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Quellcode-Bibliothek
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Datei:
gettext.h
Sprache: Isabelle
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(* 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 \ 'a \ bool" (infix "\<^bold>\" 50)
and less :: "'a \ 'a \ bool" (infix "\<^bold><" 50)
assumes order_iff: "a \<^bold>\ b \ a = a \<^bold>* b"
and strict_order_iff: "a \<^bold>< b \ a = a \<^bold>* b \ a \ b"
begin
lemma orderI: "a = a \<^bold>* b \ a \<^bold>\ b"
by (simp add: order_iff)
lemma orderE:
assumes "a \<^bold>\ b"
obtains "a = a \<^bold>* b"
using assms by (unfold order_iff)
sublocale ordering less_eq less
proof
show "a \<^bold>< b \ a \<^bold>\ b \ a \ b" for a b
by (simp add: order_iff strict_order_iff)
next
show "a \<^bold>\ a" for a
by (simp add: order_iff)
next
fix a b
assume "a \<^bold>\ b" "b \<^bold>\ 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>\ b" "b \<^bold>\ 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>\ c" by (rule orderI)
qed
lemma cobounded1 [simp]: "a \<^bold>* b \<^bold>\ a"
by (simp add: order_iff commute)
lemma cobounded2 [simp]: "a \<^bold>* b \<^bold>\ b"
by (simp add: order_iff)
lemma boundedI:
assumes "a \<^bold>\ b" and "a \<^bold>\ c"
shows "a \<^bold>\ 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>\ b \<^bold>* c"
obtains "a \<^bold>\ b" and "a \<^bold>\ c"
using assms by (blast intro: trans cobounded1 cobounded2)
lemma bounded_iff [simp]: "a \<^bold>\ b \<^bold>* c \ a \<^bold>\ b \ a \<^bold>\ 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>\ c \ a \<^bold>* b \<^bold>\ c"
by (rule trans) auto
lemma coboundedI2: "b \<^bold>\ c \ a \<^bold>* b \<^bold>\ c"
by (rule trans) auto
lemma strict_coboundedI1: "a \<^bold>< c \ 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 \ a \<^bold>* b \<^bold>< c"
using strict_coboundedI1 [of b c a] by (simp add: commute)
lemma mono: "a \<^bold>\ c \ b \<^bold>\ d \ a \<^bold>* b \<^bold>\ c \<^bold>* d"
by (blast intro: boundedI coboundedI1 coboundedI2)
lemma absorb1: "a \<^bold>\ b \ a \<^bold>* b = a"
by (rule antisym) (auto simp: refl)
lemma absorb2: "b \<^bold>\ a \ a \<^bold>* b = b"
by (rule antisym) (auto simp: refl)
lemma absorb_iff1: "a \<^bold>\ b \ a \<^bold>* b = a"
using order_iff by auto
lemma absorb_iff2: "b \<^bold>\ a \ 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)
lemma eq_neutr_iff [simp]: \<open>a \<^bold>* b = \<^bold>1 \<longleftrightarrow> a = \<^bold>1 \<and> b = \<^bold>1\<close>
by (simp add: eq_iff)
lemma neutr_eq_iff [simp]: \<open>\<^bold>1 = a \<^bold>* b \<longleftrightarrow> a = \<^bold>1 \<and> b = \<^bold>1\<close>
by (simp add: eq_iff)
end
text \<open>Interpretations for boolean operators\<close>
interpretation conj: semilattice_neutr \<open>(\<and>)\<close> True
by standard auto
interpretation disj: semilattice_neutr \<open>(\<or>)\<close> False
by standard auto
declare conj_assoc [ac_simps del] disj_assoc [ac_simps del] \<comment> \<open>already simp by default\<close>
subsection \<open>Syntactic infimum and supremum operations\<close>
class inf =
fixes inf :: "'a \ 'a \ 'a" (infixl "\" 70)
class sup =
fixes sup :: "'a \ 'a \ 'a" (infixl "\" 65)
subsection \<open>Concrete lattices\<close>
class semilattice_inf = order + inf +
assumes inf_le1 [simp]: "x \ y \ x"
and inf_le2 [simp]: "x \ y \ y"
and inf_greatest: "x \ y \ x \ z \ x \ y \ z"
class semilattice_sup = order + sup +
assumes sup_ge1 [simp]: "x \ x \ y"
and sup_ge2 [simp]: "y \ x \ y"
and sup_least: "y \ x \ z \ x \ y \ z \ 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 \ x \ a \ b \ x"
by (rule order_trans) auto
lemma le_infI2: "b \ x \ a \ b \ x"
by (rule order_trans) auto
