(* Title: HOL/Boolean_Algebras.thy
Author: Brian Huffman
Author: Florian Haftmann
*)
section \<open>Boolean Algebras\<close>
theory Boolean_Algebras
imports Lattices
begin
subsection \<open>Abstract boolean algebra\<close>
locale abstract_boolean_algebra = conj: abel_semigroup \<open>(\<^bold>\<sqinter>)\<close> + disj: abel_semigroup \<open>(\<^bold>\<squnion>)\<close>
for conj :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>\<^bold>\<sqinter>\<close> 70)
and disj :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>\<^bold>\<squnion>\<close> 65) +
fixes compl :: \<open>'a \<Rightarrow> 'a\<close> (\<open>\<^bold>- _\<close> [81] 80)
and zero :: \<open>'a\<close> (\<open>\<^bold>0\<close>)
and one :: \<open>'a\<close> (\<open>\<^bold>1\<close>)
assumes conj_disj_distrib: \<open>x \<^bold>\<sqinter> (y \<^bold>\<squnion> z) = (x \<^bold>\<sqinter> y) \<^bold>\<squnion> (x \<^bold>\<sqinter> z)\<close>
and disj_conj_distrib: \<open>x \<^bold>\<squnion> (y \<^bold>\<sqinter> z) = (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (x \<^bold>\<squnion> z)\<close>
and conj_one_right: \<open>x \<^bold>\<sqinter> \<^bold>1 = x\<close>
and disj_zero_right: \<open>x \<^bold>\<squnion> \<^bold>0 = x\<close>
and conj_cancel_right [simp]: \<open>x \<^bold>\<sqinter> \<^bold>- x = \<^bold>0\<close>
and disj_cancel_right [simp]: \<open>x \<^bold>\<squnion> \<^bold>- x = \<^bold>1\<close>
begin
sublocale conj: semilattice_neutr \<open>(\<^bold>\<sqinter>)\<close> \<open>\<^bold>1\<close>
proof
show "x \<^bold>\<sqinter> \<^bold>1 = x" for x
by (fact conj_one_right)
show "x \<^bold>\<sqinter> x = x" for x
proof -
have "x \<^bold>\<sqinter> x = (x \<^bold>\<sqinter> x) \<^bold>\<squnion> \<^bold>0"
by (simp add: disj_zero_right)
also have "\<dots> = (x \<^bold>\<sqinter> x) \<^bold>\<squnion> (x \<^bold>\<sqinter> \<^bold>- x)"
by simp
also have "\<dots> = x \<^bold>\<sqinter> (x \<^bold>\<squnion> \<^bold>- x)"
by (simp only: conj_disj_distrib)
also have "\<dots> = x \<^bold>\<sqinter> \<^bold>1"
by simp
also have "\<dots> = x"
by (simp add: conj_one_right)
finally show ?thesis .
qed
qed
sublocale disj: semilattice_neutr \<open>(\<^bold>\<squnion>)\<close> \<open>\<^bold>0\<close>
proof
show "x \<^bold>\<squnion> \<^bold>0 = x" for x
by (fact disj_zero_right)
show "x \<^bold>\<squnion> x = x" for x
proof -
have "x \<^bold>\<squnion> x = (x \<^bold>\<squnion> x) \<^bold>\<sqinter> \<^bold>1"
by simp
also have "\<dots> = (x \<^bold>\<squnion> x) \<^bold>\<sqinter> (x \<^bold>\<squnion> \<^bold>- x)"
by simp
also have "\<dots> = x \<^bold>\<squnion> (x \<^bold>\<sqinter> \<^bold>- x)"
by (simp only: disj_conj_distrib)
also have "\<dots> = x \<^bold>\<squnion> \<^bold>0"
by simp
also have "\<dots> = x"
by (simp add: disj_zero_right)
finally show ?thesis .
