src/HOL/Library/Boolean_Algebra.thy
changeset 24332 e3a2b75b1cf9
child 24357 d42cf77da51f
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Boolean_Algebra.thy	Mon Aug 20 00:22:18 2007 +0200
@@ -0,0 +1,276 @@
+(* 
+  ID:     $Id$
+  Author: Brian Huffman
+
+  Boolean algebras as locales.
+*)
+
+header {* Boolean Algebras *}
+
+theory Boolean_Algebra
+imports Main
+begin
+
+locale boolean =
+  fixes conj :: "'a => 'a => 'a" (infixr "\<sqinter>" 70)
+  fixes disj :: "'a => 'a => 'a" (infixr "\<squnion>" 65)
+  fixes compl :: "'a => 'a" ("\<sim> _" [81] 80)
+  fixes zero :: "'a" ("\<zero>")
+  fixes one  :: "'a" ("\<one>")
+  assumes conj_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
+  assumes disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
+  assumes conj_commute: "x \<sqinter> y = y \<sqinter> x"
+  assumes disj_commute: "x \<squnion> y = y \<squnion> x"
+  assumes conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+  assumes disj_conj_distrib: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+  assumes conj_one_right: "x \<sqinter> \<one> = x"
+  assumes disj_zero_right: "x \<squnion> \<zero> = x"
+  assumes conj_cancel_right: "x \<sqinter> \<sim> x = \<zero>"
+  assumes disj_cancel_right: "x \<squnion> \<sim> x = \<one>"
+begin
+
+lemmas disj_ac =
+  disj_assoc disj_commute
+  mk_left_commute [of "disj", OF disj_assoc disj_commute]
+
+lemmas conj_ac =
+  conj_assoc conj_commute
+  mk_left_commute [of "conj", OF conj_assoc conj_commute]
+
+lemma dual: "boolean disj conj compl one zero"
+apply (rule boolean.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
+
+text {* Complement *}
+
+lemma complement_unique:
+  assumes 1: "a \<sqinter> x = \<zero>"
+  assumes 2: "a \<squnion> x = \<one>"
+  assumes 3: "a \<sqinter> y = \<zero>"
+  assumes 4: "a \<squnion> y = \<one>"
+  shows "x = y"
+proof -
+  have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)" using 1 3 by simp
+  hence "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)" using conj_commute by simp
+  hence "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)" using conj_disj_distrib by simp
+  hence "x \<sqinter> \<one> = y \<sqinter> \<one>" using 2 4 by simp
+  thus "x = y" using conj_one_right by simp
+qed
+
+lemma compl_unique: "[| x \<sqinter> y = \<zero>; x \<squnion> y = \<one> |] ==> \<sim> x = y"
+by (rule complement_unique [OF conj_cancel_right disj_cancel_right])
+
+lemma double_compl [simp]: "\<sim> (\<sim> x) = x"
+proof (rule compl_unique)
+  from conj_cancel_right show "\<sim> x \<sqinter> x = \<zero>" by (simp add: conj_commute)
+  from disj_cancel_right show "\<sim> x \<squnion> x = \<one>" by (simp add: disj_commute)
+qed
+
+lemma compl_eq_compl_iff [simp]: "(\<sim> x = \<sim> y) = (x = y)"
+by (rule inj_eq [OF inj_on_inverseI], rule double_compl)
+
+text {* Conjunction *}
+
+lemma conj_absorb: "x \<sqinter> x = x"
+proof -
+  have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>" using disj_zero_right by simp
+  also have "... = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp
+  also have "... = x \<sqinter> (x \<squnion> \<sim> x)" using conj_disj_distrib by simp
+  also have "... = x \<sqinter> \<one>" using disj_cancel_right by simp
+  also have "... = x" using conj_one_right by simp
+  finally show ?thesis .
+qed
+
+lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>"
+proof -
+  have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp
+  also have "... = (x \<sqinter> x) \<sqinter> \<sim> x" using conj_assoc by simp
+  also have "... = x \<sqinter> \<sim> x" using conj_absorb by simp
+  also have "... = \<zero>" using conj_cancel_right by simp
+  finally show ?thesis .
