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author | wenzelm |

Thu, 19 Oct 2000 21:22:05 +0200 | |

changeset 10276 | 75e2c6cb4153 |

parent 10275 | 558f7569026e |

child 10277 | 081c8641aa11 |

added theory for HOL type definitions;

--- 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