qed
qed
subsubsection \<open>Complement\<close>
lemma complement_unique:
assumes 1: "a \<^bold>\<sqinter> x = \<^bold>0"
assumes 2: "a \<^bold>\<squnion> x = \<^bold>1"
assumes 3: "a \<^bold>\<sqinter> y = \<^bold>0"
assumes 4: "a \<^bold>\<squnion> y = \<^bold>1"
shows "x = y"
proof -
from 1 3 have "(a \<^bold>\<sqinter> x) \<^bold>\<squnion> (x \<^bold>\<sqinter> y) = (a \<^bold>\<sqinter> y) \<^bold>\<squnion> (x \<^bold>\<sqinter> y)"
by simp
then have "(x \<^bold>\<sqinter> a) \<^bold>\<squnion> (x \<^bold>\<sqinter> y) = (y \<^bold>\<sqinter> a) \<^bold>\<squnion> (y \<^bold>\<sqinter> x)"
by (simp add: ac_simps)
then have "x \<^bold>\<sqinter> (a \<^bold>\<squnion> y) = y \<^bold>\<sqinter> (a \<^bold>\<squnion> x)"
by (simp add: conj_disj_distrib)
with 2 4 have "x \<^bold>\<sqinter> \<^bold>1 = y \<^bold>\<sqinter> \<^bold>1"
by simp
then show "x = y"
by simp
qed
lemma compl_unique: "x \<^bold>\<sqinter> y = \<^bold>0 \<Longrightarrow> x \<^bold>\<squnion> y = \<^bold>1 \<Longrightarrow> \<^bold>- x = y"
by (rule complement_unique [OF conj_cancel_right disj_cancel_right])
lemma double_compl [simp]: "\<^bold>- (\<^bold>- x) = x"
proof (rule compl_unique)
show "\<^bold>- x \<^bold>\<sqinter> x = \<^bold>0"
by (simp only: conj_cancel_right conj.commute)
show "\<^bold>- x \<^bold>\<squnion> x = \<^bold>1"
by (simp only: disj_cancel_right disj.commute)
qed
lemma compl_eq_compl_iff [simp]:
\<open>\<^bold>- x = \<^bold>- y \<longleftrightarrow> x = y\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
proof
assume \<open>?Q\<close>
then show ?P by simp
next
assume \<open>?P\<close>
then have \<open>\<^bold>- (\<^bold>- x) = \<^bold>- (\<^bold>- y)\<close>
by simp
then show ?Q
by simp
qed
subsubsection \<open>Conjunction\<close>
lemma conj_zero_right [simp]: "x \<^bold>\<sqinter> \<^bold>0 = \<^bold>0"
using conj.left_idem conj_cancel_right by fastforce
lemma compl_one [simp]: "\<^bold>- \<^bold>1 = \<^bold>0"
by (rule compl_unique [OF conj_zero_right disj_zero_right])
lemma conj_zero_left [simp]: "\<^bold>0 \<^bold>\<sqinter> x = \<^bold>0"
by (subst conj.commute) (rule conj_zero_right)
lemma conj_cancel_left [simp]: "\<^bold>- x \<^bold>\<sqinter> x = \<^bold>0"
by (subst conj.commute) (rule conj_cancel_right)
lemma conj_disj_distrib2: "(y \<^bold>\<squnion> z) \<^bold>\<sqinter> x = (y \<^bold>\<sqinter> x) \<^bold>\<squnion> (z \<^bold>\<sqinter> x)"
by (simp only: conj.commute conj_disj_distrib)
lemmas conj_disj_distribs = conj_disj_distrib conj_disj_distrib2
subsubsection \<open>Disjunction\<close>
context
begin
interpretation dual: abstract_boolean_algebra \<open>(\<^bold>\<squnion>)\<close> \<open>(\<^bold>\<sqinter>)\<close> compl \<open>\<^bold>1\<close> \<open>\<^bold>0\<close>
apply standard
apply (rule disj_conj_distrib)
apply (rule conj_disj_distrib)
apply simp_all
done
lemma disj_one_right [simp]: "x \<^bold>\<squnion> \<^bold>1 = \<^bold>1"
by (fact dual.conj_zero_right)
lemma compl_zero [simp]: "\<^bold>- \<^bold>0 = \<^bold>1"