lemma le_infI: "x \ a \ x \ b \ x \ a \ b"
by (fact inf_greatest) (* FIXME: duplicate lemma *)
lemma le_infE: "x \ a \ b \ (x \ a \ x \ b \ P) \ P"
by (blast intro: order_trans inf_le1 inf_le2)
lemma le_inf_iff: "x \ y \ z \ x \ y \ x \ z"
by (blast intro: le_infI elim: le_infE)
lemma le_iff_inf: "x \ y \ x \ y = x"
by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1] simp add: le_inf_iff)
lemma inf_mono: "a \ c \ b \ d \ a \ b \ c \ d"
by (fast intro: inf_greatest le_infI1 le_infI2)
lemma mono_inf: "mono f \ f (A \ B) \ f A \ f B" for f :: "'a \ 'b::semilattice_inf"
by (auto simp add: mono_def intro: Lattices.inf_greatest)
end
context semilattice_sup
begin
lemma le_supI1: "x \ a \ x \ a \ b"
by (rule order_trans) auto
lemma le_supI2: "x \ b \ x \ a \ b"
by (rule order_trans) auto
lemma le_supI: "a \ x \ b \ x \ a \ b \ x"
by (fact sup_least) (* FIXME: duplicate lemma *)
lemma le_supE: "a \ b \ x \ (a \ x \ b \ x \ P) \ P"
by (blast intro: order_trans sup_ge1 sup_ge2)
lemma le_sup_iff: "x \ y \ z \ x \ z \ y \ z"
by (blast intro: le_supI elim: le_supE)
lemma le_iff_sup: "x \ y \ x \ y = y"
by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1] simp add: le_sup_iff)
lemma sup_mono: "a \ c \ b \ d \ a \ b \ c \ d"
by (fast intro: sup_least le_supI1 le_supI2)
lemma mono_sup: "mono f \ f A \ f B \ f (A \ B)" for f :: "'a \ '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 \ b) \ c = a \ (b \ c)"
by (rule antisym) (auto intro: le_infI1 le_infI2 simp add: le_inf_iff)
show "a \ b = b \ a"
by (rule antisym) (auto simp add: le_inf_iff)
show "a \ 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 \ y) \ z = x \ (y \ z)"
by (fact inf.assoc)
lemma inf_commute: "(x \ y) = (y \ x)"
by (fact inf.commute)
lemma inf_left_commute: "x \ (y \ z) = y \ (x \ z)"
by (fact inf.left_commute)
lemma inf_idem: "x \ x = x"
by (fact inf.idem) (* already simp *)
lemma inf_left_idem: "x \ (x \ y) = x \ y"
by (fact inf.left_idem) (* already simp *)
lemma inf_right_idem: "(x \ y) \ y = x \ y"
by (fact inf.right_idem) (* already simp *)
lemma inf_absorb1: "x \ y \ x \ y = x"
by (rule antisym) auto
lemma inf_absorb2: "y \ x \ x \ 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 \ b) \ c = a \ (b \ c)"
by (rule antisym) (auto intro: le_supI1 le_supI2 simp add: le_sup_iff)
show "a \ b = b \ a"
by (rule antisym) (auto simp add: le_sup_iff)
show "a \ 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 \ y) \ z = x \ (y \ z)"
by (fact sup.assoc)
lemma sup_commute: "(x \ y) = (y \ x)"
by (fact sup.commute)
lemma sup_left_commute: "x \ (y \ z) = y \ (x \ z)"
by (fact sup.left_commute)
lemma sup_idem: "x \ x = x"
by (fact sup.idem) (* already simp *)
lemma sup_left_idem [simp]: "x \ (x \ y) = x \ y"
by (fact sup.left_idem)
lemma sup_absorb1: "y \ x \ x \ y = x"
by (rule antisym) auto
lemma sup_absorb2: "x \ y \ x \ 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 (\) (>) 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 \ (x \ y) = x"
by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
lemma sup_inf_absorb [simp]: "x \ (x \ 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 \ (y \ z) \ (x \ y) \ (x \ z)"
by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
lemma distrib_inf_le: "(x \ y) \ (x \ z) \ x \ (y \ 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: "\x y z. x \ (y \ z) = (x \ y) \ (x \ z)"
shows "x \ (y \ z) = (x \ y) \ (x \ z)"
proof-
have "x \ (y \ z) = (x \ (x \ z)) \ (y \ z)"
by simp
also have "\ = x \ (z \ (x \ y))"
by (simp add: distrib inf_commute sup_assoc del: sup_inf_absorb)
also have "\ = ((x \ y) \ x) \ ((x \ y) \ z)"
by (simp add: inf_commute)
also have "\ = (x \ y) \ (x \ z)" by(simp add:distrib)
finally show ?thesis .
