--- a/src/HOL/Library/Boolean_Algebra.thy Sat Apr 20 18:02:21 2019 +0000
+++ b/src/HOL/Library/Boolean_Algebra.thy Sat Apr 20 18:02:22 2019 +0000
@@ -8,49 +8,59 @@
imports Main
begin
-locale boolean_algebra =
- fixes conj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<sqinter>" 70)
- and disj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<squnion>" 65)
- and compl :: "'a \<Rightarrow> 'a" ("\<sim> _" [81] 80)
+locale boolean_algebra = conj: abel_semigroup "(\<sqinter>)" + disj: abel_semigroup "(\<squnion>)"
+ for conj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<sqinter>" 70)
+ and disj :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<squnion>" 65) +
+ fixes compl :: "'a \<Rightarrow> 'a" ("\<sim> _" [81] 80)
and zero :: "'a" ("\<zero>")
and one :: "'a" ("\<one>")
- assumes conj_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
- and disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
- and conj_commute: "x \<sqinter> y = y \<sqinter> x"
- and disj_commute: "x \<squnion> y = y \<squnion> x"
- and conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+ assumes conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
and disj_conj_distrib: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
- and conj_one_right [simp]: "x \<sqinter> \<one> = x"
- and disj_zero_right [simp]: "x \<squnion> \<zero> = x"
+ and conj_one_right: "x \<sqinter> \<one> = x"
+ and disj_zero_right: "x \<squnion> \<zero> = x"
and conj_cancel_right [simp]: "x \<sqinter> \<sim> x = \<zero>"
and disj_cancel_right [simp]: "x \<squnion> \<sim> x = \<one>"
begin
-sublocale conj: abel_semigroup conj
- by standard (fact conj_assoc conj_commute)+
-
-sublocale disj: abel_semigroup disj
- by standard (fact disj_assoc disj_commute)+
-
-lemmas conj_left_commute = conj.left_commute
-lemmas disj_left_commute = disj.left_commute
-
-lemmas conj_ac = conj.assoc conj.commute conj.left_commute
-lemmas disj_ac = disj.assoc disj.commute disj.left_commute
+sublocale conj: semilattice_neutr "(\<sqinter>)" "\<one>"
+proof
+ show "x \<sqinter> \<one> = x" for x
+ by (fact conj_one_right)
+ show "x \<sqinter> x = x" for x
+ proof -
+ have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>"
+ by (simp add: disj_zero_right)
+ also have "\<dots> = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)"
+ by simp
+ also have "\<dots> = x \<sqinter> (x \<squnion> \<sim> x)"
+ by (simp only: conj_disj_distrib)
+ also have "\<dots> = x \<sqinter> \<one>"
+ by simp
+ also have "\<dots> = x"
+ by (simp add: conj_one_right)
+ finally show ?thesis .
+ qed
+qed
-lemma dual: "boolean_algebra disj conj compl one zero"
- apply (rule boolean_algebra.intro)
- apply (rule disj_assoc)
- apply (rule conj_assoc)
- apply (rule disj_commute)
- apply (rule conj_commute)
- apply (rule disj_conj_distrib)
- apply (rule conj_disj_distrib)
- apply (rule disj_zero_right)
- apply (rule conj_one_right)
- apply (rule disj_cancel_right)
- apply (rule conj_cancel_right)
- done
+sublocale disj: semilattice_neutr "(\<squnion>)" "\<zero>"
+proof
+ show "x \<squnion> \<zero> = x" for x
+ by (fact disj_zero_right)
+ show "x \<squnion> x = x" for x
+ proof -
+ have "x \<squnion> x = (x \<squnion> x) \<sqinter> \<one>"
+ by simp
+ also have "\<dots> = (x \<squnion> x) \<sqinter> (x \<squnion> \<sim> x)"
+ by simp
+ also have "\<dots> = x \<squnion> (x \<sqinter> \<sim> x)"
+ by (simp only: disj_conj_distrib)
+ also have "\<dots> = x \<squnion> \<zero>"
+ by simp
+ also have "\<dots> = x"
+ by (simp add: disj_zero_right)
+ finally show ?thesis .
+ qed
+qed
subsection \<open>Complement\<close>
@@ -65,7 +75,7 @@
from 1 3 have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)"
by simp
then have "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)"
- by (simp add: conj_commute)
+ by (simp add: ac_simps)
then have "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)"
by (simp add: conj_disj_distrib)
with 2 4 have "x \<sqinter> \<one> = y \<sqinter> \<one>"
@@ -80,9 +90,9 @@
lemma double_compl [simp]: "\<sim> (\<sim> x) = x"
proof (rule compl_unique)
show "\<sim> x \<sqinter> x = \<zero>"
- by (simp only: conj_cancel_right conj_commute)
+ by (simp only: conj_cancel_right conj.commute)
show "\<sim> x \<squnion> x = \<one>"
- by (simp only: disj_cancel_right disj_commute)
+ by (simp only: disj_cancel_right disj.commute)
qed
lemma compl_eq_compl_iff [simp]: "\<sim> x = \<sim> y \<longleftrightarrow> x = y"
@@ -91,27 +101,12 @@
subsection \<open>Conjunction\<close>
-lemma conj_absorb [simp]: "x \<sqinter> x = x"
-proof -
- have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>"
- by simp
- also have "\<dots> = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)"
- by simp
- also have "\<dots> = x \<sqinter> (x \<squnion> \<sim> x)"
- by (simp only: conj_disj_distrib)
- also have "\<dots> = x \<sqinter> \<one>"
- by simp
- also have "\<dots> = x"
- by simp
- finally show ?thesis .