by (fact dual.compl_one)
lemma disj_one_left [simp]: "\<^bold>1 \<^bold>\<squnion> x = \<^bold>1"
by (fact dual.conj_zero_left)
lemma disj_cancel_left [simp]: "\<^bold>- x \<^bold>\<squnion> x = \<^bold>1"
by (fact dual.conj_cancel_left)
lemma disj_conj_distrib2: "(y \<^bold>\<sqinter> z) \<^bold>\<squnion> x = (y \<^bold>\<squnion> x) \<^bold>\<sqinter> (z \<^bold>\<squnion> x)"
by (fact dual.conj_disj_distrib2)
lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2
end
subsubsection \<open>De Morgan's Laws\<close>
lemma de_Morgan_conj [simp]: "\<^bold>- (x \<^bold>\<sqinter> y) = \<^bold>- x \<^bold>\<squnion> \<^bold>- y"
proof (rule compl_unique)
have "(x \<^bold>\<sqinter> y) \<^bold>\<sqinter> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y) = ((x \<^bold>\<sqinter> y) \<^bold>\<sqinter> \<^bold>- x) \<^bold>\<squnion> ((x \<^bold>\<sqinter> y) \<^bold>\<sqinter> \<^bold>- y)"
by (rule conj_disj_distrib)
also have "\<dots> = (y \<^bold>\<sqinter> (x \<^bold>\<sqinter> \<^bold>- x)) \<^bold>\<squnion> (x \<^bold>\<sqinter> (y \<^bold>\<sqinter> \<^bold>- y))"
by (simp only: ac_simps)
finally show "(x \<^bold>\<sqinter> y) \<^bold>\<sqinter> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y) = \<^bold>0"
by (simp only: conj_cancel_right conj_zero_right disj_zero_right)
next
have "(x \<^bold>\<sqinter> y) \<^bold>\<squnion> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y) = (x \<^bold>\<squnion> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y)) \<^bold>\<sqinter> (y \<^bold>\<squnion> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y))"
by (rule disj_conj_distrib2)
also have "\<dots> = (\<^bold>- y \<^bold>\<squnion> (x \<^bold>\<squnion> \<^bold>- x)) \<^bold>\<sqinter> (\<^bold>- x \<^bold>\<squnion> (y \<^bold>\<squnion> \<^bold>- y))"
by (simp only: ac_simps)
finally show "(x \<^bold>\<sqinter> y) \<^bold>\<squnion> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y) = \<^bold>1"
by (simp only: disj_cancel_right disj_one_right conj_one_right)
qed
context
begin
interpretation dual: abstract_boolean_algebra \<open>(\<^bold>\<squnion>)\<close> \<open>(\<^bold>\<sqinter>)\<close> compl \<open>\<^bold>1\<close> \<open>\<^bold>0\<close>
apply standard
apply (rule disj_conj_distrib)
apply (rule conj_disj_distrib)
apply simp_all
done
lemma de_Morgan_disj [simp]: "\<^bold>- (x \<^bold>\<squnion> y) = \<^bold>- x \<^bold>\<sqinter> \<^bold>- y"
by (fact dual.de_Morgan_conj)
end
end
subsection \<open>Symmetric Difference\<close>
locale abstract_boolean_algebra_sym_diff = abstract_boolean_algebra +
fixes xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>\<^bold>\<ominus>\<close> 65)
assumes xor_def : \<open>x \<^bold>\<ominus> y = (x \<^bold>\<sqinter> \<^bold>- y) \<^bold>\<squnion> (\<^bold>- x \<^bold>\<sqinter> y)\<close>
begin
sublocale xor: comm_monoid xor \<open>\<^bold>0\<close>
proof
fix x y z :: 'a