qed
lemma distrib_imp2:
assumes distrib: "\x y z. x \ (y \ z) = (x \ y) \ (x \ z)"
shows "x \ (y \ z) = (x \ y) \ (x \ z)"
proof-
have "x \ (y \ z) = (x \ (x \ z)) \ (y \ z)"
by simp
also have "\ = x \ (z \ (x \ y))"
by (simp add: distrib sup_commute inf_assoc del: inf_sup_absorb)
also have "\ = ((x \ y) \ x) \ ((x \ y) \ z)"
by (simp add: sup_commute)
also have "\ = (x \ y) \ (x \ z)" by (simp add:distrib)
finally show ?thesis .
qed
end
subsubsection \<open>Strict order\<close>
context semilattice_inf
begin
lemma less_infI1: "a < x \ a \ b < x"
by (auto simp add: less_le inf_absorb1 intro: le_infI1)
lemma less_infI2: "b < x \ a \ b < x"
by (auto simp add: less_le inf_absorb2 intro: le_infI2)
end
context semilattice_sup
begin
lemma less_supI1: "x < a \ x < a \ b"
using dual_semilattice
by (rule semilattice_inf.less_infI1)
lemma less_supI2: "x < b \ x < a \ 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 \ (y \ z) = (x \ y) \ (x \ z)"
context distrib_lattice
begin
lemma sup_inf_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)"
by (simp add: sup_commute sup_inf_distrib1)
lemma inf_sup_distrib1: "x \ (y \ z) = (x \ y) \ (x \ z)"
by (rule distrib_imp2 [OF sup_inf_distrib1])
lemma inf_sup_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)"
by (simp add: inf_commute inf_sup_distrib1)
lemma dual_distrib_lattice: "class.distrib_lattice sup (\) (>) 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 \ \ = x" for x
by (rule inf_absorb1) simp
qed
lemma inf_top_left: "\ \ x = x"
by (fact inf_top.left_neutral)
lemma inf_top_right: "x \ \ = x"
by (fact inf_top.right_neutral)
lemma inf_eq_top_iff: "x \ y = \ \ x = \ \ y = \"
by (fact inf_top.eq_neutr_iff)
lemma top_eq_inf_iff: "\ = x \ y \ x = \ \ y = \"
by (fact inf_top.neutr_eq_iff)
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 \ \ = x" for x
by (rule sup_absorb1) simp
qed
lemma sup_bot_left: "\ \ x = x"
by (fact sup_bot.left_neutral)
lemma sup_bot_right: "x \ \ = x"
by (fact sup_bot.right_neutral)
lemma sup_eq_bot_iff: "x \ y = \ \ x = \ \ y = \"
by (fact sup_bot.eq_neutr_iff)
lemma bot_eq_sup_iff: "\ = x \ y \ x = \ \ y = \"
by (fact sup_bot.neutr_eq_iff)
end
class bounded_lattice_bot = lattice + order_bot
begin
subclass bounded_semilattice_sup_bot ..
lemma inf_bot_left [simp]: "\ \ x = \"
by (rule inf_absorb1) simp
lemma inf_bot_right [simp]: "x \ \ = \"
by (rule inf_absorb2) simp
end
class bounded_lattice_top = lattice + order_top
begin
subclass bounded_semilattice_inf_top ..
lemma sup_top_left [simp]: "\ \ x = \"
by (rule sup_absorb1) simp
lemma sup_top_right [simp]: "x \ \ = \"
by (rule sup_absorb2) simp
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 \ \"
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 \ - x = \"
and sup_compl_top: "x \ - x = \"
assumes diff_eq: "x - y = x \ - y"
begin
lemma dual_boolean_algebra:
"class.boolean_algebra (\x y. x \ - y) uminus sup greater_eq greater inf \ \"
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 \ x = \"
by (simp add: inf_commute inf_compl_bot)
lemma compl_sup_top [simp]: "- x \ x = \"
by (simp add: sup_commute sup_compl_top)
lemma compl_unique:
assumes "x \ y = \"
and "x \ y = \"
shows "- x = y"
proof -
have "(x \ - x) \ (- x \ y) = (x \ y) \ (- x \ y)"
using inf_compl_bot assms(1) by simp
then have "(- x \ x) \ (- x \ y) = (y \ x) \ (y \ - x)"
by (simp add: inf_commute)
then have "- x \ (x \ y) = y \ (x \ - x)"
by (simp add: inf_sup_distrib1)
then have "- x \ \ = y \ \"
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 \ 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]: "- \ = \"
proof -
from sup_compl_top have "\ \ - \ = \" .