+qed
+
+lemma compl_one [simp]: "\<sim> \<one> = \<zero>"
+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)
+
+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 add: conj_assoc [symmetric] conj_absorb)
+
+lemma conj_disj_distrib2:
+  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" 
+by (simp add: conj_commute conj_disj_distrib)
+
+lemmas conj_disj_distribs =
+   conj_disj_distrib conj_disj_distrib2
+
+text {* Disjunction *}
+
+lemma disj_absorb [simp]: "x \<squnion> x = x"
+by (rule boolean.conj_absorb [OF dual])
+
+lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>"
+by (rule boolean.conj_zero_right [OF dual])
+
+lemma compl_zero [simp]: "\<sim> \<zero> = \<one>"
+by (rule boolean.compl_one [OF dual])
+
+lemma disj_zero_left [simp]: "\<zero> \<squnion> x = x"
+by (rule boolean.conj_one_left [OF dual])
+
+lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>"
+by (rule boolean.conj_zero_left [OF dual])
+
+lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>"
+by (rule boolean.conj_cancel_left [OF dual])
+
+lemma disj_left_absorb [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
+by (rule boolean.conj_left_absorb [OF dual])
+
+lemma disj_conj_distrib2:
+  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+by (rule boolean.conj_disj_distrib2 [OF dual])
+
+lemmas disj_conj_distribs =
+   disj_conj_distrib disj_conj_distrib2
+
+text {* De Morgan's Laws *}
+
+lemma de_Morgan_conj [simp]: "\<sim> (x \<sqinter> y) = \<sim> x \<squnion> \<sim> y"
+proof (rule compl_unique)
+  have "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = ((x \<sqinter> y) \<sqinter> \<sim> x) \<squnion> ((x \<sqinter> y) \<sqinter> \<sim> y)"
+    by (rule conj_disj_distrib)
+  also have "... = (y \<sqinter> (x \<sqinter> \<sim> x)) \<squnion> (x \<sqinter> (y \<sqinter> \<sim> y))"
+    by (simp add: conj_ac)
+  finally show "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = \<zero>"
+    by (simp add: conj_cancel_right conj_zero_right disj_zero_right)
+next
+  have "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = (x \<squnion> (\<sim> x \<squnion> \<sim> y)) \<sqinter> (y \<squnion> (\<sim> x \<squnion> \<sim> y))"
+    by (rule disj_conj_distrib2)
+  also have "... = (\<sim> y \<squnion> (x \<squnion> \<sim> x)) \<sqinter> (\<sim> x \<squnion> (y \<squnion> \<sim> y))"
+    by (simp add: disj_ac)
+  finally show "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = \<one>"
+    by (simp add: disj_cancel_right disj_one_right conj_one_right)
+qed
+
+lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y"
+by (rule boolean.de_Morgan_conj [OF dual])
+
+end
+
+text {* Symmetric Difference *}
+
+locale boolean_xor = boolean +
+  fixes xor :: "'a => 'a => 'a"  (infixr "\<oplus>" 65)
+  assumes xor_def: "x \<oplus> y = (x \<sqinter> \<sim> y) \<squnion> (\<sim> x \<sqinter> y)"
+begin
+
+lemma xor_def2:
+  "x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)"
+by (simp add: xor_def conj_disj_distribs
+              disj_ac conj_ac conj_cancel_right disj_zero_left)
+
+lemma xor_commute: "x \<oplus> y = y \<oplus> x"
+by (simp add: xor_def conj_commute disj_commute)
+
+lemma xor_assoc: "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
+proof -
+  let ?t = "(x \<sqinter> y \<sqinter> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> \<sim> z) \<squnion>
+            (\<sim> x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (\<sim> x \<sqinter> \<sim> y \<sqinter> z)"
+  have "?t \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> y \<sqinter> \<sim> y) =
+        ?t \<squnion> (x \<sqinter> y \<sqinter> \<sim> y) \<squnion> (x \<sqinter> z \<sqinter> \<sim> z)"
+    by (simp add: conj_cancel_right conj_zero_right)
+  thus "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
+    apply (simp add: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+    apply (simp add: conj_disj_distribs conj_ac disj_ac)
+    done
+qed
+
+lemmas xor_ac =
+  xor_assoc xor_commute
+  mk_left_commute [of "xor", OF xor_assoc xor_commute]
+
+lemma xor_zero_right [simp]: "x \<oplus> \<zero> = x"
+by (simp add: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right)
+
+lemma xor_zero_left [simp]: "\<zero> \<oplus> x = x"
+by (subst xor_commute) (rule xor_zero_right)
+
+lemma xor_one_right [simp]: "x \<oplus> \<one> = \<sim> x"
+by (simp add: xor_def compl_one conj_zero_right conj_one_right disj_zero_left)
+
+lemma xor_one_left [simp]: "\<one> \<oplus> x = \<sim> x"
+by (subst xor_commute) (rule xor_one_right)
+
+lemma xor_self [simp]: "x \<oplus> x = \<zero>"
+by (simp add: xor_def conj_cancel_right conj_cancel_left disj_zero_right)
+
+lemma xor_left_self [simp]: "x \<oplus> (x \<oplus> y) = y"
+by (simp add: xor_assoc [symmetric] xor_self xor_zero_left)
+
+lemma xor_compl_left: "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)"
+apply (simp add: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+apply (simp add: conj_disj_distribs)
+apply (simp add: conj_cancel_right conj_cancel_left)
+apply (simp add: disj_zero_left disj_zero_right)
+apply (simp add: disj_ac conj_ac)
+done
+
+lemma xor_compl_right: "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)"
+apply (simp add: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+apply (simp add: conj_disj_distribs)
+apply (simp add: conj_cancel_right conj_cancel_left)
+apply (simp add: disj_zero_left disj_zero_right)
+apply (simp add: disj_ac conj_ac)
+done
+
+lemma xor_cancel_right [simp]: "x \<oplus> \<sim> x = \<one>"
+by (simp add: xor_compl_right xor_self compl_zero)
+
+lemma xor_cancel_left [simp]: "\<sim> x \<oplus> x = \<one>"
+by (subst xor_commute) (rule xor_cancel_right)
+
+lemma conj_xor_distrib: "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+proof -
+  have "(x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z) =
+        (y \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z)"
+    by (simp add: conj_cancel_right conj_zero_right disj_zero_left)
+  thus "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+    by (simp (no_asm_use) add:
+        xor_def de_Morgan_disj de_Morgan_conj double_compl
+        conj_disj_distribs conj_ac disj_ac)
+qed
+
+lemma conj_xor_distrib2:
+  "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)"
+proof -
+  have "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+    by (rule conj_xor_distrib)
+  thus "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)"
+    by (simp add: conj_commute)
+qed
+
+lemmas conj_xor_distribs =
+   conj_xor_distrib conj_xor_distrib2
+
+end
+
+end