let ?t = "(x \<^bold>\<sqinter> y \<^bold>\<sqinter> z) \<^bold>\<squnion> (x \<^bold>\<sqinter> \<^bold>- y \<^bold>\<sqinter> \<^bold>- z) \<^bold>\<squnion> (\<^bold>- x \<^bold>\<sqinter> y \<^bold>\<sqinter> \<^bold>- z) \<^bold>\<squnion> (\<^bold>- x \<^bold>\<sqinter> \<^bold>- y \<^bold>\<sqinter> z)"
have "?t \<^bold>\<squnion> (z \<^bold>\<sqinter> x \<^bold>\<sqinter> \<^bold>- x) \<^bold>\<squnion> (z \<^bold>\<sqinter> y \<^bold>\<sqinter> \<^bold>- y) = ?t \<^bold>\<squnion> (x \<^bold>\<sqinter> y \<^bold>\<sqinter> \<^bold>- y) \<^bold>\<squnion> (x \<^bold>\<sqinter> z \<^bold>\<sqinter> \<^bold>- z)"
by (simp only: conj_cancel_right conj_zero_right)
then show "(x \<^bold>\<ominus> y) \<^bold>\<ominus> z = x \<^bold>\<ominus> (y \<^bold>\<ominus> z)"
by (simp only: xor_def de_Morgan_disj de_Morgan_conj double_compl)
(simp only: conj_disj_distribs conj_ac ac_simps)
show "x \<^bold>\<ominus> y = y \<^bold>\<ominus> x"
by (simp only: xor_def ac_simps)
show "x \<^bold>\<ominus> \<^bold>0 = x"
by (simp add: xor_def)
qed
lemma xor_def2:
\<open>x \<^bold>\<ominus> y = (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y)\<close>
proof -
note xor_def [of x y]
also have \<open>x \<^bold>\<sqinter> \<^bold>- y \<^bold>\<squnion> \<^bold>- x \<^bold>\<sqinter> y = ((x \<^bold>\<squnion> \<^bold>- x) \<^bold>\<sqinter> (\<^bold>- y \<^bold>\<squnion> \<^bold>- x)) \<^bold>\<sqinter> (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (\<^bold>- y \<^bold>\<squnion> y)\<close>
by (simp add: ac_simps disj_conj_distribs)
also have \<open>\<dots> = (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (\<^bold>- x \<^bold>\<squnion> \<^bold>- y)\<close>
by (simp add: ac_simps)
finally show ?thesis .
qed
lemma xor_one_right [simp]: "x \<^bold>\<ominus> \<^bold>1 = \<^bold>- x"
by (simp only: xor_def compl_one conj_zero_right conj_one_right disj.left_neutral)
lemma xor_one_left [simp]: "\<^bold>1 \<^bold>\<ominus> x = \<^bold>- x"
using xor_one_right [of x] by (simp add: ac_simps)
lemma xor_self [simp]: "x \<^bold>\<ominus> x = \<^bold>0"
by (simp only: xor_def conj_cancel_right conj_cancel_left disj_zero_right)
lemma xor_left_self [simp]: "x \<^bold>\<ominus> (x \<^bold>\<ominus> y) = y"
by (simp only: xor.assoc [symmetric] xor_self xor.left_neutral)
lemma xor_compl_left [simp]: "\<^bold>- x \<^bold>\<ominus> y = \<^bold>- (x \<^bold>\<ominus> y)"
by (simp add: ac_simps flip: xor_one_left)
lemma xor_compl_right [simp]: "x \<^bold>\<ominus> \<^bold>- y = \<^bold>- (x \<^bold>\<ominus> y)"
using xor.commute xor_compl_left by auto
lemma xor_cancel_right [simp]: "x \<^bold>\<ominus> \<^bold>- x = \<^bold>1"
by (simp only: xor_compl_right xor_self compl_zero)
lemma xor_cancel_left [simp]: "\<^bold>- x \<^bold>\<ominus> x = \<^bold>1"
by (simp only: xor_compl_left xor_self compl_zero)
lemma conj_xor_distrib: "x \<^bold>\<sqinter> (y \<^bold>\<ominus> z) = (x \<^bold>\<sqinter> y) \<^bold>\<ominus> (x \<^bold>\<sqinter> z)"
proof -
have *: "(x \<^bold>\<sqinter> y \<^bold>\<sqinter> \<^bold>- z) \<^bold>\<squnion> (x \<^bold>\<sqinter> \<^bold>- y \<^bold>\<sqinter> z) =
