bounded_lattice_top = lattice + order_top
begin
subclass bounded_semilattice_inf_top ..
lemma sup_top_left [simp]: "\<top> \<squnion> x = \<top>"
by (rule sup_absorb1) simp
lemma sup_top_right [simp]: "x \<squnion> \<top> = \<top>"
by (rule sup_absorb2) simp
lemma inf_top_left: "\<top> \<sqinter> x = x"
by (fact inf_top.left_neutral)
lemma inf_top_right: "x \<sqinter> \<top> = x"
by (fact inf_top.right_neutral)
lemma inf_eq_top_iff [simp]: "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
by (simp add: eq_iff)
end
class bounded_lattice = lattice + order_bot + order_top
begin
subclass bounded_lattice_bot ..
subclass bounded_lattice_top ..
lemma dual_bounded_lattice: "class.bounded_lattice sup greater_eq greater inf \<top> \<bottom>"
by unfold_locales (auto simp add: less_le_not_le)
end
class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus +
assumes inf_compl_bot: "x \<sqinter> - x = \<bottom>"
and sup_compl_top: "x \<squnion> - x = \<top>"
assumes diff_eq: "x - y = x \<sqinter> - y"
begin
lemma dual_boolean_algebra:
"class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus sup greater_eq greater inf \<top> \<bottom>"
by (rule class.boolean_algebra.intro,
rule dual_bounded_lattice,
rule dual_distrib_lattice)
(unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
lemma compl_inf_bot [simp]: "- x \<sqinter> x = \<bottom>"
by (simp add: inf_commute inf_compl_bot)
lemma compl_sup_top [simp]: "- x \<squnion> x = \<top>"
by (simp add: sup_commute sup_compl_top)
lemma compl_unique:
assumes "x \<sqinter> y = \<bottom>"
and "x \<squnion> y = \<top>"
shows "- x = y"
proof -
have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)"
using inf_compl_bot assms(1) by simp
then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)"
by (simp add: inf_commute)
then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
by (simp add: inf_sup_distrib1)
then have "- x \<sqinter> \<top> = y \<sqinter> \<top>"
using sup_compl_top assms(2) by simp
then show "- x = y" by simp
qed
lemma double_compl [simp]: "- (- x) = x"
using compl_inf_bot compl_sup_top by (rule compl_unique)
lemma compl_eq_compl_iff [simp]: "- x = - y \<longleftrightarrow> x = y"
proof
assume "- x = - y"
then have "- (- x) = - (- y)" by (rule arg_cong)
then show "x = y" by simp
next
assume "x = y"
then show "- x = - y" by simp
qed
lemma compl_bot_eq [simp]: "- \<bottom> = \<top>"
proof -
from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
then show ?thesis by simp
qed
lemma compl_top_eq [simp]: "- \<top> = \<bottom>"
proof -
from inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
then show ?thesis by simp
qed
lemma compl_inf [simp]: "- (x \<sqinter> y) = - x \<squnion> - y"
proof (rule compl_unique)
have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
by (simp only: inf_sup_distrib inf_aci)
then show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = \<bottom>"
by (simp add: inf_compl_bot)
next
have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
by (simp only: sup_inf_distrib sup_aci)
then show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
by (simp add: sup_compl_top)
qed
lemma compl_sup [simp]: "- (x \<squnion> y) = - x \<sqinter> - y"
using dual_boolean_algebra
by (rule boolean_algebra.compl_inf)
lemma compl_mono:
assumes "x \<le> y"
shows "- y \<le> - x"
proof -
from assms have "x \<squnion> y = y" by (simp only: le_iff_sup)
then have "- (x \<squnion> y) = - y" by simp
then have "- x \<sqinter> - y = - y" by simp
then have "- y \<sqinter> - x = - y" by (simp only: inf_commute)
then show ?thesis by (simp only: le_iff_inf)
qed
lemma compl_le_compl_iff [simp]: "- x \<le> - y \<longleftrightarrow> y \<le> x"
by (auto dest: compl_mono)
lemma compl_le_swap1:
assumes "y \<le> - x"
shows "x \<le> -y"
proof -
from assms have "- (- x) \<le> - y" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed
lemma compl_le_swap2:
assumes "- y \<le> x"
shows "- x \<le> y"
proof -
from assms have "- x \<le> - (- y)" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed
lemma compl_less_compl_iff: "- x < - y \<longleftrightarrow> y < x" (* TODO: declare [simp] ? *)
by (auto simp add: less_le)
lemma compl_less_swap1:
assumes "y < - x"
shows "x < - y"
proof -
from assms have "- (- x) < - y" by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed
lemma compl_less_swap2:
assumes "- y < x"
shows "- x < y"
proof -
from assms have "- x < - (- y)"
by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed
