src/HOL/subset.thy

--- a/src/HOL/subset.thy	Thu Oct 19 21:21:41 2000 +0200
+++ b/src/HOL/subset.thy	Thu Oct 19 21:22:05 2000 +0200
@@ -2,10 +2,12 @@
     ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1994  University of Cambridge
+
+Subset lemmas and HOL type definitions.
 *)
 
 theory subset = Set
-files "Tools/typedef_package.ML":
+files "Tools/induct_attrib.ML" ("Tools/typedef_package.ML"):
 
 (*belongs to theory Ord*)
 theorems linorder_cases [case_names less equal greater] =
@@ -14,4 +16,104 @@
 (*belongs to theory Set*)
 setup Rulify.setup
 
+
+section {* HOL type definitions *}
+
+constdefs
+  type_definition :: "('a => 'b) => ('b => 'a) => 'b set => bool"
+  "type_definition Rep Abs A ==
+    (\<forall>x. Rep x \<in> A) \<and>
+    (\<forall>x. Abs (Rep x) = x) \<and>
+    (\<forall>y \<in> A. Rep (Abs y) = y)"
+  -- {* This will be stated as an axiom for each typedef! *}
+
+theorem Rep: "type_definition Rep Abs A ==> Rep x \<in> A"
+  by (unfold type_definition_def) blast
+
+theorem Rep_inverse: "type_definition Rep Abs A ==> Abs (Rep x) = x"
+  by (unfold type_definition_def) blast
+
+theorem Abs_inverse: "type_definition Rep Abs A ==> y \<in> A ==> Rep (Abs y) = y"
+  by (unfold type_definition_def) blast
+
+theorem Rep_inject: "type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)"
+proof -
+  assume tydef: "type_definition Rep Abs A"
+  show ?thesis
+  proof
+    assume "Rep x = Rep y"
+    hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
+    thus "x = y" by (simp only: Rep_inverse [OF tydef])
+  next
+    assume "x = y"
+    thus "Rep x = Rep y" by simp
+  qed
+qed
+
+theorem Rep_cases:
+  "type_definition Rep Abs A ==> y \<in> A ==> (!!x. y = Rep x ==> P) ==> P"
+proof -
+  assume tydef: "type_definition Rep Abs A"
+  assume y: "y \<in> A" and r: "(!!x. y = Rep x ==> P)"
+  show P
+  proof (rule r)
+    from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
+    thus "y = Rep (Abs y)" ..
+  qed
+qed
+
+theorem Abs_cases:
+  "type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \<in> A ==> P) ==> P"
+proof -
+  assume tydef: "type_definition Rep Abs A"
+  assume r: "!!y. x = Abs y ==> y \<in> A ==> P"
+  show P
+  proof (rule r)
+    have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
+    thus "x = Abs (Rep x)" ..
+    show "Rep x \<in> A" by (rule Rep [OF tydef])
+  qed
+qed
+
+theorem Abs_inject:
+  "type_definition Rep Abs A ==> x \<in> A ==> y \<in> A ==> (Abs x = Abs y) = (x = y)"
+proof -
+  assume tydef: "type_definition Rep Abs A"
+  assume x: "x \<in> A" and y: "y \<in> A"
+  show ?thesis
+  proof
+    assume "Abs x = Abs y"
+    hence "Rep (Abs x) = Rep (Abs y)" by simp
+    moreover note x hence "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef])
+    moreover note y hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
+    ultimately show "x = y" by (simp only:)
+  next
+    assume "x = y"
+    thus "Abs x = Abs y" by simp
+  qed
+qed
+
+theorem Rep_induct:
+  "type_definition Rep Abs A ==> y \<in> A ==> (!!x. P (Rep x)) ==> P y"
+proof -
+  assume tydef: "type_definition Rep Abs A"
+  assume "!!x. P (Rep x)" hence "P (Rep (Abs y))" .
+  moreover assume "y \<in> A" hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
+  ultimately show "P y" by (simp only:)
+qed
+
+theorem Abs_induct:
+  "type_definition Rep Abs A ==> (!!y. y \<in> A ==> P (Abs y)) ==> P x"
+proof -
+  assume tydef: "type_definition Rep Abs A"
+  assume r: "!!y. y \<in> A ==> P (Abs y)"
+  have "Rep x \<in> A" by (rule Rep [OF tydef])
+  hence "P (Abs (Rep x))" by (rule r)
+  moreover have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
+  ultimately show "P x" by (simp only:)
+qed
+
+setup InductAttrib.setup
+use "Tools/typedef_package.ML"
+
 end