(y \<^bold>\<sqinter> x \<^bold>\<sqinter> \<^bold>- x) \<^bold>\<squnion> (z \<^bold>\<sqinter> x \<^bold>\<sqinter> \<^bold>- x) \<^bold>\<squnion> (x \<^bold>\<sqinter> y \<^bold>\<sqinter> \<^bold>- z) \<^bold>\<squnion> (x \<^bold>\<sqinter> \<^bold>- y \<^bold>\<sqinter> z)"
by (simp only: conj_cancel_right conj_zero_right disj.left_neutral)
then show "x \<^bold>\<sqinter> (y \<^bold>\<ominus> z) = (x \<^bold>\<sqinter> y) \<^bold>\<ominus> (x \<^bold>\<sqinter> z)"
by (simp (no_asm_use) only:
xor_def de_Morgan_disj de_Morgan_conj double_compl
conj_disj_distribs ac_simps)
qed
lemma conj_xor_distrib2: "(y \<^bold>\<ominus> z) \<^bold>\<sqinter> x = (y \<^bold>\<sqinter> x) \<^bold>\<ominus> (z \<^bold>\<sqinter> x)"
by (simp add: conj.commute conj_xor_distrib)
lemmas conj_xor_distribs = conj_xor_distrib conj_xor_distrib2
end
subsection \<open>Type classes\<close>
class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus +
assumes inf_compl_bot: \<open>x \<sqinter> - x = \<bottom>\<close>
and sup_compl_top: \<open>x \<squnion> - x = \<top>\<close>
assumes diff_eq: \<open>x - y = x \<sqinter> - y\<close>
begin
sublocale boolean_algebra: abstract_boolean_algebra \<open>(\<sqinter>)\<close> \<open>(\<squnion>)\<close> uminus \<bottom> \<top>
apply standard
apply (rule inf_sup_distrib1)
apply (rule sup_inf_distrib1)
apply (simp_all add: ac_simps inf_compl_bot sup_compl_top)
done
lemma compl_inf_bot: "- x \<sqinter> x = \<bottom>"
by (fact boolean_algebra.conj_cancel_left)
lemma compl_sup_top: "- x \<squnion> x = \<top>"
by (fact boolean_algebra.disj_cancel_left)
lemma compl_unique:
assumes "x \<sqinter> y = \<bottom>"
and "x \<squnion> y = \<top>"
shows "- x = y"
using assms by (rule boolean_algebra.compl_unique)
lemma double_compl: "- (- x) = x"
by (fact boolean_algebra.double_compl)
lemma compl_eq_compl_iff: "- x = - y \<longleftrightarrow> x = y"
by (fact boolean_algebra.compl_eq_compl_iff)
lemma compl_bot_eq: "- \<bottom> = \<top>"
by (fact boolean_algebra.compl_zero)
lemma compl_top_eq: "- \<top> = \<bottom>"
by (fact boolean_algebra.compl_one)
lemma compl_inf: "- (x \<sqinter> y) = - x \<squnion> - y"
by (fact boolean_algebra.de_Morgan_conj)
lemma compl_sup: "- (x \<squnion> y) = - x \<sqinter> - y"
by (fact boolean_algebra.de_Morgan_disj)
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 [simp]: "- x < - y \<longleftrightarrow> y < x"
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: \<open>x \<squnion> a \<squnion> (- x \<squnion> b) = \<top>\<close>
by (simp add: ac_simps)
lemma sup_cancel_left2: \<open>- x \<squnion> a \<squnion> (x \<squnion> b) = \<top>\<close>
by (simp add: ac_simps)
lemma inf_cancel_left1: \<open>x \<sqinter> a \<sqinter> (- x \<sqinter> b) = \<bottom>\<close>
by (simp add: ac_simps)
lemma inf_cancel_left2: \<open>- x \<sqinter> a \<sqinter> (x \<sqinter> b) = \<bottom>\<close>
by (simp add: ac_simps)
lemma sup_compl_top_left1 [simp]: \<open>- x \<squnion> (x \<squnion> y) = \<top>\<close>
by (simp add: sup_assoc [symmetric])