lemma sup_cancel_left1: "sup (sup x a) (sup (- x) b) = top"
by (simp add: inf_sup_aci sup_compl_top)
lemma sup_cancel_left2: "sup (sup (- x) a) (sup x b) = top"
by (simp add: inf_sup_aci sup_compl_top)
lemma inf_cancel_left1: "inf (inf x a) (inf (- x) b) = bot"
by (simp add: inf_sup_aci inf_compl_bot)
lemma inf_cancel_left2: "inf (inf (- x) a) (inf x b) = bot"
by (simp add: inf_sup_aci inf_compl_bot)
declare inf_compl_bot [simp]
and sup_compl_top [simp]
lemma sup_compl_top_left1 [simp]: "sup (- x) (sup x y) = top"
by (simp add: sup_assoc[symmetric])
lemma sup_compl_top_left2 [simp]: "sup x (sup (- x) y) = top"
using sup_compl_top_left1[of "- x" y] by simp
lemma inf_compl_bot_left1 [simp]: "inf (- x) (inf x y) = bot"
by (simp add: inf_assoc[symmetric])
lemma inf_compl_bot_left2 [simp]: "inf x (inf (- x) y) = bot"
using inf_compl_bot_left1[of "- x" y] by simp
lemma inf_compl_bot_right [simp]: "inf x (inf y (- x)) = bot"
by (subst inf_left_commute) simp
end
ML_file \<open>Tools/boolean_algebra_cancel.ML\<close>
simproc_setup boolean_algebra_cancel_sup ("sup a b::'a::boolean_algebra") =
\<open>fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_sup_conv\<close>
simproc_setup boolean_algebra_cancel_inf ("inf a b::'a::boolean_algebra") =
\<open>fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_inf_conv\<close>
subsection \<open>\<open>min/max\<close> as special case of lattice\<close>
context linorder
begin
sublocale min: semilattice_order min less_eq less
+ max: semilattice_order max greater_eq greater
by standard (auto simp add: min_def max_def)
lemma min_le_iff_disj: "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
unfolding min_def using linear by (auto intro: order_trans)
lemma le_max_iff_disj: "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
unfolding max_def using linear by (auto intro: order_trans)
lemma min_less_iff_disj: "min x y < z \<longleftrightarrow> x < z \<or> y < z"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
lemma less_max_iff_disj: "z < max x y \<longleftrightarrow> z < x \<or> z < y"
unfolding max_def le_less using less_linear by (auto intro: less_trans)
lemma min_less_iff_conj [simp]: "z < min x y \<longleftrightarrow> z < x \<and> z < y"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
lemma max_less_iff_conj [simp]: "max x y < z \<longleftrightarrow> x < z \<and> y < z"
unfolding max_def le_less using less_linear by (auto intro: less_trans)
lemma min_max_distrib1: "min (max b c) a = max (min b a) (min c a)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
lemma min_max_distrib2: "min a (max b c) = max (min a b) (min a c)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
lemma max_min_distrib1: "max (min b c) a = min (max b a) (max c a)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
lemma max_min_distrib2: "max a (min b c) = min (max a b) (max a c)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
lemmas min_max_distribs = min_max_distrib1 min_max_distrib2 max_min_distrib1 max_min_distrib2
lemma split_min [no_atp]: "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
by (simp add: min_def)
lemma split_max [no_atp]: "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
by (simp add: max_def)
lemma min_of_mono: "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)" for f :: "'a \<Rightarrow> 'b::linorder"
by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
lemma max_of_mono: "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)" for f :: "'a \<Rightarrow> 'b::linorder"
by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
end
lemma max_of_antimono: "antimono f \<Longrightarrow> max (f x) (f y) = f (min x y)"
and min_of_antimono: "antimono f \<Longrightarrow> min (f x) (f y) = f (max x y)"
for f::"'a::linorder \<Rightarrow> 'b::linorder"
by (auto simp: antimono_def Orderings.max_def min_def intro!: antisym)
lemma inf_min: "inf = (min :: 'a::{semilattice_inf,linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (auto intro: antisym simp add: min_def fun_eq_iff)
lemma sup_max: "sup = (max :: 'a::{semilattice_sup,linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (auto intro: antisym simp add: max_def fun_eq_iff)
subsection \<open>Uniqueness of inf and sup\<close>
lemma (in semilattice_inf) inf_unique:
fixes f (infixl "\<triangle>" 70)
assumes le1: "\<And>x y. x \<triangle> y \<le> x"
and le2: "\<And>x y. x \<triangle> y \<le> y"