-qed
-
lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>"
proof -
from conj_cancel_right have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)"
by simp
also from conj_assoc have "\<dots> = (x \<sqinter> x) \<sqinter> \<sim> x"
- by (simp only:)
+ by (simp only: ac_simps)
also from conj_absorb have "\<dots> = x \<sqinter> \<sim> x"
by simp
also have "\<dots> = \<zero>"
@@ -123,51 +118,80 @@
by (rule compl_unique [OF conj_zero_right disj_zero_right])
lemma conj_zero_left [simp]: "\<zero> \<sqinter> x = \<zero>"
- by (subst conj_commute) (rule conj_zero_right)
-
-lemma conj_one_left [simp]: "\<one> \<sqinter> x = x"
- by (subst conj_commute) (rule conj_one_right)
+ by (subst conj.commute) (rule conj_zero_right)
lemma conj_cancel_left [simp]: "\<sim> x \<sqinter> x = \<zero>"
- by (subst conj_commute) (rule conj_cancel_right)
-
-lemma conj_left_absorb [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
- by (simp only: conj_assoc [symmetric] conj_absorb)
+ by (subst conj.commute) (rule conj_cancel_right)
lemma conj_disj_distrib2: "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
- by (simp only: conj_commute conj_disj_distrib)
+ by (simp only: conj.commute conj_disj_distrib)
lemmas conj_disj_distribs = conj_disj_distrib conj_disj_distrib2
+lemma conj_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
+ by (fact ac_simps)
+
+lemma conj_commute: "x \<sqinter> y = y \<sqinter> x"
+ by (fact ac_simps)
+
+lemmas conj_left_commute = conj.left_commute
+lemmas conj_ac = conj.assoc conj.commute conj.left_commute
+
+lemma conj_one_left: "\<one> \<sqinter> x = x"
+ by (fact conj.left_neutral)
+
+lemma conj_left_absorb: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
+ by (fact conj.left_idem)
+
+lemma conj_absorb: "x \<sqinter> x = x"
+ by (fact conj.idem)
+
subsection \<open>Disjunction\<close>
-lemma disj_absorb [simp]: "x \<squnion> x = x"
- by (rule boolean_algebra.conj_absorb [OF dual])
+interpretation dual: boolean_algebra "(\<squnion>)" "(\<sqinter>)" compl \<one> \<zero>
+ apply standard
+ apply (rule disj_conj_distrib)
+ apply (rule conj_disj_distrib)
+ apply simp_all
+ done
+
+lemma compl_zero [simp]: "\<sim> \<zero> = \<one>"
+ by (fact dual.compl_one)
lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>"
- by (rule boolean_algebra.conj_zero_right [OF dual])
-
-lemma compl_zero [simp]: "\<sim> \<zero> = \<one>"
- by (rule boolean_algebra.compl_one [OF dual])
-
-lemma disj_zero_left [simp]: "\<zero> \<squnion> x = x"
- by (rule boolean_algebra.conj_one_left [OF dual])
+ by (fact dual.conj_zero_right)
lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>"
- by (rule boolean_algebra.conj_zero_left [OF dual])
+ by (fact dual.conj_zero_left)
lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>"
- by (rule boolean_algebra.conj_cancel_left [OF dual])
-
-lemma disj_left_absorb [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
- by (rule boolean_algebra.conj_left_absorb [OF dual])
+ by (rule dual.conj_cancel_left)
lemma disj_conj_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
- by (rule boolean_algebra.conj_disj_distrib2 [OF dual])
+ by (rule dual.conj_disj_distrib2)
lemmas disj_conj_distribs = disj_conj_distrib disj_conj_distrib2
+lemma disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
+ by (fact ac_simps)
+
+lemma disj_commute: "x \<squnion> y = y \<squnion> x"
+ by (fact ac_simps)
+
+lemmas disj_left_commute = disj.left_commute
+
+lemmas disj_ac = disj.assoc disj.commute disj.left_commute
+
+lemma disj_zero_left: "\<zero> \<squnion> x = x"
+ by (fact disj.left_neutral)
+
+lemma disj_left_absorb: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
+ by (fact disj.left_idem)
+
+lemma disj_absorb: "x \<squnion> x = x"
+ by (fact disj.idem)
+
subsection \<open>De Morgan's Laws\<close>
@@ -189,7 +213,7 @@
qed
lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y"
- by (rule boolean_algebra.de_Morgan_conj [OF dual])
+ using dual.boolean_algebra_axioms by (rule boolean_algebra.de_Morgan_conj)
subsection \<open>Symmetric Difference\<close>