lemma sup_compl_top_left2 [simp]: \<open>x \<squnion> (- x \<squnion> y) = \<top>\<close>
using sup_compl_top_left1 [of "- x" y] by simp
lemma inf_compl_bot_left1 [simp]: \<open>- x \<sqinter> (x \<sqinter> y) = \<bottom>\<close>
by (simp add: inf_assoc [symmetric])
lemma inf_compl_bot_left2 [simp]: \<open>x \<sqinter> (- x \<sqinter> y) = \<bottom>\<close>
using inf_compl_bot_left1 [of "- x" y] by simp
lemma inf_compl_bot_right [simp]: \<open>x \<sqinter> (y \<sqinter> - x) = \<bottom>\<close>
by (subst inf_left_commute) simp
end
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
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)
subsection \<open>Simproc setup\<close>
locale boolean_algebra_cancel
begin
lemma sup1: "(A::'a::semilattice_sup) \<equiv> sup k a \<Longrightarrow> sup A b \<equiv> sup k (sup a b)"
by (simp only: ac_simps)
lemma sup2: "(B::'a::semilattice_sup) \<equiv> sup k b \<Longrightarrow> sup a B \<equiv> sup k (sup a b)"
by (simp only: ac_simps)
lemma sup0: "(a::'a::bounded_semilattice_sup_bot) \<equiv> sup a bot"
by simp
lemma inf1: "(A::'a::semilattice_inf) \<equiv> inf k a \<Longrightarrow> inf A b \<equiv> inf k (inf a b)"
by (simp only: ac_simps)
lemma inf2: "(B::'a::semilattice_inf) \<equiv> inf k b \<Longrightarrow> inf a B \<equiv> inf k (inf a b)"
by (simp only: ac_simps)
lemma inf0: "(a::'a::bounded_semilattice_inf_top) \<equiv> 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>
context boolean_algebra
begin
lemma shunt1: "(x \<sqinter> y \<le> z) \<longleftrightarrow> (x \<le> -y \<squnion> z)"
proof
assume "x \<sqinter> y \<le> z"
hence "-y \<squnion> (x \<sqinter> y) \<le> -y \<squnion> z"
using sup.mono by blast
hence "-y \<squnion> x \<le> -y \<squnion> z"
by (simp add: sup_inf_distrib1)
thus "x \<le> -y \<squnion> z"
by simp
next
assume "x \<le> -y \<squnion> z"
hence "x \<sqinter> y \<le> (-y \<squnion> z) \<sqinter> y"
using inf_mono by auto
thus "x \<sqinter> y \<le> z"
using inf.boundedE inf_sup_distrib2 by auto
qed
lemma shunt2: "(x \<sqinter> -y \<le> z) \<longleftrightarrow> (x \<le> y \<squnion> z)"
by (simp add: shunt1)
lemma inf_shunt: "(x \<sqinter> y = \<bottom>) \<longleftrightarrow> (x \<le> - y)"
by (simp add: order.eq_iff shunt1)
lemma sup_shunt: "(x \<squnion> y = \<top>) \<longleftrightarrow> (- x \<le> y)"
using inf_shunt [of \<open>- x\<close> \<open>- y\<close>, symmetric]
by (simp flip: compl_sup compl_top_eq)
lemma diff_shunt_var: "(x - y = \<bottom>) \<longleftrightarrow> (x \<le> y)"
by (simp add: diff_eq inf_shunt)
lemma sup_neg_inf:
\<open>p \<le> q \<squnion> r \<longleftrightarrow> p \<sqinter> -q \<le> r\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
proof
assume ?P
then have \<open>p \<sqinter> - q \<le> (q \<squnion> r) \<sqinter> - q\<close>
by (rule inf_mono) simp
then show ?Q
by (simp add: inf_sup_distrib2)
next
assume ?Q
then have \<open>p \<sqinter> - q \<squnion> q \<le> r \<squnion> q\<close>
by (rule sup_mono) simp
then show ?P
by (simp add: sup_inf_distrib ac_simps)
qed
end
end