and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
shows "x \<sqinter> y = x \<triangle> y"
proof (rule antisym)
show "x \<triangle> y \<le> x \<sqinter> y"
by (rule le_infI) (rule le1, rule le2)
have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
by (blast intro: greatest)
show "x \<sqinter> y \<le> x \<triangle> y"
by (rule leI) simp_all
qed
lemma (in semilattice_sup) sup_unique:
fixes f (infixl "\<nabla>" 70)
assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y"
and ge2: "\<And>x y. y \<le> x \<nabla> y"
and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x"
shows "x \<squnion> y = x \<nabla> y"
proof (rule antisym)
show "x \<squnion> y \<le> x \<nabla> y"
by (rule le_supI) (rule ge1, rule ge2)
have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z"
by (blast intro: least)
show "x \<nabla> y \<le> x \<squnion> y"
by (rule leI) simp_all
qed
subsection \<open>Lattice on \<^typ>\<open>bool\<close>\<close>
instantiation bool :: boolean_algebra
begin
definition bool_Compl_def [simp]: "uminus = Not"
definition bool_diff_def [simp]: "A - B \<longleftrightarrow> A \<and> \<not> B"
definition [simp]: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
definition [simp]: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
instance by standard auto
end
lemma sup_boolI1: "P \<Longrightarrow> P \<squnion> Q"
by simp
lemma sup_boolI2: "Q \<Longrightarrow> P \<squnion> Q"
by simp
lemma sup_boolE: "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by auto
subsection \<open>Lattice on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>
instantiation "fun" :: (type, semilattice_sup) semilattice_sup
begin
definition "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
lemma sup_apply [simp, code]: "(f \<squnion> g) x = f x \<squnion> g x"
by (simp add: sup_fun_def)
instance
by standard (simp_all add: le_fun_def)
end
instantiation "fun" :: (type, semilattice_inf) semilattice_inf
begin
definition "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
lemma inf_apply [simp, code]: "(f \<sqinter> g) x = f x \<sqinter> g x"
by (simp add: inf_fun_def)
instance by standard (simp_all add: le_fun_def)
end
instance "fun" :: (type, lattice) lattice ..
instance "fun" :: (type, distrib_lattice) distrib_lattice
by standard (rule ext, simp add: sup_inf_distrib1)
instance "fun" :: (type, bounded_lattice) bounded_lattice ..
instantiation "fun" :: (type, uminus) uminus
begin
definition fun_Compl_def: "- A = (\<lambda>x. - A x)"
lemma uminus_apply [simp, code]: "(- A) x = - (A x)"
by (simp add: fun_Compl_def)
instance ..
end
instantiation "fun" :: (type, minus) minus
begin
definition fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
lemma minus_apply [simp, code]: "(A - B) x = A x - B x"
by (simp add: fun_diff_def)
instance ..
end
instance "fun" :: (type, boolean_algebra) boolean_algebra
by standard (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+
subsection \<open>Lattice on unary and binary predicates\<close>
lemma inf1I: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x"
by (simp add: inf_fun_def)
lemma inf2I: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y"
by (simp add: inf_fun_def)
lemma inf1E: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: inf_fun_def)
lemma inf2E: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: inf_fun_def)
lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x"
by (rule inf1E)
lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y"
by (rule inf2E)
lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x"
by (rule inf1E)
lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y"
by (rule inf2E)
lemma sup1I1: "A x \<Longrightarrow> (A \<squnion> B) x"
by (simp add: sup_fun_def)
lemma sup2I1: "A x y \<Longrightarrow> (A \<squnion> B) x y"
by (simp add: sup_fun_def)
lemma sup1I2: "B x \<Longrightarrow> (A \<squnion> B) x"
by (simp add: sup_fun_def)
lemma sup2I2: "B x y \<Longrightarrow> (A \<squnion> B) x y"
by (simp add: sup_fun_def)
lemma sup1E: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: sup_fun_def) iprover
lemma sup2E: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: sup_fun_def) iprover
text \<open> \<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs \<open>B\<close>.\<close>
lemma sup1CI: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x"
by (auto simp add: sup_fun_def)
lemma sup2CI: "(\<not> B x y \<Longrightarrow> A x y) \<Longrightarrow> (A \<squnion> B) x y"
by (auto simp add: sup_fun_def)
end