then show ?thesis by simp
qed
lemma compl_top_eq [simp]: "- \ = \"
proof -
from inf_compl_bot have "\ \ - \ = \" .
then show ?thesis by simp
qed
lemma compl_inf [simp]: "- (x \ y) = - x \ - y"
proof (rule compl_unique)
have "(x \ y) \ (- x \ - y) = (y \ (x \ - x)) \ (x \ (y \ - y))"
by (simp only: inf_sup_distrib inf_aci)
then show "(x \ y) \ (- x \ - y) = \"
by (simp add: inf_compl_bot)
next
have "(x \ y) \ (- x \ - y) = (- y \ (x \ - x)) \ (- x \ (y \ - y))"
by (simp only: sup_inf_distrib sup_aci)
then show "(x \ y) \ (- x \ - y) = \"
by (simp add: sup_compl_top)
qed
lemma compl_sup [simp]: "- (x \ y) = - x \ - y"
using dual_boolean_algebra
by (rule boolean_algebra.compl_inf)
lemma compl_mono:
assumes "x \ y"
shows "- y \ - x"
proof -
from assms have "x \ y = y" by (simp only: le_iff_sup)
then have "- (x \ y) = - y" by simp
then have "- x \ - y = - y" by simp
then have "- y \ - x = - y" by (simp only: inf_commute)
then show ?thesis by (simp only: le_iff_inf)
qed
lemma compl_le_compl_iff [simp]: "- x \ - y \ y \ x"
by (auto dest: compl_mono)
lemma compl_le_swap1:
assumes "y \ - x"
shows "x \ -y"
proof -
from assms have "- (- x) \ - y" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed
lemma compl_le_swap2:
assumes "- y \ x"
shows "- x \ y"
proof -
from assms have "- x \ - (- y)" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed
lemma compl_less_compl_iff: "- x < - y \ 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
locale boolean_algebra_cancel
begin
lemma sup1: "(A::'a::semilattice_sup) \ sup k a \ sup A b \ sup k (sup a b)"
by (simp only: ac_simps)
lemma sup2: "(B::'a::semilattice_sup) \ sup k b \ sup a B \ sup k (sup a b)"
by (simp only: ac_simps)
lemma sup0: "(a::'a::bounded_semilattice_sup_bot) \ sup a bot"
by simp
lemma inf1: "(A::'a::semilattice_inf) \ inf k a \ inf A b \ inf k (inf a b)"
by (simp only: ac_simps)
lemma inf2: "(B::'a::semilattice_inf) \ inf k b \ inf a B \ inf k (inf a b)"
by (simp only: ac_simps)
lemma inf0: "(a::'a::bounded_semilattice_inf_top) \ inf a top"
by 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 \ z \ x \ z \ y \ z"
unfolding min_def using linear by (auto intro: order_trans)
lemma le_max_iff_disj: "z \ max x y \ z \ x \ z \ y"
unfolding max_def using linear by (auto intro: order_trans)
lemma min_less_iff_disj: "min x y < z \ x < z \ y < z"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
lemma less_max_iff_disj: "z < max x y \ z < x \ 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 \ z < x \ 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 \ x < z \ 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) \ (i \ j \ P i) \ (\ i \ j \ P j)"
by (simp add: min_def)
lemma split_max [no_atp]: "P (max i j) \ (i \ j \ P j) \ (\ i \ j \ P i)"
by (simp add: max_def)
lemma split_min_lin [no_atp]:
\<open>P (min a b) \<longleftrightarrow> (b = a \<longrightarrow> P a) \<and> (a < b \<longrightarrow> P a) \<and> (b < a \<longrightarrow> P b)\<close>
by (cases a b rule: linorder_cases) (auto simp add: min.absorb1 min.absorb2)
lemma split_max_lin [no_atp]:
\<open>P (max a b) \<longleftrightarrow> (b = a \<longrightarrow> P a) \<and> (a < b \<longrightarrow> P b) \<and> (b < a \<longrightarrow> P a)\<close>
by (cases a b rule: linorder_cases) (auto simp add: max.absorb1 max.absorb2)
lemma min_of_mono: "mono f \ min (f m) (f n) = f (min m n)" for f :: "'a \ 'b::linorder"
