--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Quickcheck_Lattice_Examples.thy Wed Jul 21 18:11:51 2010 +0200
@@ -0,0 +1,139 @@
+(* Title: HOL/ex/Quickcheck_Lattice_Examples.thy
+ Author: Lukas Bulwahn
+ Copyright 2010 TU Muenchen
+*)
+
+theory Quickcheck_Lattice_Examples
+imports Quickcheck_Types
+begin
+
+text {* We show how other default types help to find counterexamples to propositions if
+ the standard default type @{typ int} is insufficient. *}
+
+notation
+ less_eq (infix "\<sqsubseteq>" 50) and
+ less (infix "\<sqsubset>" 50) and
+ top ("\<top>") and
+ bot ("\<bottom>") and
+ inf (infixl "\<sqinter>" 70) and
+ sup (infixl "\<squnion>" 65)
+
+subsection {* Distributive lattices *}
+
+lemma sup_inf_distrib2:
+ "((y :: 'a :: distrib_lattice) \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+ quickcheck
+by(simp add: inf_sup_aci sup_inf_distrib1)
+
+lemma sup_inf_distrib2_1:
+ "((y :: 'a :: lattice) \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+ quickcheck
+ oops
+
+lemma sup_inf_distrib2_2:
+ "((y :: 'a :: distrib_lattice) \<sqinter> z') \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+ quickcheck
+ oops
+
+lemma inf_sup_distrib1_1:
+ "(x :: 'a :: distrib_lattice) \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x' \<sqinter> z)"
+ quickcheck
+ oops
+
+lemma inf_sup_distrib2_1:
+ "((y :: 'a :: distrib_lattice) \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (y \<sqinter> x)"
+ quickcheck
+ oops
+
+subsection {* Bounded lattices *}
+
+lemma inf_bot_left [simp]:
+ "\<bottom> \<sqinter> (x :: 'a :: bounded_lattice_bot) = \<bottom>"
+ quickcheck
+ by (rule inf_absorb1) simp
+
+lemma inf_bot_left_1:
+ "\<bottom> \<sqinter> (x :: 'a :: bounded_lattice_bot) = x"
+ quickcheck
+ oops
+
+lemma inf_bot_left_2:
+ "y \<sqinter> (x :: 'a :: bounded_lattice_bot) = \<bottom>"
+ quickcheck
+ oops
+
+lemma inf_bot_left_3:
+ "x \<noteq> \<bottom> ==> y \<sqinter> (x :: 'a :: bounded_lattice_bot) \<noteq> \<bottom>"
+ quickcheck
+ oops
+
+lemma inf_bot_right [simp]:
+ "(x :: 'a :: bounded_lattice_bot) \<sqinter> \<bottom> = \<bottom>"
+ quickcheck
+ by (rule inf_absorb2) simp
+
+lemma inf_bot_right_1:
+ "x \<noteq> \<bottom> ==> (x :: 'a :: bounded_lattice_bot) \<sqinter> \<bottom> = y"
+ quickcheck
+ oops
+
+lemma inf_bot_right_2:
+ "(x :: 'a :: bounded_lattice_bot) \<sqinter> \<bottom> ~= \<bottom>"
+ quickcheck
+ oops
+
+lemma inf_bot_right [simp]:
+ "(x :: 'a :: bounded_lattice_bot) \<squnion> \<bottom> = \<bottom>"
+ quickcheck
+ oops
+
+lemma sup_bot_left [simp]:
+ "\<bottom> \<squnion> (x :: 'a :: bounded_lattice_bot) = x"
+ quickcheck
+ by (rule sup_absorb2) simp
+
+lemma sup_bot_right [simp]:
+ "(x :: 'a :: bounded_lattice_bot) \<squnion> \<bottom> = x"
+ quickcheck
+ by (rule sup_absorb1) simp
+
+lemma sup_eq_bot_iff [simp]:
+ "(x :: 'a :: bounded_lattice_bot) \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
+ quickcheck
+ by (simp add: eq_iff)
+
+lemma sup_top_left [simp]:
+ "\<top> \<squnion> (x :: 'a :: bounded_lattice_top) = \<top>"
+ quickcheck
+ by (rule sup_absorb1) simp
+
+lemma sup_top_right [simp]:
+ "(x :: 'a :: bounded_lattice_top) \<squnion> \<top> = \<top>"
+ quickcheck
+ by (rule sup_absorb2) simp
+
+lemma inf_top_left [simp]:
+ "\<top> \<sqinter> x = (x :: 'a :: bounded_lattice_top)"
+ quickcheck
+ by (rule inf_absorb2) simp
+
+lemma inf_top_right [simp]:
+ "x \<sqinter> \<top> = (x :: 'a :: bounded_lattice_top)"
+ quickcheck
+ by (rule inf_absorb1) simp
+
+lemma inf_eq_top_iff [simp]:
+ "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
+ quickcheck
+ by (simp add: eq_iff)
+
+
+no_notation
+ less_eq (infix "\<sqsubseteq>" 50) and
+ less (infix "\<sqsubset>" 50) and
+ inf (infixl "\<sqinter>" 70) and
+ sup (infixl "\<squnion>" 65) and
+ top ("\<top>") and
+ bot ("\<bottom>")
+
+end