by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
lemma max_of_mono: "mono f \ max (f m) (f n) = f (max m n)" for f :: "'a \ 'b::linorder"
by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
end
lemma max_of_antimono: "antimono f \ max (f x) (f y) = f (min x y)"
and min_of_antimono: "antimono f \ min (f x) (f y) = f (max x y)"
for f::"'a::linorder \ 'b::linorder"
by (auto simp: antimono_def Orderings.max_def min_def intro!: antisym)
lemma inf_min: "inf = (min :: 'a::{semilattice_inf,linorder} \ 'a \ 'a)"
by (auto intro: antisym simp add: min_def fun_eq_iff)
lemma sup_max: "sup = (max :: 'a::{semilattice_sup,linorder} \ 'a \ '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 "\" 70)
assumes le1: "\x y. x \ y \ x"
and le2: "\x y. x \ y \ y"
and greatest: "\x y z. x \ y \ x \ z \ x \ y \ z"
shows "x \ y = x \ y"
proof (rule antisym)
show "x \ y \ x \ y"
by (rule le_infI) (rule le1, rule le2)
have leI: "\x y z. x \ y \ x \ z \ x \ y \ z"
by (blast intro: greatest)
show "x \ y \ x \ y"
by (rule leI) simp_all
qed
lemma (in semilattice_sup) sup_unique:
fixes f (infixl "\" 70)
assumes ge1 [simp]: "\x y. x \ x \ y"
and ge2: "\x y. y \ x \ y"
and least: "\x y z. y \ x \ z \ x \ y \ z \ x"
shows "x \ y = x \ y"
proof (rule antisym)
show "x \ y \ x \ y"
by (rule le_supI) (rule ge1, rule ge2)
have leI: "\x y z. x \ z \ y \ z \ x \ y \ z"
by (blast intro: least)
show "x \ y \ x \ 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 \ A \ \ B"
definition [simp]: "P \ Q \ P \ Q"
definition [simp]: "P \ Q \ P \ Q"
instance by standard auto
end
lemma sup_boolI1: "P \ P \ Q"
by simp
lemma sup_boolI2: "Q \ P \ Q"
by simp
lemma sup_boolE: "P \ Q \ (P \ R) \ (Q \ R) \ R"
by auto
subsection \<open>Lattice on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>
instantiation "fun" :: (type, semilattice_sup) semilattice_sup
begin
definition "f \ g = (\x. f x \ g x)"
lemma sup_apply [simp, code]: "(f \ g) x = f x \ 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 \ g = (\x. f x \ g x)"
lemma inf_apply [simp, code]: "(f \ g) x = f x \ 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 = (\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 = (\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 \ B x \ (A \ B) x"
by (simp add: inf_fun_def)
lemma inf2I: "A x y \ B x y \ (A \ B) x y"
by (simp add: inf_fun_def)
lemma inf1E: "(A \ B) x \ (A x \ B x \ P) \ P"
by (simp add: inf_fun_def)
lemma inf2E: "(A \ B) x y \ (A x y \ B x y \ P) \ P"
by (simp add: inf_fun_def)
lemma inf1D1: "(A \ B) x \ A x"
by (rule inf1E)
lemma inf2D1: "(A \ B) x y \ A x y"
by (rule inf2E)
lemma inf1D2: "(A \ B) x \ B x"
by (rule inf1E)
lemma inf2D2: "(A \ B) x y \ B x y"
by (rule inf2E)
lemma sup1I1: "A x \ (A \ B) x"
by (simp add: sup_fun_def)
lemma sup2I1: "A x y \ (A \ B) x y"
by (simp add: sup_fun_def)
lemma sup1I2: "B x \ (A \ B) x"
by (simp add: sup_fun_def)
lemma sup2I2: "B x y \ (A \ B) x y"
by (simp add: sup_fun_def)
lemma sup1E: "(A \ B) x \ (A x \ P) \ (B x \ P) \ P"
by (simp add: sup_fun_def) iprover
lemma sup2E: "(A \ B) x y \ (A x y \ P) \ (B x y \ P) \ 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: "(\ B x \ A x) \ (A \ B) x"
by (auto simp add: sup_fun_def)
lemma sup2CI: "(\ B x y \ A x y) \ (A \ B) x y"
by (auto simp add: sup_fun_def)
end
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